# Monodromy and mapping class groups
of 3-dimensional hypersurfaces

Oscar Randal-Williams<EMAIL_ADDRESS>Centre for
Mathematical Sciences
Wilberforce Road
Cambridge CB3 0WB
UK

###### Abstract.

We describe the subgroup of the mapping class group of a hypersurface in
$\mathbb{CP}^{4}$ consisting of those diffeomorphisms which can be realised by
monodromy.

###### 2010 Mathematics Subject Classification:

14M10, 14D05, 57R15, 57R50, 20E26

###### Contents

  1. 1 Introduction
  2. 2 Some recollections
  3. 3 Upper bounds: tangential structures and quadratic forms
  4. 4 Lower bounds: automorphisms of the Fermat hypersurface
  5. 5 Further upper bounds: cobordism considerations
  6. 6 Proof of Theorem B: Comparison to $W_{g,1}$

##  1\. Introduction

Let $X_{d}\subset\mathbb{C}\mathbb{P}^{4}$ denote the degree $d$ Fermat
hypersurface. It resides in the universal family
$\mathcal{X}_{d}\to\mathcal{U}_{d}$ of smooth 3-dimensional degree $d$
hypersurfaces, whose base is the open subspace
$\mathcal{U}_{d}\subset\mathbb{P}H^{0}(\mathbb{C}\mathbb{P}^{4};\mathcal{O}(d))$
of degree $d$ homogeneous polynomials in 5 variables whose zero locus is
smooth. We write

$\mathrm{Mon}_{d}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{1}(\mathcal{U}_{d},X_{d})$

for the monodromy group of this family.

On the other hand $X_{d}$ can be considered as an oriented 6-manifold, and we
write

$\mathrm{MCG}_{d}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{0}\mathrm{Diff}^{+}(X_{d})$

for the oriented mapping class group of this manifold: the group of isotopy
classes of orientation-preserving diffeomorphisms. Fibre transport for the
universal family yields a group homomorphism

$\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Mon}_{d}\longrightarrow\mathrm{MCG}_{d}.$

A presentation for the group $\mathrm{Mon}_{d}$ has been given by Lönne
[L0̈9], and a quite complete description of $\mathrm{MCG}_{d}$ has been given
by Kreck and Su [KS20]. Using the latter, Hain [Hai23] has explained how the
methods of Sullivan [Sul77] show that the image of $\alpha$ has infinite index
in $\mathrm{MCG}_{d}$. Our goal is to completely describe the image of
$\alpha$, in terms of Kreck and Su’s description of $\mathrm{MCG}_{d}$.

Most of the answer will be described in terms of automorphisms of
$\pi_{3}(X_{d})$ respecting certain structures. There is a canonical extension

$0\longrightarrow\mathbb{Z}/d\longrightarrow\pi_{3}(X_{d})\longrightarrow
H_{3}(X_{d};\mathbb{Z})\longrightarrow 0,$

and the intersection $\mathbb{Z}$-valued form of $X_{d}$ induces an
antisymmetric form $\lambda$ on $\pi_{3}(X_{d})$, whose radical is
$\mathbb{Z}/d\leq\pi_{3}(X_{d})$. We write
$\mathrm{Aut}(\pi_{3}(X_{d}),\lambda)$ for the group of automorphisms of
$\pi_{3}(X_{d})$ which preserve $\lambda$ _and which induce the identity on
the radical $\mathbb{Z}/d\leq\pi_{3}(X_{d})$ of $\lambda$_. We will explain
that $\lambda$ has a quadratic refinement

$\mu\mathrel{\mathop{\mathchar
58\relax}}\pi_{3}(X_{d})\longrightarrow\mathbb{Z}/2,$

induced by the embedding $X_{d}\subset\mathbb{C}\mathbb{P}^{4}$, and that this
quadratic refinement is $\mathrm{Mon}_{d}$-invariant. By a quadratic
refinement of $\lambda$ we mean that $\mu$ is a function satisfying

$\mu(a+b)=\mu(a)+\mu(b)+\lambda(a,b)\mod 2.$

We write $\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$ for the subgroup of
$\mathrm{Aut}(\pi_{3}(X_{d}),\lambda)$ consisting of those automorphisms which
preserve $\mu$.

######  Theorem A.

Suppose $d\geq 3$.

  1. (i)

The map $\mathrm{Mon}_{d}\to\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$ is
surjective.

  2. (ii)

The kernel of the surjection
$\mathrm{Im}(\alpha)\to\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$ consists
precisely of those diffeomorphisms of $X_{d}$ which up to isotopy may be
supported in a disc.

The latter is a quotient of
$\pi_{0}(\mathrm{Diff}_{\partial}(D^{6}))=\Theta_{7}=\mathbb{Z}/28$, and may
be extracted from the work of Kreck and Su (see Lemma 2.2). It is given by

$\begin{cases}0&d\equiv 2,4,6,10,12,14\mod 16\\\ \mathbb{Z}/2&d\equiv
3,5,8,11,13\mod 16\\\ \mathbb{Z}/4&d\equiv 0,1,7,9,15\mod
16\end{cases}\oplus\begin{cases}0&d\not\equiv 0\mod 7\\\ \mathbb{Z}/7&d\equiv
0\mod 7.\end{cases}$

Combining Theorem A with existing work on diffeomorphism groups of the
manifolds $W_{g,1}\mathrel{\mathop{\mathchar 58\relax}}=\\#^{g}S^{3}\times
S^{3}\setminus\mathrm{int}(D^{6})$ yields the following, where we recall that
the _finite residual_ of a group is the intersection of all of its finite-
index subgroups.

######  Theorem B.

The finite residual of $\mathrm{Im}(\alpha)$ is
$\mathrm{Ker}(\mathrm{Im}(\alpha)\to\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu))$.

It follows that $\mathrm{Im}(\alpha)$ is often not residually finite, so
neither is $\mathrm{MCG}_{d}$.

######  Remark 1.1.

We use the assumption $d\geq 3$ to guarantee that $X_{d}$ contains a modest
number of $S^{3}\times S^{3}$ connect-summands ($d\geq 3$ implies that it
contains at least 5). If $d=1$ or $2$ then $X_{d}$ contains no $S^{3}\times
S^{3}$ connect summands. The work of Kreck and Su shows that
$\mathrm{MCG}_{1}=\mathbb{Z}/4$ (see also [Bru71, Remark II.11]) and
$\mathrm{MCG}_{2}=0$. On the other hand $\mathrm{Mon}_{1}=0$ because _all_
degree 1 hypersurfaces are smooth, so $\mathrm{Im}(\alpha)$ is trivial if
$d\leq 2$.

The map $\alpha$ naturally factors through the symplectic mapping class group

$\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Mon}_{d}\longrightarrow\pi_{0}\mathrm{Symp}(X_{d})\longrightarrow\mathrm{MCG}_{d}.$

This work began in discussions with Ailsa Keating and Ivan Smith, exploring
the possibility of using this factorisation to investigate
$\pi_{0}\mathrm{Symp}(X_{d})$. While $\mathrm{Im}(\alpha)$ clearly gives a
lower bound for $\mathrm{Im}(\pi_{0}\mathrm{Symp}(X_{d})\to\mathrm{MCG}_{d})$,
trying to go any further seemed to produce more questions than answers on the
symplectic side:

  1. (i)

Is the quadratic refinement $\mu$ invariant under the action of
$\pi_{0}\mathrm{Symp}(X_{d})$?

  2. (ii)

Does the “distorsion of the first Pontrjagin class” vanish for
symplectomorphisms? (See Section 3.3 for this notion.)

  3. (iii)

Do $\mathrm{Mon}_{d}$ and $\pi_{0}\mathrm{Symp}(X_{d})$ have the same image in
$\mathrm{MCG}_{d}$?

  4. (iv)

Is $\mathrm{Mon}_{d}\to\pi_{0}\mathrm{Symp}(X_{d})$ surjective?

If the answer to (iv) is “yes” then it is for the other questions too, but I
am told it is not currently accesssible. With regards to (i), $\mu$ is not
$\mathrm{MCG}_{d}$-invariant, and neither is it invariant for the subgroup of
$\mathrm{MCG}_{d}$ which preserves the almost complex structure of $X_{d}$
(because this is the whole of $\mathrm{MCG}_{d}$, see Remark 3.3).

Strategy. The strategy for identifying the image of
$\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Mod}_{d}\to\mathrm{MCG}_{d}$ is fairly clear: one must
produce constraints on mapping classes—which are satisfied by monodromy—to cut
down the image, and then one must then show that enough mapping classes are
realised by monodromy to see that the constraints are sharp. This is broadly
what we will do.

The first constraint, developed in Section 3, is based on the observation that
the universal family of smooth hypersurfaces over $\mathcal{U}_{d}$ is
equipped with a certain (stable) tangential structure
$\ell^{\mathrm{hyp}}_{X_{d}}$, so that $\alpha$ lands in the stabiliser
$\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})$ of this
tangential structure. This stabiliser can be analysed following Krannich
[Kra20] or Kupers and the author [KRW21], and in particular it is shown to
preserve a quadratic refinement $\mu$ on $\pi_{3}(X_{d})$, which with the work
of Kreck and Su [KS20] leads to a central extension

$0\longrightarrow\\{\begin{subarray}{c}\text{diffeomorphisms supported}\\\
\text{near $S^{2}\subset
X_{d}$}\end{subarray}\\}\longrightarrow\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})\longrightarrow\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)\longrightarrow
0.$

The structure of the subgroup of diffeomorphisms supported near $S^{2}\subset
X_{d}$, which is a sphere generating $\pi_{2}(X_{d})\cong\mathbb{Z}$, has been
completely determined by Kreck and Su. In particular it is finite and abelian.
Implanting those diffeomorphisms which are supported on a disc gives a
homomorphism

$\Phi\mathrel{\mathop{\mathchar
58\relax}}\pi_{0}(\mathrm{Diff}_{\partial}(D^{6}))=\Theta_{7}\longrightarrow\\{\begin{subarray}{c}\text{diffeomorphisms
supported}\\\ \text{near $S^{2}\subset X_{d}$}\end{subarray}\\}.$

The second constraint, developed in Section 5, is the construction of a
surjective map $\kappa\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})\to\mathrm{Coker}(\Phi)$
such that $\kappa\circ\alpha$ is trivial. This map is obtained indirectly,
essentially by calculating the abelianisation of
$\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})$. This is done
by relating this stabiliser to a certain moduli space of manifolds
(diffeomorphic to $X_{d}$ and equipped with a certain tangential structure),
describing the first homology of the latter in terms of Thom spectra using the
work of Galatius and the author [GRW17], and then calculating using various
techniques from stable homotopy theory.

We get lower bounds for the image of $\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Mon}_{d}\to\mathrm{MCG}_{d}$ from three sources. Firstly,
work of Krylov [Kry03] quickly shows that the subgroup $\Phi(\Theta_{7})$ can
be realised by monodromy. Secondly, work of Beauville [Bea86] (for which we
supply a missing detail) fully describes the automorphisms of
$H_{3}(X_{d};\mathbb{Z})$ that can be realised by monodromy. Finally, in
Section 4 we explain how work of Pham [Pha65] and of Looijenga [Loo10]
concerning the action of groups of $d$th roots of unity on $X_{d}$ can be used
to upgrade Beauville’s work to fully describe the automorphisms of
$\pi_{3}(X_{d})$ that can be realised by monodromy.

Acknowledgements. I am grateful to A. Keating and I. Smith for many useful
discussions surrounding this work and their feedback on earlier versions, and
to R. Hain and D. Ranganathan for comments. I was supported by the ERC under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 756444).

##  2\. Some recollections

###  2.1. Algebraic topology of hypersurfaces

Let us explain our notation for the cohomology of a degree $d$ hypersurface
$X\subset\mathbb{C}\mathbb{P}^{4}$. Writing $x\in H^{2}(X;\mathbb{Z})$ for the
restriction from $\mathbb{C}\mathbb{P}^{4}$ of the hyperplane class, by
Poincaré duality there is a unique class $y\in H^{4}(X;\mathbb{Z})$ such that
$\int_{X}x\cdot y=1$. As $\int_{X}x^{3}=d$, it follows that $x^{2}=dy$. Then
the even cohomology of $X$ is

$H^{0}(X;\mathbb{Z})=\mathbb{Z}\\{1\\}\quad
H^{2}(X;\mathbb{Z})=\mathbb{Z}\\{x\\}\quad
H^{4}(X;\mathbb{Z})=\mathbb{Z}\\{y\\}\quad
H^{6}(X;\mathbb{Z})=\mathbb{Z}\\{xy\\}.$

The only odd cohomology is $H^{3}(X;\mathbb{Z})$, which is equipped with the
nondegenerate antisymmetric form $(a,b)\mapsto\int_{X}a\cdot b$.

The inclusion $X\subset\mathbb{C}\mathbb{P}^{4}$ is 3-connected by the
Lefschetz hyperplane theorem. Combined with the fact that
$\pi_{3}(\mathbb{C}\mathbb{P}^{4})=\pi_{4}(\mathbb{C}\mathbb{P}^{4})=0$ and
the Hurewicz theorem, this gives isomorphisms

$\pi_{3}(X)\overset{\sim}{\longleftarrow}\pi_{4}(\mathbb{C}\mathbb{P}^{4},X)\overset{\sim}{\longrightarrow}H_{4}(\mathbb{C}\mathbb{P}^{4},X;\mathbb{Z}).$

The long exact sequence on homology for this pair takes the form

$\cdots\longrightarrow H_{4}(X;\mathbb{Z})\longrightarrow
H_{4}(\mathbb{C}\mathbb{P}^{4};\mathbb{Z})\longrightarrow
H_{4}(\mathbb{C}\mathbb{P}^{4},X;\mathbb{Z})\longrightarrow
H_{3}(X;\mathbb{Z})\longrightarrow 0$

and so gives an extension

(2.1)
$0\longrightarrow\mathbb{Z}/d\\{\eta\\}\longrightarrow\pi_{3}(X)\longrightarrow
H_{3}(X;\mathbb{Z})\longrightarrow 0.$

The subgroup $\mathbb{Z}/d$ is generated by the Hopf map
$\eta\mathrel{\mathop{\mathchar 58\relax}}S^{3}\to S^{2}$ on the generator of
$\pi_{2}(X)\cong H_{2}(X;\mathbb{Z})\cong\mathbb{Z}$ dual to $x\in
H^{2}(X;\mathbb{Z})$. The intersection form of $X$ induces an antisymmetric
form $\lambda$ on $\pi_{3}(X)$, whose radical is precisely the subgroup
$\mathbb{Z}/d\\{\eta\\}$.

The definition of $X$ as a hypersurface gives an isomorphism of complex vector
bundles
$TX\oplus\mathcal{O}(d)|_{X}\oplus\underline{\mathbb{C}}=\mathcal{O}(1)^{\oplus
5}|_{X}$, and so its total Chern class is

$\displaystyle c(TX)=\tfrac{(1+x)^{5}}{(1+dx)}$
$\displaystyle=1+(5-d)x+(d^{2}-5d+10)x^{2}+(-d^{3}+5d^{2}-10d+10)x^{3}$
$\displaystyle=1+(5-d)x+(d^{2}-5d+10)dy+(-d^{3}+5d^{2}-10d+10)dxy.$

As the third Chern class of $TX$ is also the Euler class of this vector
bundle, we see that $\chi(X)=(-d^{3}+5d^{2}-10d+10)d$ and so

$\mathrm{rank}\,H^{3}(X;\mathbb{Z})=4-(-d^{3}+5d^{2}-10d+10)d=d^{4}-5d^{3}+10d^{2}-10d+4.$

We write $2g$ for this number: by a theorem of Wall [Wal66, Theorem 1] there
is a decomposition $X\cong X^{\prime}\\#g(S^{3}\times S^{3})$; it will be
useful to know that $g\geq 5$ as long as $d\geq 3$. The class
$c_{1}(TX)=(5-d)x$ reduces modulo 2 to $w_{2}(TX)$, so $X$ is Spin if and only
if $d$ is odd. Its first Pontrjagin class is

$\displaystyle p_{1}(TX)$
$\displaystyle=-c_{2}(TX\otimes\mathbb{C})=-c_{2}(TX\oplus\overline{TX})=-(2c_{2}(TX)-c_{1}(TX)^{2})$
$\displaystyle=(5-d^{2})dy.$

###  2.2. Monodromy

Beauville [Bea86] has studied the monodromy action of the universal family of
degree $d$ hypersurfaces on the middle cohomology, and his results in
particular apply to the composition

(2.2)
$\mathrm{Mon}_{d}\overset{\alpha}{\longrightarrow}\mathrm{MCG}_{d}\longrightarrow\mathrm{Aut}(H^{3}(X_{d};\mathbb{Z}),(a,b)\mapsto
a\cdot b).$

He explains that the collection $\Delta_{X_{d}}\subset
H^{3}(X_{d};\mathbb{Z})$ of vanishing cycles for a Lefschetz fibration
containing $X_{d}$ forms a “réseau évanescent”, and that for $d\geq 3$ $X_{d}$
admits a deformation to a $E_{6}$-singularity, so [Bea86, Théorème 3] applies
to say that (2.2) is either

  1. (i)

surjective, or

  2. (ii)

has image the stabiliser of some quadratic refinement of $-\cdot-$.

It is an easy algebraic exercise to see that distinct quadratic refinements
have distinct stabilisers, so in the latter case there is a canonical
quadratic refinement $q_{X_{d}}\mathrel{\mathop{\mathchar
58\relax}}H^{3}(X_{d};\mathbb{Z})\to\mathbb{Z}/2$ associated to $X_{d}$.

Beauville then argues that these two cases correspond to $d$ being even or odd
respectively. For $d$ even he shows that there is indeed no
$\mathrm{Mon}_{d}$-invariant quadratic refinement, so case (i) holds. For $d$
odd he refers to work of Browder [Bro79] or Wood [Woo75] for the existence of
a quadratic refinement, however we are concerned with dimension 3 and in the
exceptional dimensions $1$, $3$, and $7$ those results do not apply111This is
explicit in [Bro79, Theorem B]. On the other hand [Woo75, Theorem 2]
calculates the Kervaire invariant of $X_{d}$ without seeming to concern itself
with whether this invariant is defined, but the definition at the end of §1
defines the zero quadratic function in dimensions 1, 3, or 7. However, see
Remark 3.8.. In fact, as Beauville explains [Bea86, p. 15] there is no
$\mathrm{MCG}_{d}$-invariant quadratic refinement, so any construction of
$q_{X_{d}}$ cannot be by plain differential topology. Nonetheless Beauville is
correct that case (ii) holds when $d$ is odd: one contribution of this paper
is to produce, in Section 3.4, the required $\mathrm{Mon}_{d}$-invariant
quadratic refinement. Pending this detail, we record Beauville’s result as
follows.

######  Theorem 2.1 (Beauville [Bea86]).

For $d\geq 3$ the map

$\mathrm{Mon}_{d}\longrightarrow\begin{cases}\mathrm{Aut}(H^{3}(X_{d};\mathbb{Z}),(a,b)\mapsto
a\cdot b,q_{X_{d}})&d\text{ odd}\\\
\mathrm{Aut}(H^{3}(X_{d};\mathbb{Z}),(a,b)\mapsto a\cdot b)&d\text{
even}\end{cases}$

is surjective.∎

###  2.3. The mapping class group

Kreck and Su [KS20] have made a detailed study of the mapping class groups of
6-manifolds which “look like 3-dimensional complete intersections”: of course
a 3-dimensional hypersurface such as $X_{d}$ is such a manifold. For us the
most useful form of their result is as follows [KS20, Theorem 2.6]. There are
extensions

(2.3)
${1}$${\mathrm{SMCG}_{d}}$${\mathrm{MCG}_{d}}$${\mathrm{Aut}(\pi_{3}(X_{d}),\lambda)}$${1}$${1}$${\mathrm{K}_{d}}$${\mathrm{SMCG}_{d}}$${H^{3}(X_{d};\mathbb{Z})}$${1.}$$\scriptstyle{v_{p}}$

The first defines the subgroup $\mathrm{SMCG}_{d}$, and expresses the fact
that all automorphisms of $\pi_{3}(X_{d})$ which preserve $\lambda$ are are
the identity on its radical $\mathbb{Z}/d\\{\eta\\}$ can be realised by
diffeomorphisms. The second describes $\mathrm{SMCG}_{d}$ as an extension of
$H^{3}(X_{d};\mathbb{Z})$ by a certain finite abelian group $\mathrm{K}_{d}$
(we shall describe the map $v_{p}$ later: experts will understand the idea
from the phrase “distorsion of the first Pontrjagin class”). The subgroup
$\mathrm{K}_{d}$ is generated by those diffeomorphisms of $X_{d}$ that can be
supported in a neighbourhood of $S^{2}\subset X_{d}$. This includes those
diffeomorphisms wich are supported in a disc, which defines a homomorphism
$\Phi\mathrel{\mathop{\mathchar
58\relax}}\pi_{0}(\mathrm{Diff}_{\partial}(D^{6}))\cong\Theta_{7}\cong\mathbb{Z}/28\to\mathrm{K}_{d}$.
The group $\mathrm{K}_{d}$ is completely determined by Kreck and Su, and we
summarise what we need from their calculations as follows.

######  Lemma 2.2 (Kreck–Su [KS20]).

The map $\Phi\mathrel{\mathop{\mathchar 58\relax}}\Theta_{7}\to\mathrm{K}_{d}$
has

$\mathrm{Ker}(\Phi)\cong\begin{cases}\mathbb{Z}/4&d\equiv 2,4,6,10,12,14\mod
16\\\ \mathbb{Z}/2&d\equiv 3,5,8,11,13\mod 16\\\ 0&d\equiv 0,1,7,9,15\mod
16\end{cases}\oplus\begin{cases}\mathbb{Z}/7&d\not\equiv 0\mod 7\\\ 0&d\equiv
0\mod 7\end{cases}$

and

$\mathrm{Coker}(\Phi)\cong\begin{cases}\mathbb{Z}/2&d\equiv 0\mod 4\\\
0&d\not\equiv 0\mod 4\end{cases}\oplus\begin{cases}\mathbb{Z}/3&d\equiv 0\mod
3\\\ 0&d\not\equiv 0\mod 3.\end{cases}$

###### Proof sketch.

The bulk of this concerns the 2-torsion, which in the $d$ odd case, for
example, requires a case-by-case analysis of table (7.1) on p. 35 of [KS20]
reduced modulo 4, with $k\mathrel{\mathop{\mathchar
58\relax}}=\tfrac{1}{4}(5-d^{2})d$. Looking at the 4th and 5th columns in
particular, one sees that the quotient of $(\mathbb{Z}/4)^{3}$ by the
relations given by the columns of this table is generated by the class of the
first basis vector. The 7-torsion is immediate from the discussion just before
this table, as is the 3-torsion. The $d$ even case is analogous, using the
table on p. 42 of [KS20]. ∎

##  3\. Upper bounds: tangential structures and quadratic forms

The goal of this section is as follows. We shall describe a certain tangential
structure $\theta^{\mathrm{hyp}}$ and a $\theta^{\mathrm{hyp}}$-structure
$\ell^{\mathrm{hyp}}_{X_{d}}$ on $X_{d}$, which is preserved up to homotopy by
monodromy: thus there is a factorisation

$\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Mon}_{d}\longrightarrow\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})\leq\mathrm{MCG}_{d}.$

We will then use the work of Kreck and Su to analyse the subgroup
$\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})$, and show that
the extensions (2.3) simplify to a single central extension

${1}$${\mathrm{K}_{d}}$${\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})}$${\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)}$${1,}$

for $\mu\mathrel{\mathop{\mathchar 58\relax}}\pi_{3}(X_{d})\to\mathbb{Z}/2$ a
certain quadratic refinement that we will construct.

###  3.1. Tangential structures

A _stable tangential structure_ is a Serre fibration
$\bar{\theta}\mathrel{\mathop{\mathchar 58\relax}}B\to B\mathrm{O}$, and a
_$\bar{\theta}$ -structure_ on a manifold $M$ means a choice of lift
$\ell\mathrel{\mathop{\mathchar 58\relax}}M\to B$ of the map
$\tau^{s}_{M}\mathrel{\mathop{\mathchar 58\relax}}M\to B\mathrm{O}$
classifying the stable tangent bundle of $M$. Familiar examples are
$\bar{\theta}^{+}\mathrel{\mathop{\mathchar 58\relax}}B\mathrm{SO}\to
B\mathrm{O}$, for which $\bar{\theta}^{+}$-structures are orientations, or
$\bar{\theta}^{\mathrm{sfr}}\mathrel{\mathop{\mathchar
58\relax}}E\mathrm{O}\to B\mathrm{O}$, for which
$\bar{\theta}^{\mathrm{sfr}}$-structures are stable framings.

As the map $\tau_{M}$ is not literally unique—it is only unique up to coherent
homotopies—in order to study the action of $\mathrm{Diff}(M)$ on
$\bar{\theta}$-structures it is best to take another model. If $M$ is
$d$-dimensional then we define a corresponding (unstable) tangential structure
via the pullback

(3.1)
${B_{d}}$${B}$${B\mathrm{O}(d)}$${B\mathrm{O},}$$\scriptstyle{\theta}$$\scriptstyle{\bar{\theta}}$

so that there is a $d$-dimensional vector bundle $\theta^{*}\gamma_{d}\to
B_{d}$ classified by $\theta$. Then we define the space of $\theta$-structures
on $M$ to be the space of bundle maps

$\Theta(M)\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Bun}(TM,\theta^{*}\gamma_{d}),$

i.e. continuous maps $TM\to\theta^{*}\gamma_{d}$ which cover a map $M\to
B_{d}$ and are linear isomorphisms on each fibre. This has an evident right
action of $\mathrm{Diff}(M)$, by precomposing with the differential of a
diffeomorphism. We write

$\theta(M)\mathrel{\mathop{\mathchar 58\relax}}=\pi_{0}\Theta(M),$

which inherits a right $\pi_{0}\mathrm{Diff}(M)$-action.

Choosing a map $\tau_{M}\mathrel{\mathop{\mathchar 58\relax}}M\to
B\mathrm{O}(d)$ classifying the tangent bundle of $M$, it is easy to show
(using that $\mathrm{Bun}(TM,\gamma_{d})\simeq*$, which is the defining
property of the universal bundle $\gamma_{d}\to B\mathrm{O}(d)$) that
$\Theta(M)$ is homotopy equivalent to the space of lifts in the diagram

${B_{d}}$${M}$${B\mathrm{O}(d),}$$\scriptstyle{\theta}$$\scriptstyle{\tau_{M}}$

and so by cartesianness of (3.1) it is equivalent to the space of lifts of the
map $\tau^{s}_{M}$ along $\bar{\theta}$, as we originally defined a
$\bar{\theta}$-structure to be. By the cartesianness of (3.1) we therefore
blur the distinction between $\bar{\theta}$\- and $\theta$-structures.

######  Remark 3.1.

In the discussion above it is clear that it does not matter that $\theta$ is
obtained by pulling back from a fibration over $B\mathrm{O}$, and it is common
to refer to a fibration over $B\mathrm{O}(d)$ as a tangential structure.

###  3.2. The tangential structure $\theta^{\mathrm{hyp}}$

There are two natural tangential structures in the situation we are
considering. The tangent bundle of a degree $d$ hypersurface
$X\subset\mathbb{C}\mathbb{P}^{4}$ satisfies
$TX\oplus\mathcal{O}(d)|_{X}=T\mathbb{CP}^{4}|_{X}$ and so using that
$T\mathbb{CP}^{4}\oplus\underline{\mathbb{C}}\cong\mathcal{O}(1)^{\oplus 5}$
we have

(3.2)
$TX\oplus\mathcal{O}(d)|_{X}\oplus\underline{\mathbb{C}}\cong\mathcal{O}(1)^{\oplus
5}|_{X}.$

If we define maps

$\bar{\theta}^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{C}\mathbb{P}^{\infty}\xrightarrow{\mathcal{O}(1)^{\oplus
5}-\mathcal{O}(d)-\underline{\mathbb{C}}^{4}}B\mathrm{U}\overset{\bar{\theta}^{\mathbb{C}}}{\longrightarrow}B\mathrm{O},$

which we implicitly convert into Serre fibrations, then the identity (3.2)
shows that $X$ has a canonical $\theta^{\mathrm{hyp}}$-structure
$\ell_{X}^{\mathrm{hyp}}$, and so also a canonical
$\theta^{\mathbb{C}}$-structure $\ell_{X}^{\mathbb{C}}$.

More generally, the universal family of smooth hypersurfaces
$\mathcal{X}_{d}\to\mathcal{U}_{d}$ comes with a fibrewise embedding
$\mathcal{X}_{d}\subset\mathcal{U}_{d}\times\mathbb{CP}^{4}$, and the vertical
tangent bundle of this family satisfies
$T_{v}\mathcal{X}_{d}\oplus\mathcal{O}(d)|_{\mathcal{X}_{d}}\oplus\underline{\mathbb{C}}\cong\mathcal{O}(1)^{\oplus
5}|_{\mathcal{X}_{d}}$, giving a lift of the map
$\tau_{v}\mathrel{\mathop{\mathchar 58\relax}}\mathcal{X}_{d}\to B\mathrm{O}$
classifying $T_{v}\mathcal{X}_{d}$ along $\bar{\theta}^{\mathrm{hyp}}$, i.e. a
$\theta^{\mathrm{hyp}}$-structure on the whole family. This implies that
$\theta^{\mathrm{hyp}}$-structure $\ell_{X_{d}}^{\mathrm{hyp}}$ is preserved
by $\mathrm{Mon}_{d}$. Thus the $\theta^{\mathbb{C}}$-structure
$\ell_{X_{d}}^{\mathbb{C}}$ is also preserved by $\mathrm{Mon}_{d}$, which is
unsurprising as it is even preserved by the symplectic mapping class group
$\pi_{0}\mathrm{Symp}(X_{d})$. Somewhat surprisingly we have the following.

######  Lemma 3.2.

The $\theta^{\mathbb{C}}$-structure $\ell_{X_{d}}^{\mathbb{C}}$ is preserved
by $\mathrm{MCG}_{d}$.

###### Proof.

As $\bar{\theta}^{\mathbb{C}}\mathrel{\mathop{\mathchar
58\relax}}B\mathrm{U}\to B\mathrm{O}$ is a fibration of $H$-spaces (in fact of
infinite loop spaces) with fibre the $H$-space $\mathrm{O}/\mathrm{U}$, the
set $\theta^{\mathbb{C}}(X_{d})$ is a torsor for the abelian group
$[X_{d},O/U]$, with action

$-\cdot-\mathrel{\mathop{\mathchar
58\relax}}\theta^{\mathbb{C}}(X_{d})\times[X_{d},\mathrm{O}/\mathrm{U}]\longrightarrow\theta^{\mathbb{C}}(X_{d}).$

An orientation-preserving diffeomorphism $\varphi\mathrel{\mathop{\mathchar
58\relax}}X_{d}\to X_{d}$ then has $\ell^{\mathbb{C}}_{X_{d}}\circ
D\varphi=\ell^{\mathbb{C}}_{X_{d}}\cdot\delta^{\mathbb{C}}(\varphi)$ for a
unique $\delta^{\mathbb{C}}(\varphi)\in[X_{d},\mathrm{O}/\mathrm{U}]$, and
this defines a function

$\delta^{\mathbb{C}}\mathrel{\mathop{\mathchar
58\relax}}\mathrm{MCG}_{d}\longrightarrow[X_{d},\mathrm{O}/\mathrm{U}]$

which is a (right) crossed homomorphism with respect to the right action of
$\mathrm{MCG}_{d}$ on $[X_{d},\mathrm{O}/\mathrm{U}]$ by precomposition.

Using the Atiyah–Hirzebruch spectral sequence to calculate
$[X_{d},\mathrm{O}/\mathrm{U}]$, we see it has a filtration with filtration
quotients

$H^{0}(X_{d};\mathbb{Z}/2)=\mathbb{Z}/2,\quad
H^{2}(X_{d};\mathbb{Z})=\mathbb{Z}\\{x\\},\quad
H^{6}(X_{d};\mathbb{Z})=\mathbb{Z}\\{xy\\}.$

The projection of $\delta^{\mathbb{C}}(\varphi)$ to the first quotient
$H^{0}(X_{d};\mathbb{Z}/2)$ is always trivial, as this records whether
$\varphi$ preserves orientation. Neglecting this first quotient we see that
the remainder $[X_{d},\mathrm{SO}/\mathrm{U}]$ of
$[X_{d},\mathrm{O}/\mathrm{U}]$ is torsion-free, and the Chern–Dold character
gives an isomorphism

$ch\mathrel{\mathop{\mathchar
58\relax}}[X_{d},\mathrm{SO}/\mathrm{U}]\otimes\mathbb{Q}\overset{\sim}{\longrightarrow}H^{4*+2}(X_{d},\mathbb{Q}).$

In other words $\delta^{\mathbb{C}}(\varphi)$ is trivial if and only if
$\varphi$ fixes $c_{1}(X_{d})$ and $c_{3}(X_{d})$. But all orientation-
preserving diffeomorphisms act trivially on $H^{even}(X_{d};\mathbb{Z})$, so
they all preserve $c_{1}(X_{d})$ and $c_{3}(X_{d})$. ∎

######  Remark 3.3.

There is an unstable tangential structure
$\theta^{ac}\mathrel{\mathop{\mathchar 58\relax}}B\mathrm{U}(3)\to
B\mathrm{O}(6)$, classifying almost complex structures on 6-manifolds:
$\bar{\theta}^{\mathbb{C}}\mathrel{\mathop{\mathchar 58\relax}}B\mathrm{U}\to
B\mathrm{O}$ classifies _stable_ almost complex structures. Although we will
not need it, we record the fact that $\mathrm{MCG}_{d}$ also preserves any
almost complex structure refining $\ell_{X_{d}}^{\mathbb{C}}$.

There is a canonical map

$\theta^{ac}(X_{d})\longrightarrow\theta^{\mathbb{C}}(X_{d})$

and we claim that its fibres have size $\leq 2$. The homotopy fibre of
$\Theta^{ac}(X_{d})\to\Theta^{\mathbb{C}}(X_{d})$ over some stable almost
complex structure is identified with the space $\Gamma$ of lifts

${B\mathrm{U}(3)}$${X_{d}}$${B\mathrm{SO}(6)\times_{B\mathrm{SO}}B\mathrm{U}.}$

Taking the Postnikov tower of the vertical map, and choosing a lift $\ell$
(for example the one coming from the actual complex structure of $X_{d}$),
leads to a (fringed) Federer spectral sequence of signature

$E^{2}_{s,t}=H^{-s}(X_{d};\pi_{t}(\mathrm{hofib}(B\mathrm{U}(3)\to
B\mathrm{SO}(6)\times_{B\mathrm{SO}}B\mathrm{U})))\Longrightarrow\pi_{t+s}(\Gamma,\ell)$

with differentials of the form $d^{r}\mathrel{\mathop{\mathchar
58\relax}}E^{r}_{s,t}\to E^{r}_{s-r,t+r-1}$. We may calculate the coefficients
using the homotopy fibre sequence

$\mathrm{hofib}(B\mathrm{U}(3)\to
B\mathrm{SO}(6)\times_{B\mathrm{SO}}B\mathrm{U})\longrightarrow\mathrm{U}/\mathrm{U}(3)\longrightarrow\mathrm{SO}/\mathrm{SO}(6).$

Using the work of Paechter [Pae56] we see that
$\pi_{i}(\mathrm{SO}/\mathrm{SO}(6))=0,0,0,0,0,\mathbb{Z},\mathbb{Z}/4,0$ for
$i=1,2,\ldots,8$ and using the work of Gilmore [Gil67] we see that
$\pi_{i}(\mathrm{U}/\mathrm{U}(3))=0,0,0,0,0,0,\mathbb{Z},0$ for
$i=1,2,\ldots,8$. For the map between them we use that
$\pi_{6}(\mathrm{U}(3))=0$ and $\pi_{6}(\mathrm{SO}(6))=0$ from [Ker60], and
consider the diagram

${\mathbb{Z}=\pi_{7}(\mathrm{U})}$${\mathbb{Z}=\pi_{7}(\mathrm{SO})}$${\mathbb{Z}=\pi_{7}(\mathrm{U}/\mathrm{U}(3))}$${\mathbb{Z}/4=\pi_{7}(\mathrm{SO}/\mathrm{SO}(6))}$${0=\pi_{6}(\mathrm{U}(3))}$${0=\pi_{6}(\mathrm{SO}(6))}$$\scriptstyle{2}$$\scriptstyle{\cong}$

to see that the middle map has image $2\mathbb{Z}/4\leq\mathbb{Z}/4$. This
yields

$\pi_{i}(\mathrm{hofib}(B\mathrm{U}(3)\to
B\mathrm{SO}(6)\times_{B\mathrm{SO}}B\mathrm{U}))=0,0,0,0,\mathbb{Z},\mathbb{Z}/2,\mathbb{Z}\text{
for }i=1,2,\ldots,7.$

It follows that in total degree $t+s=0$ the only non-zero entry in the
spectral sequence is
$E^{2}_{-6,6}=H^{6}(X_{d};\mathbb{Z}/2)\cong\mathbb{Z}/2$, and so $\Gamma$ has
at most two path components. This proves the claim.

It follows that $\ell^{\mathbb{C}}_{X_{d}}$ has either one or two almost
complex structures refining it. If there is just one then it is clearly
preserved by $\mathrm{MCG}_{d}$ by Lemma 3.2. If there are two, then acting on
them gives a homomorphism $\mathrm{MCG}_{d}\to\mathfrak{S}_{2}$. But by [KS20,
Theorem 2.7] the abelianisation of $\mathrm{MCG}_{d}$ is trivial (note $X_{d}$
is Spin precisely when $d$ is odd) and so this homomorphism must be trivial.
Thus $\mathrm{MCG}_{d}$ fixes each of the two refinements.

It is not the case that $\mathrm{MCG}_{d}$ preserves
$\ell^{\mathrm{hyp}}_{X_{d}}$, and our next goal is to determine
$\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})$. The methods
we will use are related to those of [KRW21, Kra20]. We begin by an analysis
similar to the proof of Lemma 3.2. The map
$\bar{\theta}^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{CP}^{\infty}\to B\mathrm{O}$ is no longer a map of
$H$-spaces, but we can proceed as follows. We have explained that
$\Theta^{\mathrm{hyp}}(X_{d})$ is homotopy equivalent to the space of lifts of
$\tau^{s}_{X_{d}}\mathrel{\mathop{\mathchar 58\relax}}X_{d}\to B\mathrm{O}$
along $\bar{\theta}^{\mathrm{hyp}}$, which shows that it fits into a homotopy
cartesian square

${\Theta^{\mathrm{hyp}}(X_{d})}$${\mathrm{map}(X_{d},\mathbb{C}\mathbb{P}^{\infty})}$${\\{\tau_{X_{d}}\\}}$${\mathrm{map}(X_{d},B\mathrm{O}).}$$\scriptstyle{(\bar{\theta}^{\mathrm{hyp}})_{*}}$

On homotopy groups this gives a sequence

$0=H^{1}(X_{d};\mathbb{Z})\longrightarrow
KO^{-1}(X_{d})\overset{\curvearrowright}{\longrightarrow}\theta^{\mathrm{hyp}}(X_{d})\overset{\hat{x}}{\longrightarrow}H^{2}(X_{d};\mathbb{Z})\overset{(\bar{\theta}^{\mathrm{hyp}})_{*}}{\longrightarrow}KO^{0}(X_{d})$

which is exact exact in the sense of groups and pointed sets: the two leftmost
terms are groups, the map marked $\curvearrowright$ is an action, and the
orbits of this action are in bijection with
$((\bar{\theta}^{\mathrm{hyp}})_{*})^{-1}(0)$. The map $\hat{x}$ assigns to a
lift $\ell^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}X_{d}\to\mathbb{C}\mathbb{P}^{\infty}$ the class
$(\ell^{\mathrm{hyp}})^{*}(\iota_{2})\in H^{2}(X_{d};\mathbb{Z})$.

As an orientation-preserving diffeomorphism $\varphi\mathrel{\mathop{\mathchar
58\relax}}X_{d}\to X_{d}$ acts trivially on $H^{2}(X_{d};\mathbb{Z})$ it
follows that $\ell^{\mathrm{hyp}}_{X_{d}}\circ D\varphi$ and
$\ell^{\mathrm{hyp}}_{X_{d}}$ lie in the same fibre of $\hat{x}$: thus there
is a unique $\delta^{\mathrm{hyp}}(\varphi)\in KO^{-1}(X_{d})$ such that
$\ell^{\mathrm{hyp}}_{X_{d}}\circ
D\varphi=\ell^{\mathrm{hyp}}_{X_{d}}\cdot\delta^{\mathrm{hyp}}(\varphi)$. As
in the proof of Lemma 3.2 this defines a function

$\delta^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}\mathrm{MCG}_{d}\longrightarrow KO^{-1}(X_{d})$

which is a crossed homomorphism with respect to the right action of
$\mathrm{MCG}_{d}$ on $KO^{-1}(X_{d})$ by contravariant functoriality. Using
the Atiyah–Hirzebruch spectral sequence to calculate $KO^{-1}(X_{d})$, we see
it has a filtration with filtration quotients

$H^{0}(X_{d};\pi_{0}(\mathrm{O}))=\mathbb{Z}^{\times},\quad
H^{3}(X_{d};\pi_{3}(\mathrm{O})).$

As in the lemma, projection of $\delta^{\mathrm{hyp}}(\varphi)$ to the first
quotient detects whether $\varphi$ preserves orientation, which it does, so
$\delta^{\mathrm{hyp}}$ refines to a crossed homomorphism

$\delta^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}\mathrm{MCG}_{d}\longrightarrow
H^{3}(X_{d};\pi_{3}(\mathrm{O}))\subset KO^{-1}(X_{d}).$

We then have

$\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})=\mathrm{Ker}(\delta^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}\mathrm{MCG}_{d}\to H^{3}(X_{d};\pi_{3}(\mathrm{O}))).$

######  Remark 3.4.

The stable tangential structure
$\bar{\theta}^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{CP}^{\infty}\to B\mathrm{O}$ is not the most sophisticated
structure we can endow a hypersurface $X\subset\mathbb{CP}^{4}$ with. The
equation $TX\oplus\mathcal{O}(d)|_{X}=T\mathbb{CP}^{4}|_{X}$ shows that it may
be endowed with the (unstable) tangential structure given by the homotopy
equaliser

${B}$${B\mathrm{U}(3)\times\mathbb{CP}^{4}}$${B\mathrm{U}(4)}$

of the maps classifying $\gamma_{3}^{\mathbb{C}}\oplus\mathcal{O}(d)$ and
$T\mathbb{CP}^{4}$ respectively, made into a tangential structure via the
natural maps $B\to B\mathrm{U}(3)\times\mathbb{CP}^{4}\to B\mathrm{U}(3)\to
B\mathrm{O}(6)$. It is more complicated to analyse the set of such structures
on $X_{d}$, and it seems likely that for reasons similar to Remark 3.3 there
is no advantage in doing so.

###  3.3. Distorsion of the first Pontrjagin class

We wish to relate $\delta^{\mathrm{hyp}}$ to Kreck and Su’s map
$v_{p}\mathrel{\mathop{\mathchar 58\relax}}\mathrm{SMCG}_{d}\to
H^{3}(X_{d};\mathbb{Z})$ from (2.3), which is described in terms of Sullivan’s
“distorsion of the first Pontrjagin class”, see [Sul77, §13]. As explained by
Hain [Hai23, §5.2], in the case at hand this may be implemented as follows.

Let $\varphi\in\mathrm{MCG}_{d}$ act trivially on cohomology, i.e. lie in the
_Torelli subgroup_ $\mathrm{TMCG}_{d}\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Ker}(\mathrm{MCG}_{d}\to\mathrm{Aut}(H^{3}(X_{d};\mathbb{Z})))$,
and let $T_{\varphi}$ denote its mapping torus. The long exact sequence on
cohomology for the pair $(T_{\varphi},X_{d})$, using excision, takes the form

$\cdots\longrightarrow
H^{3}(X_{d};\mathbb{Z})\overset{Id-\varphi^{*}=0}{\longrightarrow}H^{3}(X_{d};\mathbb{Z})\overset{\partial}{\longrightarrow}H^{4}(T_{\varphi};\mathbb{Z})\longrightarrow
H^{4}(X_{d};\mathbb{Z})\longrightarrow\cdots.$

There is a unique $\bar{x}\in H^{2}(T_{\varphi};\mathbb{Z})$ which restricts
to $x\in H^{2}(X_{d};\mathbb{Z})$. On $X_{d}$ we have
$p_{1}(TX_{d})=(5-d^{2})x^{2}$, so the class
$p_{1}(T(T_{\varphi}))-(5-d^{2})\bar{x}^{2}\in H^{4}(T_{\varphi};\mathbb{Z})$
vanishes on $X_{d}$ and so comes from a unique class
$\Delta_{p_{1}}(\varphi)\in H^{3}(X_{d};\mathbb{Z})$. This defines a function

$\Delta_{p_{1}}\mathrel{\mathop{\mathchar
58\relax}}\mathrm{TMCG}_{d}\longrightarrow H^{3}(X_{d};\mathbb{Z}),$

which is easily checked to be a homomorphism.

######  Lemma 3.5.

We have

$\Delta_{p_{1}}\mathrel{\mathop{\mathchar
58\relax}}\mathrm{TMCG}_{d}\overset{\delta^{\mathrm{hyp}}|_{\mathrm{TMCG}_{d}}}{\longrightarrow}H^{3}(X_{d};\pi_{3}(\mathrm{O}))\overset{(p_{1})_{*}}{\longrightarrow}H^{3}(X_{d};\mathbb{Z}).$

Under the usual identification $\pi_{3}(\mathrm{O})=\mathbb{Z}$, the latter
map is multiplication by 2.

###### Proof.

For $\varphi$ in the Torelli group, we describe
$\delta^{\mathrm{hyp}}(\varphi)\in KO^{-1}(X_{d})$ in a similar way to
$\Delta_{p_{1}}$. The long exact sequence on $KO$-theory for the pair
$(T_{\varphi},X_{d})$, using excision, takes the form

$\cdots\longrightarrow
KO^{-1}(X_{d})\overset{Id-\varphi^{*}}{\longrightarrow}KO^{-1}(X_{d})\overset{\partial}{\longrightarrow}KO^{0}(T_{\varphi})\longrightarrow
KO^{0}(X_{d})\longrightarrow\cdots.$

The short exact sequence $0\to H^{3}(X_{d};\mathbb{Z})\to KO^{-1}(X_{d})\to
H^{0}(X_{d};\mathbb{Z}/2)\to 0$ is functorially split by $*\to X_{d}\to*$, so
the fact that $\varphi$ acts as the identity on cohomology shows that the
left-hand map in the above sequence is zero. Using the map
$\bar{x}\mathrel{\mathop{\mathchar 58\relax}}T_{\varphi}\to
K(\mathbb{Z},2)=\mathbb{CP}^{\infty}$ we can form the class

$(T(T_{\varphi})-\varepsilon^{7})-\bar{x}^{*}\bar{\theta}^{\mathrm{hyp}}\in
KO^{0}(T_{\varphi}).$

This vanishes on $X_{d}$, and so comes from a unique class in
$KO^{-1}(X_{d})$: this is $\delta^{\mathrm{hyp}}(\varphi)$.

The first Pontrjagin class gives a homomorphism
$p_{1}\mathrel{\mathop{\mathchar 58\relax}}KO^{0}(T_{\varphi})\to
H^{4}(T_{\varphi};\mathbb{Z})$, which extends to a map between the two exact
sequences above. The composition $H^{3}(X_{d};\pi_{3}(\mathrm{O}))\subset
KO^{-1}(X_{d})\overset{p_{1}}{\to}H^{3}(X_{d};\mathbb{Z})$ is the map induced
by $p_{1}\mathrel{\mathop{\mathchar
58\relax}}\pi_{3}(\mathrm{O})\to\mathbb{Z}$, which is well-known to be an
isomorphism onto the subgroup $2\mathbb{Z}$. ∎

We now relate this to the work of Kreck and Su, namely to the map $v_{p}$ in
the central extension

$0\longrightarrow\mathrm{K}_{d}\longrightarrow\mathrm{SMCG}_{d}\overset{v_{p}}{\longrightarrow}H^{3}(X_{d};\mathbb{Z})\longrightarrow
0,$

where $\mathrm{K}_{d}$ is a certain finite abelian group, and
$\mathrm{SMCG}_{d}\leq\mathrm{MCG}_{d}$ is the subgroup of those
diffeomorphisms which act trivially on $\pi_{3}(X_{d})$. This map is made
explicit by the diagram after the statement of [KS20, Theorem 2.6] and the
definition of the map $v_{x,p}$ just before [KS20, Proposition 4.6]. Together
those show that:

######  Lemma 3.6.

We have $\Delta_{p_{1}}|_{\mathrm{SMCG}_{d}}=4\cdot v_{p}$.∎

Restricted to $\mathrm{SMCG}_{d}$ the crossed homomorphism
$\delta^{\mathrm{hyp}}\mathrel{\mathop{\mathchar 58\relax}}\mathrm{MCG}_{d}\to
H^{3}(X_{d};\mathbb{Z})$ is a homomorphism, so it annihilates the finite group
$\mathrm{K}_{d}$ and hence descends to a homomorphism

$\delta^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}\mathrm{SMCG}_{d}/\mathrm{K}_{d}\longrightarrow
H^{3}(X_{d};\pi_{3}(\mathrm{O})).$

######  Corollary 3.7.

This is an isomorphism onto $2\cdot H^{3}(X_{d};\pi_{3}(\mathrm{O}))$.

###### Proof.

Postcomposing with the map $(p_{1})_{*}\mathrel{\mathop{\mathchar
58\relax}}H^{3}(X_{d};\pi_{3}(\mathrm{O}))\to H^{3}(X_{d};\mathbb{Z})$, which
is an isomorphism onto $2\cdot H^{3}(X_{d};\mathbb{Z})$, gives the map
$\Delta_{p_{1}}$ by Lemma 3.5, which is the map $4\cdot v_{p}$ by Lemma 3.6,
so is an isomorphism onto $4\cdot H^{3}(X_{d};\mathbb{Z})$ as $v_{p}$ is
surjective. ∎

In particular
$\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})=\mathrm{Ker}(\delta^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}\mathrm{MCG}_{d}\to H^{3}(X_{d};\pi_{3}(\mathrm{O})))$ intersects
$\mathrm{SMCG}_{d}$ precisely in $\mathrm{K}_{d}$, giving a half-exact
sequence

(3.3)
$1\longrightarrow\mathrm{K}_{d}\longrightarrow\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})\longrightarrow\mathrm{Aut}(\pi_{3}(X_{d}),\lambda).$

###  3.4. The surgery kernel

The action of $\mathrm{MCG}_{d}$ on the set $\theta^{\mathrm{hyp}}(X_{d})$
preserves the subset $\theta^{\mathrm{hyp}}(X_{d};x,+)$ of those
$\theta^{\mathrm{hyp}}$-structures which map to $x\in H^{2}(X_{d};\mathbb{Z})$
under $\hat{x}$ and which induce the standard orientation of $X_{d}$. This
action descends to an action of $\mathrm{Aut}(\pi_{3}(X_{d}),\lambda)$ on the
set

$\theta^{\mathrm{hyp}}(X_{d};x,+)/\mathrm{SMCG}_{d},$

and the discussion of the previous section shows that this set is a torsor for
$H^{3}(X_{d};\mathbb{Z}/2)=H^{3}(X_{d};\pi_{3}(\mathrm{O}))/\mathrm{Im}(\delta^{\mathrm{hyp}}|_{\mathrm{SMCG}_{d}})$.
We wish to identify this set with something more meaningful. Associated to a
$\theta^{\mathrm{hyp}}$-structure $\ell$ there is a quadratic form
$\mu_{\ell}\mathrel{\mathop{\mathchar 58\relax}}\pi_{3}(X_{d})\to\mathbb{Z}/2$
on

$\pi_{3}(X_{d})\overset{\sim}{\longleftarrow}\pi_{4}(\mathbb{C}\mathbb{P}^{\infty},X_{d})\overset{\sim}{\longrightarrow}H_{4}(\mathbb{C}\mathbb{P}^{\infty},X_{d};\mathbb{Z})$

obtained by considering this as the surgery kernel for the normal map
$\ell\mathrel{\mathop{\mathchar
58\relax}}X_{d}\to\mathbb{C}\mathbb{P}^{\infty}$ covered by the corresponding
bundle map, see [Kre99, p. 728]. This $\mu_{\ell}$ is a quadratic refinement
of the intersection form $\lambda$ in the sense that

$\mu_{\ell}(a+b)=\mu_{\ell}(a)+\mu_{\ell}(b)+\lambda(a,b)\mod 2.$

In terms of the extension (2.1), as the subgroup
$\mathbb{Z}/d\\{\eta\\}\leq\pi_{3}(X_{d})$ is radical with respect to
$\lambda$, the restriction
$\mu_{\ell}|_{\mathbb{Z}/d}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}/d\to\mathbb{Z}/2$ is a homomorphism. In particular if $d$
is odd then this homomorphism must be trivial, and so $\mu_{\ell}$ descends to
a quadratic refinement of the intersection form on $H_{3}(X_{d};\mathbb{Z})$.

######  Remark 3.8.

The quadratic refinement $\mu_{\ell^{\mathrm{hyp}}_{X_{d}}}$ is
$\mathrm{Mon}_{d}$-invariant, and when $d$ is odd the induced quadratic
refinement on $H_{3}(X_{d};\mathbb{Z})$ supplies the missing ingredient in
Beauville’s Theorem 2.1. The argument of [Woo75, Section 2] applies to this
quadratic refinement, showing that it has Arf invariant $0$ if $d\equiv\pm
1\mod 8$ and Arf invariant $1$ if $d\equiv\pm 3\mod 8$.

If $d$ is even then this is not the case:

######  Lemma 3.9.

If $d$ is even then the homomorphism
$\mu_{\ell}|_{\mathbb{Z}/d}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}/d\to\mathbb{Z}/2$ is surjective.

###### Proof.

If not then it would descend to a quadratic refinement of the intersection
form on $H_{3}(X_{d};\mathbb{Z})$, which would be
$\mathrm{Mon}_{d}$-invariant, contradicting Beauville’s Theorem 2.1 in the
case $d$ even. ∎

Let us write $\mathrm{Quad}(\pi_{3}(X_{d}),\lambda)$ for the set of quadratic
refinements $\mu$ of $(\pi_{3}(X_{d}),\lambda)$ whose restriction to
$\mathbb{Z}/d\leq\pi_{3}(X_{d})$ is zero if $d$ is odd, and non-zero if $d$ is
even. If $\mu$ and $\mu^{\prime}$ are such quadratic refinements then
$\mu-\mu^{\prime}\mathrel{\mathop{\mathchar
58\relax}}\pi_{3}(X_{d})\to\mathbb{Z}/2$ vanishes on
$\mathbb{Z}/d\leq\pi_{3}(X_{d})$ so descends to a homomorphism
$H_{3}(X_{d};\mathbb{Z})\to\mathbb{Z}/2$, and hence
$\mathrm{Quad}(\pi_{3}(X_{d}),\lambda)$ forms a
$H^{3}(X_{d};\mathbb{Z}/2)$-torsor.

The construction above defines a function

$\displaystyle\Phi\mathrel{\mathop{\mathchar
58\relax}}\theta^{\mathrm{hyp}}(X_{d};x,+)$
$\displaystyle\longrightarrow\mathrm{Quad}(\pi_{3}(X_{d}),\lambda)$
$\displaystyle\ell$ $\displaystyle\longmapsto\mu_{\ell}.$

This is equivariant when the right-hand side is equipped with the right
$\mathrm{Aut}(\pi_{3}(X_{d}),\lambda)$-action by precomposition, so it
descends to a $\mathrm{Aut}(\pi_{3}(X_{d}),\lambda)$-equivariant function

$\Phi\mathrel{\mathop{\mathchar
58\relax}}\theta^{\mathrm{hyp}}(X_{d};x,+)/\mathrm{SMCG}_{d}\longrightarrow\mathrm{Quad}(\pi_{3}(X_{d}),\lambda).$

This is easily checked to be a map of $H^{3}(X_{d};\mathbb{Z}/2)$-torsors, so
is a bijection.

######  Corollary 3.10.

The image of $\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})$
in $\mathrm{Aut}(\pi_{3}(X_{d}),\lambda)$ is the stabiliser
$\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$ of the quadratic form
$\mu\mathrel{\mathop{\mathchar 58\relax}}=\mu_{\ell^{\mathrm{hyp}}_{X_{d}}}$.∎

We can therefore improve the half-exact sequence (3.3) to an exact sequence

(3.4)
$1\longrightarrow\mathrm{K}_{d}\longrightarrow\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})\longrightarrow\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)\longrightarrow
1.$

##  4\. Lower bounds: automorphisms of the Fermat hypersurface

The goal of this section is to show that the image of

$\mathrm{Mon}_{d}\longrightarrow\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})$

is not too small, by showing that it contains
$\mathrm{Im}(\Phi\mathrel{\mathop{\mathchar
58\relax}}\Theta_{7}\to\mathrm{K}_{d})$, and surjects onto
$\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$. The first of these is an immediate
consequence of the work of Krylov [Kry03], and is very general: it just uses
that $X_{d}$ is a 3-fold which admits a deformation to an $A_{2}$-singularity.
The latter is more specific, and proceeds by analysing the effect of
automorphisms of the Fermat hypersurface on $\pi_{3}(X_{d})$, following Pham
[Pha65] and Looijenga [Loo10].

###  4.1. Dehn twists and the Milnor sphere

The following simple consequence of the work of Krylov is rather surprising.
It presumably admits generalisations to higher dimensions, and can perhaps be
thought of as the analogue of the construction of $bP$-spheres as Brieskorn
varieties, but for automorphisms.

######  Theorem 4.1.

For $d\geq 3$ the subgroup $\mathrm{Im}(\Phi\mathrel{\mathop{\mathchar
58\relax}}\Theta_{7}\to\mathrm{K}_{d})\leq\mathrm{MCG}_{d}$ is contained in
the image of $\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Mon}_{d}\to\mathrm{MCG}_{d}$.

###### Proof.

Under the given condition $X_{d}$ admits a deformation to an
$A_{2}$-singularity, meaning that there is an orientation-preserving embedding
$T^{*}S^{3}\natural T^{*}S^{3}\subset X_{d}$ of the plumbing such that the
Dehn twists around each $S^{3}$ lies in the image of $\alpha$. Now
$T^{*}S^{3}\natural T^{*}S^{3}\cong S^{3}\times S^{3}\setminus D^{6}\subset
S^{3}\times S^{3}$ and, considering $S^{3}\subset\mathbb{H}$ as the unit
quaternions, inside the latter manifold the two Dehn twists are isotopic to

$Y\mathrel{\mathop{\mathchar 58\relax}}(u,v)\mapsto(u,uv)\quad\text{ and
}\quad U\mathrel{\mathop{\mathchar 58\relax}}(u,v)\mapsto(v^{-1}u,v).$

By [Kry03, Theorem 3] these satisfy
$(YUY)^{4}=\Sigma_{\text{Milnor}}^{-1}\in\Theta_{7}\leq\pi_{0}(\mathrm{Diff}^{+}(S^{3}\times
S^{3}))$. On the other hand it follows from [Kre79, p. 657 and Lemma 3 b)]
that the map

$\pi_{0}(\mathrm{Diff}_{c}(T^{*}S^{3}\natural
T^{*}S^{3}))\cong\pi_{0}(\mathrm{Diff}_{c}(S^{3}\times S^{3}\setminus
D^{6}))\longrightarrow\pi_{0}(\mathrm{Diff}^{+}(S^{3}\times S^{3})),$

which extends by the identity over $D^{6}$, is an isomorphism. Thus the given
expression represents $\Sigma_{\text{Milnor}}^{-1}\in\Theta_{7}$ in the
plumbing, and so when put inside $X_{d}$ also represents the generator
$[\Sigma_{\text{Milnor}}^{-1}]\in\mathrm{Im}(\Theta_{7}\to K_{d})$. ∎

###  4.2. Automorphisms of $\pi_{3}(X_{d})$ realised by monodromy

By the discussion in Section 3.4 the quadratic refinement $\mu$ on
$\pi_{3}(X_{d})$ descends to a quadratic refinement on
$H_{3}(X_{d};\mathbb{Z})$ if and only if $d$ is odd. The map

$\rho\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)\longrightarrow\begin{cases}\mathrm{Aut}(H_{3}(X_{d};\mathbb{Z}),\lambda,\mu)&d\text{
odd}\\\ \mathrm{Aut}(H_{3}(X_{d};\mathbb{Z}),\lambda)&d\text{
even}\end{cases}$

is surjective, and its kernel is identified with the group of automorphisms of
the extension

$0\longrightarrow\mathbb{Z}/d\\{\eta\\}\longrightarrow\pi_{3}(X_{d})\longrightarrow
H_{3}(X_{d};\mathbb{Z})\longrightarrow 0$

which are the identity on the outer terms, and preserve $\mu$. There is an
isomorphism

$\mathrm{Ker}(\rho)\overset{\sim}{\longrightarrow}\mathrm{Hom}\left(H_{3}(X_{d};\mathbb{Z}),\begin{cases}\mathbb{Z}/d&d\text{
odd}\\\ \mathrm{ker}(\mu\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}/d\to\mathbb{Z}/2)&d\text{ even}\end{cases}\right)$

given as follows: for $\phi\in\mathrm{Ker}(\rho)$ and $z\in
H_{3}(X_{d};\mathbb{Z})$, choose a lift $\bar{z}\in\pi_{3}(X_{d})$ and form
$\phi(\bar{z})-\bar{z}\in\mathrm{Ker}(\pi_{3}(X_{d})\to
H_{3}(X_{d};\mathbb{Z}))=\mathbb{Z}/d\\{\eta\\}$, which does not depend on the
choice of lift $\bar{z}$ of $z$. Then
$\mu(\phi(\bar{z})-\bar{z})=\mu(\phi(\bar{z}))+\mu(\bar{z})-\lambda(\phi(\bar{z}),\bar{z})=0$
as the first two terms cancel (because $\phi$ preserves $\mu$) and the latter
is $\lambda(z,z)=0$ (because $\lambda$ factors over $H_{3}(X_{d};\mathbb{Z})$
and is antisymmetric). This map is easily checked to be an isomorphism. To
avoid distinguishing cases, we can write the target above as
$H^{3}(X_{d};2\cdot\mathbb{Z}/d)$.

######  Lemma 4.2.

If $G\leq\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$ is a subgroup which
satisfies $\rho(G)=\rho(\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu))$, then
$\mathrm{Ker}(\rho|_{G})=k\cdot H^{3}(X_{d};2\cdot\mathbb{Z}/d)$ for some
$k\in\mathbb{Z}$, under the identification given above.

###### Proof.

Under the surjection $1\mapsto 2\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}\to 2\cdot\mathbb{Z}/d$, the preimage of
$\mathrm{Ker}(\rho|_{G})$ is a subgroup of $H^{3}(X_{d};\mathbb{Z})$, and is
preserved by the action of $\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$ on this
group via $\rho$. It therefore suffices to show that all subgroups $I\leq
H^{3}(X_{d};\mathbb{Z})$ which are preserved by
$\mathrm{Aut}(H_{3}(X_{d};\mathbb{Z}),\lambda,q)$ and some quadratic
refinement $q$ are of the form $k\cdot H^{3}(X_{d};\mathbb{Z})$.

The group $\mathrm{Aut}(H_{3}(X_{d};\mathbb{Z}),\lambda,q)$ acts transitively
on the set of unimodular elements of the same $q$-length (this is not hard to
show by hand, but it follows from [Fri17, Corollary 3.13] using that
$\mathrm{usr}(\mathbb{Z})=2$ and $d\geq 3$ so that
$(H_{3}(X_{d};\mathbb{Z}),\lambda)$ contains $\geq 4$ hyperbolic forms and
hence $(H_{3}(X_{d};\mathbb{Z}),\lambda,q)$ contains $\geq 3$ hyperbolic
forms). Choose a symplectic basis $\\{e_{1},f_{1},\ldots,e_{g},f_{g}\\}$ for
$(H_{3}(X_{d};\mathbb{Z}),\lambda)$, such that $q(e_{i})=q(f_{i})=0$ for
$i>1$, and $q(e_{1})=q(f_{1})=\mathrm{Arf}(q)$.

Let $k\mathrel{\mathop{\mathchar 58\relax}}=\min\\{|\lambda(a,b)|\neq
0\,\mathrel{\mathop{\mathchar 58\relax}}\,a\in I,b\in
H^{3}(X_{d};\mathbb{Z})\\}$, then all elements of $I$ are divisible by $k$ in
$H^{3}(X_{d};\mathbb{Z})$, so $I\leq k\cdot H^{3}(X_{d};\mathbb{Z})$. Choose
an $a_{0}\in I$ such that $\lambda(a_{0},b_{0})=k$. Then $a_{0}=k\cdot
a_{0}^{\prime}$ for a unimodular $a^{\prime}_{0}\in H^{3}(X_{d};\mathbb{Z})$.

If $q(a_{0}^{\prime})=0$ then there are automorphisms sending $a_{0}^{\prime}$
to: (i) $e_{i}$ or $f_{i}$ for any $i>1$, (ii)
$e_{1}+\mathrm{Arf}(q)(e_{2}+f_{2})$ or $f_{1}+\mathrm{Arf}(q)(e_{2}+f_{2})$.
Thus $I$ contains $k\cdot e_{i}$ and $k\cdot f_{i}$ for all $i$, so $k\cdot
H^{3}(X_{d};\mathbb{Z})=I$.

If $q(a_{0}^{\prime})=1$ then there are automorphisms sending $a_{0}^{\prime}$
to: (i) $e_{1}+f_{1}$, (ii) $e_{1}+f_{1}+e_{i}$ or $e_{1}+f_{1}+f_{i}$ for any
$i\geq 1$, (iii) $e_{1}+(1-\mathrm{Arf}(q))(e_{2}+f_{2})$ or
$f_{1}+(1-\mathrm{Arf}(q))(e_{2}+f_{2})$. Thus $I$ contains $k\cdot e_{i}$ and
$k\cdot f_{i}$ for all $i$, so $k\cdot H^{3}(X_{d};\mathbb{Z})=I$. ∎

######  Proposition 4.3.

If $G\leq\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$ is a subgroup which
satisfies $\rho(G)=\rho(\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu))$, then

$H_{0}(G;\pi_{3}(X_{d}))\cong\mathbb{Z}/(k,d)\\{\eta\\},$

for $k$ as in Lemma 4.2. Thus $G=\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$ if
and only if the composition

$\mathbb{Z}/d\\{\eta\\}\longrightarrow\pi_{3}(X_{d})\longrightarrow
H_{0}(G;\pi_{3}(X_{d}))$

is zero.

###### Proof.

Let us abbreviate $A\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$, and
$H_{3}(X_{d})=H_{3}(X_{d};\mathbb{Z})$. Then

$\displaystyle H_{0}(A;\pi_{3}(X_{d}))$
$\displaystyle=H_{0}(\mathrm{Im}(\rho);H_{0}(\mathrm{Ker}(\rho);\pi_{3}(X_{d})))$
$\displaystyle H_{0}(G;\pi_{3}(X_{d}))$
$\displaystyle=H_{0}(\mathrm{Im}(\rho);H_{0}(\mathrm{Ker}(\rho|_{G});\pi_{3}(X_{d}))).$

and so first consider the long exact sequences for the action of
$\mathrm{Ker}(\rho)$ and $\mathrm{Ker}(\rho|_{G})$ on the extension describing
$\pi_{3}(X_{d})$:

${H_{1}(\mathrm{Ker}(\rho|_{G});H_{3}(X_{d}))}$${\mathbb{Z}/d}$${H_{0}(\mathrm{Ker}(\rho|_{G});\pi_{3}(X_{d}))}$${H_{0}(\mathrm{Ker}(\rho|_{G});H_{3}(X_{d}))}$${H_{1}(\mathrm{Ker}(\rho);H_{3}(X_{d}))}$${\mathbb{Z}/d}$${H_{0}(\mathrm{Ker}(\rho);\pi_{3}(X_{d}))}$${H_{0}(\mathrm{Ker}(\rho);H_{3}(X_{d})).}$$\scriptstyle{\partial}$$\scriptstyle{\partial}$

Under the identification $H_{1}(\mathrm{Ker}(\rho);H_{3}(X_{d}))\cong
H^{3}(X_{d};2\cdot\mathbb{Z}/d)\otimes H_{3}(X_{d})$ the lower map $\partial$
is given by evaluation, so it has image $2\cdot\mathbb{Z}/d$. Similarly the
upper map $\partial$ has image $2k\cdot\mathbb{Z}/d$. This gives a map of
short exact sequences, and the induced map on long exact sequence for
$\mathrm{Im}(\rho)$-homology takes the form:

${H_{1}(\mathrm{Im}(\rho);H_{3}(X_{d}))}$${\mathbb{Z}/(2k,d)}$${H_{0}(G;\pi_{3}(X_{d}))}$${H_{0}(\mathrm{Im}(\rho);H_{3}(X_{d}))}$${0}$${H_{1}(\mathrm{Im}(\rho);H_{3}(X_{d}))}$${\mathbb{Z}/(2,d)}$${H_{0}(A;\pi_{3}(X_{d}))}$${H_{0}(\mathrm{Im}(\rho);H_{3}(X_{d}))}$${0}$$\scriptstyle{\partial}$$\scriptstyle{\partial}$

By [Kra20, Lemma A.2] the rightmost terms vanish, as $\mathrm{Im}(\rho)$ is a
(quadratic) symplectic group.

If $d$ is odd then $\mathbb{Z}/(2,d)$ vanishes, and if $d$ is even then it is
$\mathbb{Z}/2$. When $d$ is even we claim that the lower $\partial$ is onto.
If not, we obtain an $A$-invariant homomorphism $Q\mathrel{\mathop{\mathchar
58\relax}}\pi_{3}(X_{d})\to\mathbb{Z}/2$ which agrees with $\mu$ on
$\mathbb{Z}/d$. Then $\mu-Q$ is again a quadratic refinement of $\lambda$ on
$\pi_{3}(X_{d})$, but it vanishes on $\mathbb{Z}/d\leq\pi_{3}(X_{d})$ and so
descends to an $A$-invariant quadratic refinement of $\lambda$ on
$H_{3}(X_{d};\mathbb{Z})$. By Beauville’s theorem there is no such quadratic
refinement. For $d$ both even and odd it then follows that
$H_{0}(A;\pi_{3}(X_{d}))=0$.

The group $H_{1}(\mathrm{Im}(\rho);H_{3}(X_{d}))$ has exponent 2 by the
centre-kills trick222The group $\mathrm{Im}(\rho)$ contains the matrix
$-\mathrm{Id}$ in its centre, which acts on the module $H_{3}(X_{d})$ by $-1$;
conjugation by this matrix induces a map on homology which is both (i) the
identity, as it is inner, and (ii) multiplication by $-1$., so the image of
the top $\partial$ lands in the subgroup of $\mathbb{Z}/(2k,d)$ of elements of
order 2, i.e. the subgroup generated by $[k]$. If $d$ is odd this means it
vanishes. If $d$ is even it must hit $[k]$, by commutativity of the left-hand
square.333This shows that $\mathbb{Z}/(2k,d)\to\mathbb{Z}/(2,d)$ is split,
i.e. that $\mathbb{Z}/(k,d/2)$ has odd order. This is a bit surprising, but is
just saying that the subgroups $\mathrm{Ker}(\rho|_{G})\leq
H^{3}(X_{d};2\cdot\mathbb{Z}/d)$ are even more constrained than Lemma 4.2
indicates. It follows that $H_{0}(G;\pi_{3}(X_{d}))\cong\mathbb{Z}/(k,d)$. ∎

######  Theorem 4.4.

The composition

$\mathrm{Mon}_{d}\longrightarrow\mathrm{MCG}_{d}\longrightarrow\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$

is surjective.

###### Proof.

By Beauville’s theorem, the image $G$ of this composition satisfies
$\rho(G)=\rho(\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu))$, and so by
Proposition 4.3 the composition is surjective if and only if the composition
$\mathbb{Z}/d\\{\eta\\}\to\pi_{3}(X_{d})\to
H_{0}(\mathrm{Mon}_{d};\pi_{3}(X_{d}))$ is zero.

Let us write $X_{d}=X_{d}^{3}\subset\mathbb{C}\mathbb{P}^{4}$: the
decomposition
$\mathbb{C}\mathbb{P}^{3}\subset\mathbb{C}\mathbb{P}^{4}\supset\mathbb{A}^{4}$
induces a corresponding decomposition $X_{d}^{2}\subset X_{d}^{3}\supset
A_{d}^{3}$, where $A_{d}^{3}$ is the corresponding affine Fermat hypersurface,
and $X_{d}^{2}\subset\mathbb{C}\mathbb{P}^{3}$ is the 2-dimensional Fermat
hypersurface. The normal bundle of $X_{d}^{2}\subset X_{d}^{3}$ is
$\mathcal{O}(1)$. The long exact sequence on homology for
$(X_{d}^{3},A_{d}^{3})$, along with excision and the Thom isomorphism, gives
the bottom part of the following commutative diagram:

${\mathbb{Z}/d\\{\eta\\}}$${\pi_{3}(A_{d}^{3})}$${\pi_{3}(X_{d}^{3})}$${H_{4}(X_{d}^{3};\mathbb{Z})}$${H_{4}(X_{d}^{3},A_{d}^{3};\mathbb{Z})}$${H_{3}(A_{d}^{3};\mathbb{Z})}$${H_{3}(X_{d}^{3};\mathbb{Z})}$${0}$${\mathbb{Z}}$${H_{2}(X_{d}^{2};\mathbb{Z})}$${\mathbb{Z}}$$\scriptstyle{\simeq}$$\scriptstyle{\partial}$$\scriptstyle{\simeq}$$\scriptstyle{d\cdot-}$$\scriptstyle{h}$

The top part comes from the fact that $A_{d}^{3}$ is homotopy equivalent to a
wedge of $3$-spheres, and teh defining extension for $\pi_{3}(X_{d})$.

Claim: The map $\pi_{3}(A_{d}^{3})\to\pi_{3}(X_{d}^{3})$ is surjective.

###### Proof of Claim.

We just have to show that it hits $\eta$. If $f\mathrel{\mathop{\mathchar
58\relax}}S^{3}\to X_{d}^{3}$ is a homotopy class which is homologically
trivial, then it is some multiple of $\eta$. This multiple may be found as
follows: choose a 4-chain bounding $f$ in $X_{d}^{3}$, and choose a 4-disc
bounding $f$ in $\mathbb{C}\mathbb{P}^{4}$, then intersect the union of these
with $\mathbb{C}\mathbb{P}^{2}\subset\mathbb{C}\mathbb{P}^{4}$ (equivalently,
evaluate $h^{2}$ on this cycle). The result is well-defined only modulo $d$
because $h^{2}$ evaluated on a 4-cycle in $X_{d}^{3}$ lies in $d\mathbb{Z}$.

By the Lefschetz hyperplane theorem the inclusion
$X_{d}^{2}\to\mathbb{C}\mathbb{P}^{3}$ is 2-connected, and in particular there
is an $[a]\in H_{2}(X_{d}^{2};\mathbb{Z})$ such that $\langle h,[a]\rangle=1$.
Under the Thom isomorphism in the diagram above this corresponds to an
$[\bar{a}]\in H_{4}(X_{d}^{3},A_{d}^{3};\mathbb{Z})$, and
$\partial([\bar{a}])$ can be represented by a sphere in $A_{d}^{3}$ and hence
a $[f]\in\pi_{3}(X_{d}^{3})$ which is homologically trivial. By construction
$\bar{a}$ is a 4-chain in $X_{d}^{3}$ which bounds $f$, and we can choose the
disc filling $f$ to be in $\mathbb{A}^{4}$. Thus the intersection of this
cycle with $\mathbb{C}\mathbb{P}^{2}$ is the same as the intersection of
$\bar{a}\cap\mathbb{C}\mathbb{P}^{3}=a$ with
$\mathbb{C}\mathbb{P}^{1}\subset\mathbb{C}\mathbb{P}^{3}$, which is $\langle
h,[a]\rangle=1$ by design. ∎

We now use the work of Pham [Pha65] describing $\pi_{3}(A_{d}^{3})\cong
H_{3}(A_{d}^{3};\mathbb{Z})$, and its extension by Looijenga [Loo10]
describing $H_{3}(X_{d}^{3};\mathbb{Z})$. These are described in terms of the
action of the abelian group $\mu_{d}^{4}=\langle
t_{1},t_{2},t_{3},t_{4}\,|\,t_{1}^{d},t_{2}^{d},t_{3}^{d},t_{4}^{d}\rangle$ of
4-tuples of $d$-th roots of unity, which acts on $\mathbb{C}\mathbb{P}^{4}$ by

$t_{1}^{a_{1}}\cdots t_{4}^{a_{4}}\cdot[z_{0}\mathrel{\mathop{\mathchar
58\relax}}z_{1}\mathrel{\mathop{\mathchar
58\relax}}\cdots\mathrel{\mathop{\mathchar
58\relax}}z_{4}]=[z_{0}\mathrel{\mathop{\mathchar
58\relax}}t_{1}^{a_{1}}z_{1}\mathrel{\mathop{\mathchar
58\relax}}t_{2}^{a_{2}}z_{2}\mathrel{\mathop{\mathchar
58\relax}}t_{3}^{a_{3}}z_{3}\mathrel{\mathop{\mathchar
58\relax}}t_{4}^{a_{4}}z_{4}]$

and hence acts on $X_{d}^{3}$, and by restriction acts on $\mathbb{A}^{4}$ and
hence $A_{n}^{d}$ in the evident way. Pham showed that there is a
$\mathbb{Z}[\mu_{d}^{4}]$-module isomorphism

$H_{3}(A^{3}_{d};\mathbb{Z})\cong\mathbb{Z}[\mu_{d}^{4}]/(\sum_{k=0}^{d-1}t_{i}^{k},i=1,2,3,4),$

and Looijenga showed that the surjection $H_{3}(A_{d}^{3};\mathbb{Z})\to
H_{3}(X_{d}^{3};\mathbb{Z})$ is the quotient by the
sub-$\mathbb{Z}[\mu_{d}^{4}]$-module $I$ generated by the element
$\sum_{k=0}^{d-1}(t_{1}t_{2}t_{3}t_{4})^{k}$.

With the discussion above we get a map of extensions of
$\mathbb{Z}[\mu_{d}^{4}]$-modules:

${0}$${I}$${\mathbb{Z}[\mu_{d}^{4}]/(\sum_{k=0}^{d-1}t_{i}^{k})}$${\mathbb{Z}[\mu_{d}^{4}]/(\sum_{k=0}^{d-1}(t_{1}t_{2}t_{3}t_{4})^{k},\sum_{k=0}^{d-1}t_{i}^{k})}$${0}$${0}$${\mathbb{Z}/d\\{\eta\\}}$${\pi_{3}(X_{d}^{3})}$${H_{3}(X_{d};\mathbb{Z})}$${0.}$

The induced map of long exact sequences on $H_{*}(\mu_{d}^{4};-)$ contains the
portion

${\cdots}$${H_{0}(\mu_{d}^{4};I)}$${\mathbb{Z}/d}$${\mathbb{Z}/d}$${0}$${\cdots}$${\mathbb{Z}/d\\{\eta\\}}$${H_{0}(\mu_{d}^{4};\pi_{3}(X_{d}^{3}))}$${H_{0}(\mu_{d}^{4};H_{3}(X_{d}^{3};\mathbb{Z}))}$${0.}$

The top right map is surjective and therefore bijective as the two groups have
the same size: thus the top left map is zero: but as the vertical left map is
onto it then follows that the bottom left map is zero: i.e.
$\mathbb{Z}/d\\{\eta\\}\to\pi_{3}(X_{d}^{3})\to
H_{0}(\mu_{d}^{4};\pi_{3}(X_{d}^{3}))$ is zero.

The argument will be finished once we show that the automorphisms of
$\pi_{3}(X_{d}^{3})$ given by the $t_{i}\in\mu_{d}^{4}$ are realised by
monodromy, as then the class $\eta$ vanishes in
$H_{0}(\mathrm{Mon}_{d};\pi_{3}(X_{d}^{3}))$ too, as required. To see this,
recall that $PGL_{5}(\mathbb{C})$ acts on the space
$\mathcal{U}_{d}\subset\mathbb{P}H^{0}(\mathbb{C}\mathbb{P}^{4};\mathcal{O}(d))$
of smooth hypersurfaces, and $\mu_{d}^{4}\leq PGL_{5}(\mathbb{C})$ stabilises
$X_{d}$: thus acting on $X_{d}$ gives a based map
$PGL_{5}(\mathbb{C})/\mu_{d}^{4}\to\mathcal{U}_{d}$ and hence a homomorphism
$\pi_{1}(PGL_{5}(\mathbb{C})/\mu_{d}^{4})\to\mathrm{Mon}_{d}$. The domain fits
into an extension

$1\longrightarrow\pi_{1}(PGL_{5}(\mathbb{C}))=\mathbb{Z}/5\longrightarrow\pi_{1}(PGL_{5}(\mathbb{C})/\mu_{d}^{4})\longrightarrow\mu_{d}^{4}\longrightarrow
1$

and any lift of $t_{i}\in\mu_{d}^{4}$ to
$\pi_{1}(PGL_{5}(\mathbb{C})/\mu_{d}^{4})$ gives an element of the monodromy
group which acts on $\pi_{3}(X_{d})$ as $t_{i}$. (It might not have order $d$
in $\mathrm{Mon}_{d}$, but this is not important.) ∎

##  5\. Further upper bounds: cobordism considerations

The discussion so far gives the extension

$1\longrightarrow\mathrm{K}_{d}\longrightarrow\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})\longrightarrow\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)\longrightarrow
1$

from (3.4), and shows that the image of $\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Mon}_{d}\to\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})$
surjects to $\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$ (Theorem 4.4) and
contains the image of $\Phi\mathrel{\mathop{\mathchar
58\relax}}\Theta_{7}\to\mathrm{K}_{d}$ (Theorem 4.1). To prove Theorem A we
must therefore account for the cokernel of $\Phi$. The goal of this section
will be to make sense of, and prove, the statement that
$\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Mon}_{d}\to\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})$
does not hit the $\mathrm{Coker}(\Phi)$ part of $\mathrm{K}_{d}$. We formulate
this as follows.

######  Theorem 5.1.

There is a factorisation

$\mathrm{K}_{d}\longrightarrow\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})\overset{\kappa}{\longrightarrow}\mathrm{Coker}(\Phi)$

of the natural quotient map, such that $\kappa\circ\alpha$ is trivial.

This shows that $\alpha$ lands in
$\mathrm{Ker}(\kappa\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})\to\mathrm{Coker}(\Phi))$,
which with our earlier discussion shows that this kernel is precisely
$\mathrm{Im}(\alpha)$, giving an extension

$1\longrightarrow\Phi(\Theta_{7})\longrightarrow\mathrm{Im}(\alpha)\longrightarrow\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)\longrightarrow
1,$

which proves Theorem A.

The cokernel of $\Phi$ is described in Lemma 2.2 as

$\mathrm{Coker}(\Phi)\cong\begin{cases}\mathbb{Z}/2&d\equiv 0\mod 4\\\
0&d\not\equiv 0\mod 4\end{cases}\oplus\begin{cases}\mathbb{Z}/3&d\equiv 0\mod
3\\\ 0&d\not\equiv 0\mod 3,\end{cases}$

so our strategy for proving Theorem 5.1 will be to analyse
$H_{1}(\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}});\mathbb{Z})$,
2- and 3-locally. The method we will use is analogous to that of [GRW16],
though with significant additional difficulties.

###  5.1. Calculating the abelianisation of
$\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})$

Using the space $\Theta^{\mathrm{hyp}}(X_{d})$ of
$\theta^{\mathrm{hyp}}$-structures on $X_{d}$, we define

$\mathrm{MCG}_{d}^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{1}(\Theta^{\mathrm{hyp}}(X_{d})/\\!\\!/\mathrm{Diff}(X_{d}),\ell^{\mathrm{hyp}}_{X_{d}}).$

The long exact sequence on homotopy groups for this homotopy orbit space
contains a portion

$\pi_{1}(\mathrm{Diff}(X_{d}))\longrightarrow\pi_{1}(\Theta^{\mathrm{hyp}}(X_{d}),\ell^{\mathrm{hyp}}_{X_{d}})\longrightarrow\mathrm{MCG}_{d}^{\mathrm{hyp}}\longrightarrow\mathrm{MCG}_{d}\overset{\curvearrowright}{\longrightarrow}\theta^{\mathrm{hyp}}(X_{d})$

so gives a half-exact sequence

$\pi_{1}(\Theta^{\mathrm{hyp}}(X_{d}),\ell^{\mathrm{hyp}}_{X_{d}})\longrightarrow\mathrm{MCG}_{d}^{\mathrm{hyp}}\longrightarrow\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})\longrightarrow
1.$

####  5.1.1. Calculating the abelianisation of
$\mathrm{MCG}_{d}^{\mathrm{hyp}}$

The abelianisation of $\mathrm{MCG}_{d}^{\mathrm{hyp}}$ can be computed by
cobordism-theoretic methods, using the results of [GRW14, GRW18, GRW17]
(proceeding similarly to [GRW16]). The result is given in terms of the
corresponding unstable tangential structure defined by the homotopy cartesian
square

(5.1)
${B_{6}}$${\mathbb{C}\mathbb{P}^{\infty}}$${B\mathrm{O}(6)}$${B\mathrm{O},}$$\scriptstyle{\theta^{\mathrm{hyp}}}$$\scriptstyle{\bar{\theta}^{\mathrm{hyp}}}$

specifically the Thom spectrum
$MT\theta^{\mathrm{hyp}}(6)\mathrel{\mathop{\mathchar
58\relax}}=Th(-(\theta^{\mathrm{hyp}})^{*}\gamma_{6})$ of the
$(-6)$-dimensional virtual vector bundle
$-(\theta^{\mathrm{hyp}})^{*}\gamma_{6}$ over $B_{6}$.

######  Proposition 5.2.

There is a map

$H_{1}(\mathrm{MCG}_{d}^{\mathrm{hyp}};\mathbb{Z})\longrightarrow\pi_{1}^{s}(MT\theta^{\mathrm{hyp}}(6))$

which is an isomorphism for $d\geq 3$.

###### Proof.

The parameterised Pontrjagin–Thom construction gives a map

(5.2)
$\Theta^{\mathrm{hyp}}(X_{d})/\\!\\!/\mathrm{Diff}(X_{d})\longrightarrow\Omega^{\infty}MT\theta^{\mathrm{hyp}}(6).$

The lift $\ell^{\mathrm{hyp}}_{X_{d}}\mathrel{\mathop{\mathchar
58\relax}}X_{d}\to B_{6}$ of $\tau_{X_{d}}\mathrel{\mathop{\mathchar
58\relax}}X_{d}\to B\mathrm{O}(6)$ is 3-connected, because
$X_{d}\to\mathbb{CP}^{4}\to\mathbb{CP}^{\infty}$ is by the Lefschetz
hyperplane theorem, and $B_{6}\to\mathbb{CP}^{\infty}$ is 6-connected by the
homotopy cartesian square (5.1). Thus [GRW17, Theorem 1.8] applies to the path
component of $\ell^{\mathrm{hyp}}_{X_{d}}$ in
$\Theta^{\mathrm{hyp}}(X_{d})/\\!\\!/\mathrm{Diff}(X_{d})$ and shows that the
map (5.2) restricted to this path-component is an isomorphism on homology in
degrees satisfying
$2*\leq\bar{g}^{\mathrm{hyp}}(X_{d},\ell^{\mathrm{hyp}}_{X_{d}})-3$. To obtain
an isomorphism in degree 1, we must argue that
$\bar{g}^{\mathrm{hyp}}(X_{d},\ell^{\mathrm{hyp}}_{X_{d}})\geq 5$.

The quantity $\bar{g}^{\mathrm{hyp}}(X_{d},\ell^{\mathrm{hyp}}_{X_{d}})$ is
described in [GRW17, Section 1.3], and is bounded below by
${g}^{\mathrm{hyp}}(X_{d},\ell^{\mathrm{hyp}}_{X_{d}})$, the largest number of
disjointly embedded copies of $W_{1,1}\mathrel{\mathop{\mathchar
58\relax}}=S^{3}\times S^{3}\setminus\mathrm{int}(D^{6})$ into $X_{d}$ on
which the restriction of $\ell^{\mathrm{hyp}}_{X_{d}}$ is “admissible”
([GRW18, Definition 1.3]). We shall not need to go into the definition of
admissible, because we can apply [GRW18, Remark 7.16]: the intersection form
$(H_{3}(X_{d};\mathbb{Z}),\lambda)$ is isomorphic to a sum of
$g\mathrel{\mathop{\mathchar 58\relax}}=\tfrac{d^{4}-5d^{3}+10d^{2}-10d+4}{2}$
hyperbolic forms, and $H_{3}(B_{6};\mathbb{Z})=0$, so by that remark
${g}^{\mathrm{hyp}}(X_{d},\ell^{\mathrm{hyp}}_{X_{d}})\geq g$. For $d\geq 3$
we have $g\geq 5$, and so indeed have
$\bar{g}^{\mathrm{hyp}}(X_{d},\ell^{\mathrm{hyp}}_{X_{d}})\geq 5$. ∎

Proceeding in parallel to [GRW16, Section 5], the 6-connected map
$B_{6}\to\mathbb{CP}^{\infty}$ gives a 0-connected map

$MT\theta^{\mathrm{hyp}}(6)\longrightarrow\Sigma^{-6}MT\bar{\theta}^{\mathrm{hyp}}$

on Thomifying, where $MT\bar{\theta}^{\mathrm{hyp}}$ is the Thom spectrum of
_minus_ the 0-dimensional virtual vector bundle classified by
$\bar{\theta}^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{CP}^{\infty}\to B\mathrm{O}$. This produces a long exact
sequence

$\pi_{8}^{s}(MT\bar{\theta}^{\mathrm{hyp}})\overset{\partial}{\longrightarrow}\mathbb{Z}/4=\pi_{7}^{s}(\mathrm{SO}/\mathrm{SO}(6))\longrightarrow\pi_{1}^{s}(MT\theta^{\mathrm{hyp}}(6))\longrightarrow\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})\longrightarrow
0.$

(The 0 at the right-hand end is because
$\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$ is easily seen to be torsion and
the next term is really
$\pi_{6}^{s}(\mathrm{SO}/\mathrm{SO}(6))\cong\mathbb{Z}$.)

####  5.1.2. Calculating
$\pi_{1}(\Theta^{\mathrm{hyp}}(X_{d}),\ell^{\mathrm{hyp}}_{X_{d}})$

The group $\pi_{1}(\Theta^{\mathrm{hyp}}(X_{d}),\ell^{\mathrm{hyp}}_{X_{d}})$
can be approached by the same method as in Section 3.2, which gives an exact
sequence

$\mathbb{Z}=H^{0}(X_{d};\mathbb{Z})\overset{(\theta^{\mathrm{hyp}})_{*}}{\longrightarrow}KO^{-2}(X_{d})\longrightarrow\pi_{1}(\Theta^{\mathrm{hyp}}(X_{d}),\ell^{\mathrm{hyp}}_{X_{d}})\longrightarrow
H^{1}(X_{d};\mathbb{Z})=0.$

Thus there is an exact sequence

$KO^{-2}(X_{d})\longrightarrow
H_{1}(\mathrm{MCG}_{d}^{\mathrm{hyp}};\mathbb{Z})\longrightarrow
H_{1}(\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}});\mathbb{Z})\longrightarrow
0.$

Let us describe the composition

$KO^{-2}(X_{d})\longrightarrow
H_{1}(\mathrm{MCG}_{d}^{\mathrm{hyp}};\mathbb{Z})\overset{\sim}{\longrightarrow}\pi_{1}^{s}(MT\theta^{\mathrm{hyp}}(6)).$

In geometric terms it is given as follows. There is a map of vector bundles
$\ell^{\prime}\mathrel{\mathop{\mathchar 58\relax}}T(S^{1}\times
X_{d})\cong\underline{\mathbb{R}}\oplus\pi_{2}^{*}TX_{d}\to\underline{\mathbb{R}}\oplus(\theta^{\mathrm{hyp}})^{*}(\gamma_{6})$
induced by the $\theta^{\mathrm{hyp}}$-structure
$\ell\mathrel{\mathop{\mathchar
58\relax}}TX_{d}\to(\theta^{\mathrm{hyp}})^{*}(\gamma_{6})$ on $X_{d}$, which
via the Pontrjagin–Thom correspondence defines a class $[S^{1}\times
X_{d},\ell^{\prime}]\in\pi_{1}^{s}(MT\theta^{\mathrm{hyp}}(6))$. This is in
fact $\eta\cdot[X_{d},\ell]$. The map above is then given by

$KO^{-2}(X_{d})\overset{s\boxtimes-}{\longrightarrow}KO^{-1}(S^{1}\times
X_{d})\longrightarrow\pi_{1}^{s}(MT\theta^{\mathrm{hyp}}(6))$

where $s\in KO^{1}(S^{1})$ is the suspension class, and the second map sends a
$f\mathrel{\mathop{\mathchar 58\relax}}S^{1}\times X_{d}\to\mathrm{O}$,
considered as a stable isomorphism of the vector bundle $T(S^{1}\times
X_{d})$, to the class $[S^{1}\times X_{d},\ell^{\prime}\circ f]$. In homotopy-
theoretic terms it is given as follows. It sends the class
$[g\mathrel{\mathop{\mathchar
58\relax}}S^{1}\wedge(X_{d})_{+}\to\mathrm{O}]\in KO^{-2}(X_{d})$ to the class

${S^{1}\wedge S^{0}}$${MT\theta^{\mathrm{hyp}}(6)}$${S^{1}\wedge
Th(\nu_{X_{d}})}$${S^{1}\wedge(X_{d})_{+}\wedge
Th(\nu_{X_{d}})}$${\mathrm{O}_{+}\wedge
Th(\nu_{X_{d}})}$${Th(\nu_{X_{d}}).}$$\scriptstyle{S^{1}\wedge[X_{d}]}$$\scriptstyle{\begin{subarray}{c}Thom\\\
diag.\end{subarray}}$$\scriptstyle{g_{*}}$$\scriptstyle{g\wedge
Id}$act$\scriptstyle{Th(\ell)}$

Here $g_{*}$ is defined to be the lower composition,
$[X_{d}]\mathrel{\mathop{\mathchar 58\relax}}S^{0}\to Th(\nu_{X_{d}})$ is the
Thom collapse map for $X_{d}$, and we have used the action of $\mathrm{O}$ on
all spectra, via $J\mathrel{\mathop{\mathchar 58\relax}}\mathrm{O}\to
GL_{1}(S^{0})$.

###  5.2. Cobordism calculation: 3-torsion

We first treat the simpler case of the 3-torsion in
$\operatorname*{Coker}(\Phi)$, which occurs only when $d\equiv 0\mod 3$.

######  Proposition 5.3.

Suppose $d\equiv 0\mod 3$, $d\geq 3$. Then

  1. (i)

there is a surjection $p_{3}\mathrel{\mathop{\mathchar
58\relax}}\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(3)}\to\mathbb{Z}/3$,
such that

  2. (ii)

the composition
$KO^{-2}(X_{d})_{(3)}\to\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(3)}\overset{p_{3}}{\to}\mathbb{Z}/3$
is trivial.

We calculate $\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(3)}$ using the
Adams spectral sequence444The $E_{2}$-pages of the Adams spectral sequences in
this and the following section were computed using the
$\mathrm{Ext}$-calculator at https://spectralsequences.github.io/sseq/, and
typeset using the spectralsequences LaTeX package. Both are excellent. at
$p=3$. There is an isomorphism
$H^{*}(MT\bar{\theta}^{\mathrm{hyp}};\mathbb{F}_{3})=u\cdot
H^{*}(\mathbb{C}\mathbb{P}^{\infty};\mathbb{F}_{3})$, and the action of the
Steenrod algebra is twisted. This is well-known at $p=2$, but less well known
at odd primes so we briefly explain it. Write
$\mathcal{P}=Id+\mathcal{P}^{1}+\mathcal{P}^{2}+\cdots$. If $u$ is the Thom
class of a complex line bundle with Euler class $x$ then we have
$\mathcal{P}(u)=u+u^{p}=u\cdot(1+x^{p-1})$ by the axioms of Steenrod
operations and the definition of the Euler class. If $u$ is the Thom class of
a sum of line bundles with Euler classes $x_{1},x_{2},\ldots,x_{r}$ then this
gives

$\mathcal{P}(u)=u\cdot\prod_{i=1}^{r}(1+x_{i}^{p-1})=u\cdot\sum_{j=0}^{r}e_{j}(x_{1}^{p-1},\ldots,x_{r}^{p-1})$

where $e_{j}$ are the elementary symmetric polynomials. The expression
$e_{j}(x_{1}^{p-1},\ldots,x_{r}^{p-1})$ is again a symmetric polynomial, known
as the $(p-1)$th Frobenius of $e_{j}$, so may be expressed in terms of the
elementary symmetric polynomials $e_{k}(x_{1},\ldots,x_{r})$, i.e. the Chern
classes of the original sum of line bundles. By the splitting principle this
expression then holds for any complex vector bundle, and hence for any virtual
bundle too.

In the case at hand we have
$c(-\bar{\theta}^{\mathrm{hyp}})=1+(d+1)x+dx^{2}+x^{3}+(1+d)x^{4}+O(x^{5})\mod
3$, and so

$\displaystyle\mathcal{P}(u)$
$\displaystyle=u\cdot(1+(c_{1}^{2}-2c_{2})+(c_{2}^{2}-2c_{3}c_{1}+2c_{4})+\cdots)$
$\displaystyle=u\cdot(1+(d^{2}+1)x^{2}+d^{2}x^{4}+O(x^{5})).$

In particular when $d\equiv 0\mod 3$ we have
$\mathcal{P}(u)=u\cdot(1+x^{2}+O(x^{5}))$. Then
$H^{*}(MT\bar{\theta}^{\mathrm{hyp}};\mathbb{F}_{3})=u\cdot
H^{*}(\mathbb{C}\mathbb{P}^{\infty};\mathbb{F}_{3})$ has generating Steenrod
operations

$\mathcal{P}^{1}(u)=u\cdot x^{2},\quad\mathcal{P}^{1}(u\cdot x)=2u\cdot x^{3}$

in degrees $\leq 9$. Using this we can calculate the $E^{2}$-page of the Adams
spectral sequence in a range, which is shown in Figure 1.

[ name = p3d0mod3, Adams grading, classes=fill, x range = 48, y range = 04]

(4,1) (4,2) (4,1)(4,2) (4,3) (4,2)(4,3) (4,4) (4,3)(4,4) (4,5) (4,4)(4,5)

(6,1) (6,2) (6,1)(6,2) (6,3) (6,2)(6,3) (6,4) (6,3)(6,4) (6,5) (6,4)(6,5)

(7,1) (7,2) (7,1)(7,2) (4,1)(7,2)

(8,0) (8,1) (8,0)(8,1) (8,2) (8,1)(8,2) (8,3) (8,2)(8,3) (8,4) (8,3)(8,4)
(8,5) (8,4)(8,5)

[̣dashed]2(8,0)(7,2) [ name = p3d0mod3, page = 2 ]

Figure 1. $E_{2}$-page of the Adams spectral sequence at $p=3$ for
$MT\bar{\theta}^{\mathrm{hyp}}$, with $d\equiv 0\mod 3$.

We see that $\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(3)}$ is either
$\mathbb{Z}/9$ or $\mathbb{Z}/3$, depending on the differential coming out of
the 8-column, but no smaller. In either case, taking the quotient by those
classes of $\mathbb{F}_{3}$-Adams filtration $\geq 2$ gives a surjection

$p_{3}\mathrel{\mathop{\mathchar
58\relax}}\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(3)}\longrightarrow\mathbb{Z}/3.$

We now wish to show that the composition
$KO^{-2}(X_{d})_{(3)}\to\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(3)}\overset{p_{3}}{\to}\mathbb{Z}/3$
is trivial. To do so, we make use of the following types of classes in
$KO^{-2}(X_{d})$.

######  Lemma 5.4.

The abelian group $KO^{-2}(X_{d})$ is generated by the following classes:

  1. (i)

The pullback along the map $c\mathrel{\mathop{\mathchar 58\relax}}X_{d}\to
S^{6}$ that collapses the complement $X^{\prime}_{d}\subset X_{d}$ of a ball
of the Bott class $\Sigma^{-2}\beta\in KO^{-2}(S^{6})$,

  2. (ii)

classes pulled back along
$\ell^{\mathrm{hyp}}_{X_{d}}\mathrel{\mathop{\mathchar
58\relax}}X_{d}\to\mathbb{CP}^{\infty}$.

###### Proof.

Clearly the image of $KO^{-2}(*)\to KO^{-2}(X_{d})$ is of type (ii). The
Atiyah–Hirzebruch spectral sequence gives an extension

$0\longrightarrow
H^{6}(X_{d};\mathbb{Z})\longrightarrow\widetilde{KO}^{-2}(X_{d})\longrightarrow
H^{2}(X_{d};\mathbb{Z})\longrightarrow 0$

and the left-hand term corresponds to the classes of type (i). The map of
Atiyah–Hirzebruch spectral sequences for
$\ell^{\mathrm{hyp}}_{X_{d}}\mathrel{\mathop{\mathchar
58\relax}}X_{d}\to\mathbb{CP}^{\infty}$ shows that the right-hand term can be
saturated by classes of type (ii). ∎

For classes of type (i), we consider the commutative diagram

${\widetilde{KO}^{-2}(S^{6})}$${{KO}^{-2}(X_{d},X^{\prime}_{d})}$${\pi_{1}^{s}(Th(\nu_{X_{d}}),Th(\nu_{X_{d}}|_{X^{\prime}_{d}}))}$${\pi_{7}^{s}(S^{0})}$${{KO}^{-2}(X_{d})}$${\pi_{1}^{s}(Th(\nu_{X_{d}}))}$${\pi_{1}^{s}(MT{\theta}^{\mathrm{hyp}}(6))}$${\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}}).}$$\scriptstyle{\cong}$$\scriptstyle{\iota}$$\scriptstyle{Th(\ell)_{*}}$

Recall that the octonionic Hopf fibration $S^{15}\to S^{8}$ represents a class
$\sigma\in\pi_{7}^{s}(S^{0})$, and that
$\pi_{7}^{s}(S^{0})=\mathbb{Z}/240\\{\sigma\\}$. Hence the diagram shows that
$c^{*}(\Sigma^{-2}\beta)$ maps to a multiple of $\iota\circ\sigma$ in
$\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$, but
$\sigma\in\pi_{7}^{s}(S^{0})$ has $\mathbb{F}_{3}$-Adams filtration 2 so
$\iota\circ\sigma$ has $\mathbb{F}_{3}$-Adams filtration $\geq 2$, but this is
precisely what we divided out to form the quotient
$p_{3}\mathrel{\mathop{\mathchar
58\relax}}\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(3)}\to\mathbb{Z}/3$.

A class $[g]\in{KO}^{-2}(X_{d})$ of type (ii) is by definition pulled back
from some $[G]\in{KO}^{-2}(\mathbb{C}\mathbb{P}^{\infty})$. By the
construction at the end of Section 5.1.2 this induces a map

$G_{*}\mathrel{\mathop{\mathchar 58\relax}}S^{1}\wedge
MT\bar{\theta}^{\mathrm{hyp}}\longrightarrow MT\bar{\theta}^{\mathrm{hyp}}$

and by naturality the image of $[g]\in{KO}^{-2}(X_{d})$ in
$\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$ is given by

$S^{1}\wedge
S^{6}\overset{S^{1}\wedge[X_{d},\ell^{\mathrm{hyp}}_{X_{d}}]}{\longrightarrow}S^{1}\wedge
MT\bar{\theta}^{\mathrm{hyp}}\overset{G_{*}}{\longrightarrow}MT\bar{\theta}^{\mathrm{hyp}},$

for
$[X_{d},\ell^{\mathrm{hyp}}_{X_{d}}]\in\pi_{6}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$.

In the Adams chart for $MT\bar{\theta}^{\mathrm{hyp}}$ in Figure 1 we see that
$[X_{d},\ell^{\mathrm{hyp}}_{X_{d}}]\in\pi_{6}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$
has $\mathbb{F}_{3}$-Adams filtration $\geq 1$. Furthermore, the map $G_{*}$
is trivial on $\mathbb{F}_{3}$-cohomology, so has $\mathbb{F}_{3}$-Adams
filtration $\geq 1$. It follows that the image of $[g]$ in
$\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$ has $\mathbb{F}_{3}$-Adams
filtration $\geq 2$, but again this is precisely what we divided out to form
the quotient $p_{3}\mathrel{\mathop{\mathchar
58\relax}}\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(3)}\to\mathbb{Z}/3$.

###  5.3. Cobordism calculation: 2-torsion

We now treat the 2-torsion in $\operatorname*{Coker}(\Phi)$, which occurs only
when $d\equiv 0\mod 4$. It is parallel to the 3-torsion case just described,
but somewhat more complicated.

######  Proposition 5.5.

Suppose $d\equiv 0\mod 4$, $d\geq 4$. Then

  1. (i)

there is a surjection $p_{2}\mathrel{\mathop{\mathchar
58\relax}}\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(2)}\to\mathbb{Z}/2$,
such that

  2. (ii)

the composition
$KO^{-2}(X_{d})_{(2)}\to\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(2)}\overset{p_{2}}{\to}\mathbb{Z}/2$
is trivial.

To understand the 2-local homotopy groups of $MT\bar{\theta}^{\mathrm{hyp}}$,
we can identify the homotopy type of its 9-skeleton with a certain stunted
complex projective space.

######  Lemma 5.6.

If $d\equiv 0\mod 8$ then there is a 2-local equivalence

$Th(\mathcal{O}(d)-5\mathcal{O}(1)+4\to\mathbb{C}\mathbb{P}^{4})\simeq\Sigma^{-2(2^{6}-5)}\mathbb{C}\mathbb{P}_{2^{6}-5}^{2^{6}-1}.$

If $d\equiv 4\mod 8$ then there is a 2-local equivalence

$Th(\mathcal{O}(d)-5\mathcal{O}(1)+4\to\mathbb{C}\mathbb{P}^{4})\simeq\Sigma^{-2(2^{5}-5))}\mathbb{C}\mathbb{P}_{2^{5}-5}^{2^{5}-1}.$

###### Proof.

We claim that the 2-local spherical fibrations for the complex bundles
$\mathcal{O}(d)-5\mathcal{O}(1)+4$ and $(2^{6}-5)(\mathcal{O}(1)-1)$ or
$(2^{5}-5)(\mathcal{O}(1)-1)$ are equivalent: then the 2-local Thom spaces are
equivalent, and it is standard that
$Th(k\mathcal{O}(1)\to\mathbb{C}\mathbb{P}^{n})\simeq\mathbb{C}\mathbb{P}_{k}^{n+k}$.
This is a kind of James periodicity, and comes down to showing these bundles
agree under the 2-local $J$-homomorphism

$J_{(2)}\mathrel{\mathop{\mathchar
58\relax}}KO^{0}(\mathbb{C}\mathbb{P}^{4})\longrightarrow
J(\mathbb{C}\mathbb{P}^{4})_{(2)}.$

The latter groups have been calculated by Adams and Walker [AW65], and we
explain how to extract the specific information that we want.

Writing $x\mathrel{\mathop{\mathchar
58\relax}}=[\mathcal{O}(1)]-1\in\widetilde{K}^{0}(\mathbb{C}\mathbb{P}^{4})$
and $y\mathrel{\mathop{\mathchar
58\relax}}=r(x)\in\widetilde{KO}^{0}(\mathbb{C}\mathbb{P}^{4})$, by [AW65,
Section 2] we have $K^{0}(\mathbb{C}\mathbb{P}^{4})=\mathbb{Z}[x]/(x^{5})$ and
${KO}^{0}(\mathbb{C}\mathbb{P}^{4})=\mathbb{Z}[y]/(y^{3})$. The
complexification map $c\mathrel{\mathop{\mathchar
58\relax}}KO^{0}(\mathbb{C}\mathbb{P}^{4})\to K^{0}(\mathbb{C}\mathbb{P}^{4})$
is a ring homomorphism and commutes with Adams operations. It satisfies

$\displaystyle c(y)$ $\displaystyle=cr(x)=x+\psi^{-1}_{\mathbb{C}}(x)$
$\displaystyle=x+(1+x)^{-1}-1=\frac{x^{2}}{1+x}$
$\displaystyle=x^{2}-x^{3}+x^{4}$

and so $c(y^{2})=x^{4}$, and hence $c$ is injective. From this one can check
that $r(x^{2})=2y+y^{2}$, so $r\mathrel{\mathop{\mathchar
58\relax}}K^{0}(\mathbb{C}\mathbb{P}^{4})\to KO^{0}(\mathbb{C}\mathbb{P}^{4})$
is surjective. As $K^{0}(\mathbb{C}\mathbb{P}^{4})$ is spanned by sums of
complex line bundles, it follows that ${KO}^{0}(\mathbb{C}\mathbb{P}^{4})$ is
spanned by sums of $\mathrm{O}(2)$-bundles, and so an easy version of the
Adams conjecture [Ada63, Theorem 1.3] applies to show that for any odd $k$ and
any $z\in KO^{0}(\mathbb{C}\mathbb{P}^{4})$ the class
$(\psi^{k}_{\mathbb{R}}-1)z$ is in the kernel of $J_{(2)}$. Let us write
$a\sim b$ to mean that $a-b\in KO^{0}(\mathbb{CP}^{4})$ lies in the kernel of
$J_{(2)}$.

As

$\psi_{\mathbb{C}}^{k}(x)=(1+x)^{k}-1=kx+\binom{k}{2}x^{2}+\binom{k}{3}x^{3}+\binom{k}{4}x^{4}$

and $c$ is injective and commutes with Adams operations, we have

$\displaystyle c(\psi_{\mathbb{R}}^{k}(y))$
$\displaystyle=\psi^{k}_{\mathbb{C}}(x^{2}-x^{3}+x^{4})$
$\displaystyle=k^{2}x^{2}-k^{2}x^{3}+\tfrac{1}{12}k^{2}(k^{2}+11)x^{4}$
$\displaystyle=c(k^{2}y+\tfrac{1}{12}k^{2}(k^{2}-1)y^{2})$

giving $\psi_{\mathbb{R}}^{k}(y)=k^{2}y+\tfrac{1}{12}k^{2}(k^{2}-1)y^{2}$, and
hence $\psi_{\mathbb{R}}^{k}(y^{2})=k^{4}y^{2}$. In particular we have

$(\psi_{\mathbb{R}}^{3}-1)(y)=2^{3}y+2\cdot
3y^{2}\quad\quad(\psi_{\mathbb{R}}^{3}-1)(y^{2})=2^{4}\cdot 5y^{2}.$

As the target of $J_{(2)}$ is 2-local, it follows that $2^{4}y^{2}\sim 0$, and
hence that $2^{6}y\sim 0$.

By [AW65, Lemma A.2] realification commutes with Adams operations, so writing
$d=2^{s}\cdot t$ with $t$ odd and $s\geq 2$, we have

$\displaystyle r(\mathcal{O}(d)-1)$
$\displaystyle=r(\psi^{d}_{\mathbb{C}}(\mathcal{O}(1)-1))=\psi_{\mathbb{R}}^{d}(y)$
$\displaystyle=\psi_{\mathbb{R}}^{t}(\psi_{\mathbb{R}}^{2^{s}}(y))$
$\displaystyle\sim\psi_{\mathbb{R}}^{2^{s}}(y)=2^{2s}y+2^{2s-2}\tfrac{(2^{2s}-1)}{3}y^{2}$

Suppose first that $s\geq 3$. Then $2^{2s-2}y^{2}\sim 0$, and $2^{2s}y\sim 0$,
so $r(\mathcal{O}(d)-1)\sim 0\sim 2^{6}y$, and so
$r(\mathcal{O}(d)-5\mathcal{O}(1)+4)\sim r((2^{6}-5)(\mathcal{O}(1)-1))$ as
required.

If $s=2$ then $\tfrac{2^{2s}-1}{3}=5$ so the above is

$\displaystyle 2^{4}y+2^{2}\cdot 5y^{2}$
$\displaystyle=2^{5}y-2^{4}y+2^{2}\cdot 5y^{2}$ $\displaystyle\sim
2^{5}y-2(-2\cdot 3y^{2})+2^{2}\cdot 5y^{2}$
$\displaystyle=2^{5}y+2^{5}y^{2}\sim 2^{5}y$

and so $r(\mathcal{O}(d)-5\mathcal{O}(1)+4)\sim
r((2^{5}-5)(\mathcal{O}(1)-1))$ as required. ∎

######  Corollary 5.7.

If $d\equiv 0\mod 4$ then

$\displaystyle\pi_{5}(MT\bar{\theta}^{\mathrm{hyp}})_{(2)}$
$\displaystyle\cong\mathbb{Z}/4$
$\displaystyle\pi_{7}(MT\bar{\theta}^{\mathrm{hyp}})_{(2)}$
$\displaystyle\cong\begin{cases}\mathbb{Z}/2\oplus\mathbb{Z}/16&d\equiv 0\mod
8\\\ \mathbb{Z}/2\oplus\mathbb{Z}/8&d\equiv 4\mod 8.\end{cases}$
$\displaystyle\pi_{8}(MT\bar{\theta}^{\mathrm{hyp}})_{(2)}$
$\displaystyle\cong\mathbb{Z}_{(2)}\oplus\mathbb{Z}/4.$

###### Proof.

We have
$MT\bar{\theta}^{\mathrm{hyp}}=Th(\mathcal{O}(d)-5\mathcal{O}(1)+4\to\mathbb{C}\mathbb{P}^{\infty})$,
and by Lemma 5.6 its 9-skeleton is 2-locally equivalent to either
$\Sigma^{-2(2^{6}-5)}\mathbb{C}\mathbb{P}_{2^{6}-5}^{2^{6}-1}$ or
$\Sigma^{-2(2^{6}-5)}\mathbb{C}\mathbb{P}_{2^{6}-5}^{2^{6}-1}$, which in turn
are the 9-skeletons of
$\Sigma^{-2(2^{6}-5)}\mathbb{C}\mathbb{P}_{2^{6}-5}^{\infty}$ or
$\Sigma^{-2(2^{6}-5)}\mathbb{C}\mathbb{P}_{2^{6}-5}^{\infty}$.

The first claim follows from [Mat62, Theorem 1 c)], using the discussion in §9
of that paper and Toda’s identification
$\pi_{2n+i}(U(n))\cong\pi_{i}^{s}(\mathbb{C}\mathbb{P}_{n}^{\infty})$ for odd
$i$ in the metastable range. The second claim follows from [Mat63, Theorem 2].
The third claim follows from [Mos69, Table 2.2]. ∎

On the other hand, we could also approach the homotopy groups of
$MT\bar{\theta}^{\mathrm{hyp}}$ via the Adams spectral sequence. As modules
over the Steenrod algebra we have
$H^{*}(MT\bar{\theta}^{\mathrm{hyp}};\mathbb{F}_{2})\cong u\cdot
H^{*}(\mathbb{C}\mathbb{P}^{\infty};\mathbb{F}_{2})$ where
$H^{*}(\mathbb{C}\mathbb{P}^{\infty};\mathbb{F}_{2})$ carries its usual
Steenrod-module structure, and $\mathrm{Sq}(u)=u\cdot
w(-\theta^{\mathrm{hyp}})$, which as $d$ is even is

$\mathrm{Sq}(u)=u\cdot(1+x+x^{2}+x^{3}+O(x^{5})).$

In degrees $\leq 9$ we then have that
$H^{*}(MT\bar{\theta}^{\mathrm{hyp}};\mathbb{F}_{2})$ is a sum of modules over
the Steenrod algebra $\mathbb{F}_{2}\\{u,u\cdot x,u\cdot x^{2},u\cdot
x^{3}\\}\oplus\mathbb{F}_{2}\\{u\cdot x^{4}\\}$ with generating Steenrod
operations

$\mathrm{Sq}^{2}(u)=u\cdot x,\quad\mathrm{Sq}^{4}(u)=u\cdot
x^{2},\quad\mathrm{Sq}^{2}(u\cdot x^{2})=u\cdot x^{3}.$

The $E_{2}$-page of the Adams spectral sequence near the degrees in which we
are interested in then as in Figure 2.

[ name = p2deven, Adams grading, classes=fill, x range = 48, y range = 05]

(4,2) (4,3) (4,2)(4,3) (4,4) (4,3)(4,4) (4,5) (4,4)(4,5) (4,6) (4,5)(4,6)

(5,1) (5,2) (5,1)(5,2) (5,3) (5,2)(5,3) (4,2)(5,3)

(6,1) (6,2) (6,1)(6,2) (6,3) (6,2)(6,3) (6,4) (6,3)(6,4) (6,5) (6,4)(6,5)
(6,6) (6,5)(6,6)

(7,1) (7,2) (7,2) (7,1)(7,2,1) (6,1)(7,2,2) (7,3) (7,2,1)(7,3) (7,4)
(7,3)(7,4)

(8,0) (8,1) (8,0)(8,1) (8,2) (8,2) (8,1)(8,2,2) (8,3) (8,3) (8,2,1)(8,3,1)
(8,2,2)(8,3,2) (7,2,2)(8,3,1) (8,4) (8,3,2)(8,4) (8,5) (8,4)(8,5) (8,6)
(8,5)(8,6)

2̣(6,1)(5,3) [̣dashed]4(8,0)(7,4) [ name = p2deven, page = 2–4 ]

Figure 2. $E_{2}$-page of the Adams spectral sequence at $p=2$ for
$MT\bar{\theta}^{\mathrm{hyp}}$, with $d$ even. The class of filtration 1 in
degree 7 detects the image of $\sigma\in\pi_{7}^{s}(S^{0})$ on the bottom cell
$\iota\mathrel{\mathop{\mathchar 58\relax}}S^{0}\to
MT\bar{\theta}^{\mathrm{hyp}}$.

From the homotopy groups shown in Corollary 5.7, we see that there is a unique
pattern of differentials in this range: $d_{2}\mathrel{\mathop{\mathchar
58\relax}}E_{2}^{1,7}\to E_{2}^{3,8}$ is an isomorphism, and
$d_{4}\mathrel{\mathop{\mathchar 58\relax}}E_{4}^{0,8}\to E_{4}^{4,11}$ is an
isomorphism if and only if $d\equiv 4\mod 8$, with all other differentials
being zero. In particular, taking the quotient by the subgroup generated by
$\iota\circ\sigma\in\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(2)}$ gives a
surjection

$p_{2}\mathrel{\mathop{\mathchar
58\relax}}\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(2)}\longrightarrow\mathbb{Z}/2.$

We wish to show that the composition
$KO^{-2}(X_{d})_{(2)}\to\pi_{1}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(2)}\overset{p_{2}}{\to}\mathbb{Z}/2$
is trivial. We do this just as in the 3-torsion case, using Lemma 5.4. Namely,
classes of type (i) coming from $c\mathrel{\mathop{\mathchar
58\relax}}X_{d}\to S^{6}$ map to a multiple of
$\iota\circ\sigma\in\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$ so vanishes
under $p_{2}$. As in the 3-torsion case, for classes of type (ii) their images
may be represented as

$S^{1}\wedge
S^{6}\overset{S^{1}\wedge[X_{d},\ell^{\mathrm{hyp}}_{X_{d}}]}{\longrightarrow}S^{1}\wedge
MT\bar{\theta}^{\mathrm{hyp}}\overset{G_{*}}{\longrightarrow}MT\bar{\theta}^{\mathrm{hyp}}$

for certain maps $G_{*}$. Figure 2 shows the Adams chart for
$MT\bar{\theta}^{\mathrm{hyp}}$, and because of the differential out of the
6-column, the class
$[X_{d},\ell^{\mathrm{hyp}}_{X_{d}}]\in\pi_{6}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$
has $\mathbb{F}_{2}$-Adams filtration $\geq 2$. The map $G_{*}$ has
$\mathbb{F}_{2}$-Adams filtration $\geq 1$, so the classes obtained in this
way all have $\mathbb{F}_{2}$-Adams filtration $\geq 3$. Thus they lie in the
subgroup of $\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(2)}$ generated by
$4\iota\circ\sigma$, and so vanish under $p_{2}$.

###  5.4. Proof of Theorem 5.1

Firstly, abbreviate $\mathrm{Aut}\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)$ and consider the Serre
spectral sequence for (3.4), which gives

(5.3)
$H_{2}(\mathrm{Aut};\mathbb{Z})\overset{d_{2}}{\longrightarrow}\mathrm{K}_{d}\longrightarrow
H_{1}(\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}});\mathbb{Z})\longrightarrow
H_{1}(\mathrm{Aut};\mathbb{Z})\longrightarrow 0.$

The group $\mathrm{Aut}$ participates in an extension

$1\longrightarrow
H^{3}(X_{d};2\cdot\mathbb{Z}/d)\longrightarrow\mathrm{Aut}\overset{\rho}{\longrightarrow}\begin{cases}\mathrm{Aut}(H_{3}(X_{d};\mathbb{Z}),\lambda,\mu)&d\text{
odd}\\\ \mathrm{Aut}(H_{3}(X_{d};\mathbb{Z}),\lambda)&d\text{
even}\end{cases}\longrightarrow 1,$

where the (outer) action of the quotient on the kernel is the usual one.
Writing $\mathrm{Aut}^{\prime}$ for the right-hand term, the Serre spectral
sequence for this extension gives

$H_{0}(\mathrm{Aut}^{\prime};H^{3}(X_{d};2\cdot\mathbb{Z}/d))\longrightarrow
H_{1}(\mathrm{Aut};\mathbb{Z})\longrightarrow
H_{1}(\mathrm{Aut}^{\prime};\mathbb{Z})\longrightarrow 1.$

Using that $d\geq 3$ and so $g\geq 5$, the right-hand term is $\mathbb{Z}/4$
if $d$ is odd and $0$ if $d$ is even by e.g. [KRW21, Section 4.1.1].
Similarly, if $d$ is even, or $d$ is odd and $\mu$ has Arf invariant 0, then
the left-hand term vanishes by [Kra20, Lemma A.2]. If $d$ is odd and $\mu$ has
Arf invariant 1 then it still vanishes, though we do not know a specific
reference: a similar argument to [Kra20, Lemma A.2] works. The overall
conclusion is that $H_{1}(\mathrm{Aut};\mathbb{Z})$ is $\mathbb{Z}/4$ if $d$
is odd and $0$ if $d$ is even.

By the discussion in Sections 5.1, 5.2, and 5.3, for $d\geq 3$ we have the
diagram

${KO^{-2}(X_{d})}$${H_{1}(\mathrm{MCG}_{d}^{\mathrm{hyp}};\mathbb{Z})}$${H_{1}(\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}});\mathbb{Z})}$${0}$${\pi_{1}^{s}(MT{\theta}^{\mathrm{hyp}}(6))}$${\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})}$${\mathrm{Coker}(\Phi)^{\prime}}$$\scriptstyle{\cong}$$\scriptstyle{\kappa}$$\scriptstyle{p_{2}\oplus
p_{3}}$

where the top row is exact and the lower composition is trivial, so the dashed
surjection $\kappa$ exists. Here we have written
$\mathrm{Coker}(\Phi)^{\prime}$ for the abstract group

$\begin{cases}\mathbb{Z}/2&d\equiv 0\mod 4\\\ 0&d\not\equiv 0\mod
4\end{cases}\oplus\begin{cases}\mathbb{Z}/3&d\equiv 0\mod 3\\\ 0&d\not\equiv
0\mod 3.\end{cases}$

######  Lemma 5.8.

The composition

$\mathrm{Mon}_{d}\overset{\alpha}{\longrightarrow}\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})\longrightarrow
H_{1}(\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}});\mathbb{Z})\overset{\kappa}{\longrightarrow}\mathrm{Coker}(\Phi)^{\prime}$

is trivial.

###### Proof.

$\mathcal{U}_{d}\subset\mathbb{P}H^{0}(\mathbb{CP}^{4};\mathcal{O}(d))$ is the
complement of the discriminant locus, which is irreducible, so its fundamental
group $\mathrm{Mon}_{d}$ is normally generated by a single element, which can
be taken to be a symplectic Dehn twist $\tau$. As $\mathrm{Coker}(\Phi)$ is an
abelian group, it is enough to show that this Dehn twist maps to zero. The map
$\alpha$ naturally lifts to $\mathrm{MCG}_{d}^{\mathrm{hyp}}$, as the
universal family of smooth hypersurfaces admits a fibrewise
$\theta^{\mathrm{hyp}}$-structure as discussed in Section 3.2.

By choosing a degeneration of $X_{d}$ to an $A_{2}$-singularity, we may find
an embedding $T^{*}S^{3}\natural T^{*}S^{3}\subset X_{d}$ of the plumbing such
that $\tau$ is the Dehn twist around one of the spheres. There is a
diffeomorphism $T^{*}S^{3}\natural T^{*}S^{3}\cong S^{3}\times S^{3}\setminus
D^{6}=\mathrm{int}(W_{1,1})$ and so the mapping torus $T_{\tau}$ can be
expressed as

$T_{\tau}=T_{\tau|_{W_{1,1}}}\bigcup_{S^{1}\times
S^{5}}(S^{1}\times(X_{d}\setminus W_{1,1})),$

equipped with a $\theta^{\mathrm{hyp}}$-structure
$\ell^{\mathrm{hyp}}_{T_{\tau}}$ that agrees with the pullback of
$\ell^{\mathrm{hyp}}_{X_{d}}$ on the second term. Via
$D^{2}\times(X_{d}\setminus W_{1,1})$, the second term is
$\theta^{\mathrm{hyp}}$-cobordant rel boundary to $D^{2}\times S^{5}$, so
$T_{\tau}$ is $\theta^{\mathrm{hyp}}$-cobordant to the manifold

$M\mathrel{\mathop{\mathchar
58\relax}}=T_{\tau|_{W_{1,1}}}\bigcup_{S^{1}\times S^{5}}D^{2}\times S^{5}$

with some $\theta^{\mathrm{hyp}}$-structure
$\ell_{M}^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}M\to\mathbb{CP}^{\infty}$. But $M$ is easily checked to be
2-connected, so $\ell^{\mathrm{hyp}}_{M}$ may be lifted along
$E\mathrm{O}\to\mathbb{CP}^{\infty}$ to a stable framing
$\ell_{M}^{\mathrm{sfr}}$. It follows that
$[T_{\tau},\ell^{\mathrm{hyp}}_{T_{\tau}}]\in\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$
lies in the image of $\iota_{*}\mathrel{\mathop{\mathchar
58\relax}}\pi_{7}^{s}(S^{0})\to\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$.
Both 3-locally and 2-locally the image of this map lies in the subgroup that
we divided out to form the map $p_{2}\oplus p_{3}\mathrel{\mathop{\mathchar
58\relax}}\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})_{(2)}\to\mathrm{Coker}(\Phi)^{\prime}$.
∎

The argument of this lemma can be understood in a more general context by the
discussion in Section 6, see Remark 6.5.

######  Lemma 5.9.

The map $q\mathrel{\mathop{\mathchar 58\relax}}K_{d}\to
H_{1}(\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}});\mathbb{Z})\overset{\kappa}{\to}\mathrm{Coker}(\Phi)^{\prime}$
is the quotient by $\mathrm{Im}(\Phi)$.

###### Proof.

For this it suffices to show that it is 2- and 3-locally surjective, and that
it vanishes on precomposing with $\Phi\mathrel{\mathop{\mathchar
58\relax}}\Theta_{7}\to K_{d}$. Working 2-locally there is only anything to
show when $d\equiv 0\mod 4$, in which case $H_{1}(\mathrm{Aut};\mathbb{Z})=0$
and so $K_{d}\to
H_{1}(\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}});\mathbb{Z})$
is surjective, so $q$ is surjective too. Working 3-locally there is only
anything to show when $d\equiv 0\mod 3$. Then $H_{1}(\mathrm{Aut};\mathbb{Z})$
is 0 or $\mathbb{Z}/4$ so is 3-locally trivial and so $K_{d}\to
H_{1}(\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}});\mathbb{Z})$
is 3-locally surjective, and so $q$ is 3-locally surjective too.

To see that
$\Theta_{7}\overset{\Phi}{\to}K_{d}\overset{q}{\to}\mathrm{Coker}(\Phi)^{\prime}$
is trivial, we use that $\Phi(\Theta_{7})$ lies in the image of $\alpha$ by
Theorem 4.1, then apply Lemma 5.8. ∎

##  6\. Proof of Theorem B: Comparison to $W_{g,1}$

In order to prove Theorem B we will compare some of our calculations with
analogous calculations for mapping class groups of connect-sums of
$S^{3}\times S^{3}$’s, which have been studied in some detail, especially
recently. Recall from Section 2.1 that by a theorem of Wall we may find an
embedding

$e\mathrel{\mathop{\mathchar 58\relax}}W_{g,1}\mathrel{\mathop{\mathchar
58\relax}}=\\#^{g}(S^{3}\times
S^{3})\setminus\mathrm{int}(D^{6})\longrightarrow X_{d}$

which induces an isomorphism on $H_{3}(-;\mathbb{Z})$. By our assumption
$d\geq 3$ we have $g\geq 5$. Let
$\ell^{\mathrm{hyp}}_{W_{g,1}}=e^{*}\ell^{\mathrm{hyp}}_{X_{d}}$ be the
induced $\theta^{\mathrm{hyp}}$-structure, which in particular induces a
$\theta^{\mathrm{hyp}}$-structure $\ell^{\mathrm{hyp}}_{\partial W_{g,1}}$ on
the boundary $\partial W_{g,1}$. Recall that the universal principal
$\mathrm{O}$-bundle $\bar{\theta}^{\mathrm{sfr}}\mathrel{\mathop{\mathchar
58\relax}}E\mathrm{O}\to B\mathrm{O}$ defines a stable tangential structure
that we call a _stable framing_.

######  Lemma 6.1.

There is a boundary condition on stable framings
$\ell^{\mathrm{sfr}}_{\partial W_{g,1}}$ and a
$\mathrm{Diff}_{\partial}(W_{g,1})$-equivariant homotopy equivalence

$u_{*}\mathrel{\mathop{\mathchar
58\relax}}\Theta^{\mathrm{sfr}}(W_{g,1};\ell^{\mathrm{sfr}}_{\partial
W_{g,1}})\overset{\sim}{\longrightarrow}\Theta^{\mathrm{hyp}}(W_{g,1};\ell^{\mathrm{hyp}}_{\partial
W_{g,1}}).$

###### Proof.

The map $\ell^{\mathrm{hyp}}_{\partial W_{g,1}}\mathrel{\mathop{\mathchar
58\relax}}\partial W_{g,1}=S^{5}\to\mathbb{C}\mathbb{P}^{\infty}$ is
nullhomotopic, giving a lift along $u\mathrel{\mathop{\mathchar
58\relax}}E\mathrm{O}\to\mathbb{C}\mathbb{P}^{\infty}$ and hence a boundary
condition $\ell^{\mathrm{sfr}}_{\partial W_{g,1}}$ for stable framings (and a
canonical isomorphism from $u_{*}\ell^{\mathrm{sfr}}_{\partial W_{g,1}}$ to
$\ell^{\mathrm{hyp}}_{\partial W_{g,1}}$). The map $u$ then induces a map
$u_{*}$ between the two spaces of structures. Seeing that it is surjective in
$\pi_{0}$ means showing that the relative lifting problem

${\partial
W_{g,1}}$${E\mathrm{O}}$${W_{g,1}}$${\mathbb{C}\mathbb{P}^{\infty}}$$\scriptstyle{\ell^{\mathrm{sfr}}_{\partial
W_{g,1}}}$$\scriptstyle{u}$$\scriptstyle{\ell^{\mathrm{hyp}}}$

can always be solved: it can because $H^{2}(W_{g,1},\partial
W_{g,1};\mathbb{Z})=0$. Moreover it can be solved uniquely up to homotopy as
all the lower relative cohomology groups also vanish, which shows that the
fibres of $u_{*}$ are contractible too. ∎

This lemma means that $\ell^{\mathrm{hyp}}_{W_{g,1}}$ corresponds to a
canonical stable framing $\ell^{\mathrm{sfr}}_{W_{g,1}}$. In Section 3.4 we
constructed a quadratic refinement $\mu=\mu_{\ell^{\mathrm{hyp}}_{X_{d}}}$ on
$\pi_{3}(X_{d})$, which restricts along $e$ to a quadratic refinement
$e^{*}\mu$ on $\pi_{3}(W_{g,1})=H_{3}(W_{g,1};\mathbb{Z})$. In [KRW21, Section
6.1] it was shown that under the action of
$\Gamma_{g,1}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{0}\mathrm{Diff}_{\partial}(W_{g,1})$ the set
$\theta^{\mathrm{sfr}}(W_{g,1};\ell^{\mathrm{sfr}}_{\partial
W_{g,1}})\mathrel{\mathop{\mathchar
58\relax}}=\pi_{0}\Theta^{\mathrm{sfr}}(W_{g,1};\ell^{\mathrm{sfr}}_{\partial
W_{g,1}})$ has two orbits, distinguished by the Arf invariant of the
corresponding quadratic form. However, we will not need to know what the Arf
invariant of $e^{*}\mu$ is.

###  6.1. The groups $\Gamma_{g,1}^{\mathrm{hyp}}$ and
$\mathrm{Stab}_{\Gamma_{g,1}}(\ell^{\mathrm{hyp}}_{W_{g,1}})$

If we define

$\Gamma_{g,1}^{\mathrm{hyp}}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{1}(\Theta^{\mathrm{hyp}}(W_{g,1};\ell^{\mathrm{hyp}}_{\partial
W_{g,1}})/\\!\\!/\mathrm{Diff}_{\partial}(W_{g,1}),\ell^{\mathrm{hyp}}_{W_{g,1}})$

then the long exact sequence of homotopy groups for this homotopy quotient has
a portion

$\pi_{1}(\Theta^{\mathrm{hyp}}(W_{g,1};\ell^{\mathrm{hyp}}_{\partial
W_{g,1}}),\ell^{\mathrm{hyp}}_{W_{g,1}})\longrightarrow\Gamma_{g,1}^{\mathrm{hyp}}\longrightarrow\mathrm{Stab}_{\Gamma_{g,1}}(\ell^{\mathrm{hyp}}_{W_{g,1}})\longrightarrow
1,$

where $\mathrm{Stab}_{\Gamma_{g,1}}(\ell^{\mathrm{hyp}}_{W_{g,1}})$ denotes
the stabiliser of
$[\ell^{\mathrm{hyp}}_{W_{g,1}}]\in\pi_{0}\Theta^{\mathrm{hyp}}(W_{g,1};\ell^{\mathrm{hyp}}_{\partial
W_{g,1}})$ under the $\Gamma_{g,1}$-action. Using Lemma 6.1,
$\Gamma_{g,1}^{\mathrm{hyp}}$ is identified with the analogous stably framed
mapping class group $\Gamma_{g,1}^{\mathrm{sfr}}$, and similarly for the
stabiliser. Considered as the stabiliser of a stable framing, the latter has
been determined up to an extension in [KRW21]. Specifically, combining the
discussion in Section 6.1 of that paper with its Proposition 5.1 and Section
3.4 gives a central extension

(6.1)
$1\longrightarrow\Theta_{7}\longrightarrow\mathrm{Stab}_{\Gamma_{g,1}}(\ell^{\mathrm{hyp}}_{W_{g,1}})\longrightarrow\mathrm{Sp}_{2g}^{\text{q
or a}}(\mathbb{Z})\longrightarrow 1,$

where $\mathrm{Sp}_{2g}^{\text{q or a}}(\mathbb{Z})$ denotes the stabiliser of
a quadratic refinement of Arf invariant 0 or 1 respectively. (Depending on
what the Arf invariant of $e^{*}\mu$ is.)

In [GRW16, Lemma 7.5] there is described555Strictly speaking that source only
discusses $\mathrm{Sp}_{2g}^{\text{q}}(\mathbb{Z})$, but in a stable range the
cohomology of $\mathrm{Sp}_{2g}^{\text{q}}(\mathbb{Z})$ and
$\mathrm{Sp}_{2g}^{\text{a}}(\mathbb{Z})$ agree. a class $\mu\in
H^{2}(\mathrm{Sp}_{2g}^{\text{q or a}}(\mathbb{Z});\mathbb{Z})$, characterised
by two properties:

  1. (i)

$\mu$ vanishes when restricted to the $\mathbb{Z}/4$ subgroup generated by
$\big{(}\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\big{)}\in\mathrm{Sp}_{2}^{\text{q}}(\mathbb{Z})$,

  2. (ii)

the map $\mu_{*}\mathrel{\mathop{\mathchar
58\relax}}H_{2}(\mathrm{Sp}_{2g}^{\text{q or
a}}(\mathbb{Z});\mathbb{Z})\to\mathbb{Z}$ is surjective as long as $g\geq 2$
in the case $q$ and $g\geq 3$ in the case $a$666See [Kra20, Lemma 3.5] for
$\mathrm{Sp}_{4}^{\text{q}}(\mathbb{Z})$; for
$\mathrm{Sp}_{6}^{\text{a}}(\mathbb{Z})$ observe it contains
$\mathrm{Sp}_{4}^{\text{q}}(\mathbb{Z})$., and 8 times it is the signature
map. It generates $\mathrm{Hom}(H_{2}(\mathrm{Sp}_{2g}^{\text{q or
a}}(\mathbb{Z});\mathbb{Z}),\mathbb{Z})$.

This class $\mu$ can be multiplied by $\Sigma_{\mathrm{Milnor}}\in\Theta_{7}$
to give a class $\mu\cdot\Sigma_{\mathrm{Milnor}}\in
H^{2}(\mathrm{Sp}_{2g}^{\text{q or a}}(\mathbb{Z});\Theta_{7})$, corresponding
to a central extension

(6.2) $0\longrightarrow\Theta_{7}\longrightarrow
E\longrightarrow\mathrm{Sp}_{2g}^{\text{q or a}}(\mathbb{Z})\longrightarrow
1.$

######  Theorem 6.2.

The extension (6.1) is isomorphic to the extension (6.2).

###### Proof.

Including $\mathrm{Stab}_{\Gamma_{g,1}}(\ell^{\mathrm{hyp}}_{W_{g,1}})$ into
$\Gamma_{g,1}$, we obtain a map of central extensions

${1}$${\Theta_{7}}$${\mathrm{Stab}_{\Gamma_{g,1}}(\ell^{\mathrm{hyp}}_{W_{g,1}})}$${\mathrm{Sp}_{2g}^{\text{q
or
a}}(\mathbb{Z})}$${1}$${1}$${\Theta_{7}}$${\Gamma_{g,1}}$${\Gamma_{g,1}/\Theta_{7}}$${1.}$$\scriptstyle{i}$$\scriptstyle{j}$

Krannich [Kra20] has analysed the lower extension, as follows (we implicitly
specialise his results to the case $2n=6$ without further comment).

Using the notation $\Gamma_{g,1/2}=\Gamma_{g,1}/\Theta_{7}$, in [Kra20, (1.7)]
he gives an extension777Our map $j$ splits this extension over the subgroup
$\mathrm{Sp}_{2g}^{\text{q or a}}(\mathbb{Z})$, which does not contradict the
fact [Kra20, Theorem 2.2] that it is not split over
$\mathrm{Sp}_{2g}(\mathbb{Z})$.

$1\longrightarrow
H^{3}(W_{g,1};S\pi_{3}(\mathrm{SO}(3)))\longrightarrow\Gamma_{g,1}/\Theta_{7}\overset{p}{\longrightarrow}\mathrm{Sp}_{2g}(\mathbb{Z})\longrightarrow
0.$

The set of stable framings
$\pi_{0}\Theta^{\mathrm{sfr}}(W_{g,1};\ell^{\mathrm{sfr}}_{\partial W_{g,1}})$
is a $[W_{g,1}/\partial,\mathrm{SO}]_{*}$-torsor, and acting on the stable
framing $\ell^{\mathrm{sfr}}_{W_{g,1}}$ corresponding via Lemma 6.1 to
$\ell^{\mathrm{hyp}}_{W_{g,1}}$ gives a crossed homomorphism

$s\mathrel{\mathop{\mathchar
58\relax}}\Gamma_{g,1}\longrightarrow[W_{g,1}/\partial,\mathrm{SO}]_{*}=H^{3}(W_{g,1};\pi_{3}(\mathrm{SO})),$

which descends to $\Gamma_{g,1}/\Theta_{7}$; this is completely parallel to
the discussion in Section 3.1. Together the maps $s$ and $p$ give a
homomorphism

$(s,p)\mathrel{\mathop{\mathchar
58\relax}}\Gamma_{g,1}/\Theta_{7}\longrightarrow
H^{3}(W_{g,1};\pi_{3}(\mathrm{SO}))\rtimes\mathrm{Sp}_{2g}(\mathbb{Z})=\mathbb{Z}^{2g}\rtimes\mathrm{Sp}_{2g}(\mathbb{Z}).$

This is not an isomorphism, because
$S\pi_{3}(\mathrm{SO}(3))\to\pi_{3}(\mathrm{SO})$ is not (this is related to
Corollary 3.7), but it is injective. Krannich shows that
$\Gamma_{g,1}/\Theta_{7}$ has trivial abelianisation, and then defines a
cohomology class “$\tfrac{\chi^{2}-\mathrm{sgn}}{8}$” $\in
H^{2}(\Gamma_{g,1}/\Theta_{7};\mathbb{Z})$ by showing in [Kra20, Lemma 3.19
(iii)] that the composition

$H_{2}(\Gamma_{g,1}/\Theta_{7};\mathbb{Z})\overset{(s,p)_{*}}{\longrightarrow}H_{2}(\mathbb{Z}^{2g}\rtimes\mathrm{Sp}_{2g}(\mathbb{Z});\mathbb{Z})\overset{\chi^{2}-\mathrm{sgn}}{\longrightarrow}\mathbb{Z}$

has image $8\cdot\mathbb{Z}$. Here $\mathrm{sgn}$ is the signature map,
factoring over $H_{2}(\mathrm{Sp}_{2g}(\mathbb{Z});\mathbb{Z})$, and
$\chi^{2}$ is obtained by cup-squaring the canonical twisted cohomology class
$\chi\in
H^{1}(\mathbb{Z}^{2g}\rtimes\mathrm{Sp}_{2g}(\mathbb{Z});\mathbb{Z}^{2g})$
then applying $\lambda\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}^{2g}\otimes\mathbb{Z}^{2g}\to\mathbb{Z}$ to the
coefficients. In [Kra20, Theorem 3.22] (using [Kra20, Lemma 3.4]) he shows
that the lower extension is classified by the cohomology class

$-\tfrac{\chi^{2}-\mathrm{sgn}}{8}\cdot\Sigma_{\mathrm{Milnor}}\in
H^{2}(\Gamma_{g,1}/\Theta_{7};\Theta_{7}).$

Pulling this back along $j\mathrel{\mathop{\mathchar
58\relax}}\mathrm{Sp}_{2g}^{\text{q or
a}}(\mathbb{Z})\to\mathrm{Sp}_{2g}(\mathbb{Z})\to\Gamma_{g,1}/\Theta_{7}$, and
using that $\chi$ vanishes on this subgroup, it follows that the top extension
is classified by $\mu\cdot\Sigma_{\mathrm{Milnor}}\in
H^{2}(\mathrm{Sp}_{2g}^{\text{q or a}}(\mathbb{Z});\Theta_{7})$. ∎

We record the following consequences:

######  Corollary 6.3.

Suppose that $g\geq 3$.

  1. (i)

In the Leray–Hochschild–Serre spectral sequence for the extension (6.1) the
differential $d_{2}\mathrel{\mathop{\mathchar
58\relax}}H_{2}(\mathrm{Sp}_{2g}^{\text{q or
a}}(\mathbb{Z});\mathbb{Z})\to\Theta_{7}$ is surjective.

  2. (ii)

The finite residual of
$\mathrm{Stab}_{\Gamma_{g,1}}(\ell^{\mathrm{hyp}}_{W_{g,1}})$ is the central
subgroup $\Theta_{7}$.

###### Proof.

Item (i) follows from the facts that $\mu\mathrel{\mathop{\mathchar
58\relax}}H_{2}(\mathrm{Sp}_{2g}^{\text{q or
a}}(\mathbb{Z});\mathbb{Z})\to\mathbb{Z}$ is surjective and
$\Sigma_{\mathrm{Milnor}}$ generates $\Theta_{7}$. Item (ii) follows from the
discussion in [KRW20] near equation (6). ∎

###  6.2. Calculating the abelianisation of $\Gamma_{g,1}^{\mathrm{hyp}}$

The abelianisation of
$\Gamma_{g,1}^{\mathrm{hyp}}=\Gamma_{g,1}^{\mathrm{sfr}}$ can be calculated in
a way completely parallel to Section 5.1.1, using
$\bar{\theta}^{\mathrm{sfr}}\mathrel{\mathop{\mathchar
58\relax}}E\mathrm{O}\to B\mathrm{O}$ and its pullback
$\theta^{\mathrm{sfr}}\mathrel{\mathop{\mathchar
58\relax}}\mathrm{SO}/\mathrm{SO}(6)\to B\mathrm{SO}(6)$ to an unstable
tangential structure. The corresponding Thom spectrum is now
$MT\theta^{\mathrm{sfr}}(6)=\Sigma^{\infty-6}\mathrm{SO}/\mathrm{SO}(6)_{+}$,
so there is a map

$H_{1}(\Gamma_{g,1}^{\mathrm{hyp}};\mathbb{Z})\longrightarrow\pi_{1}^{s}(\Sigma^{\infty-6}\mathrm{SO}/\mathrm{SO}(6)_{+})$

which is an isomorphism as long as $g\geq 5$. The corresponding long exact
sequence simplifies to a split short exact sequence

$0\to\mathbb{Z}/4=\pi_{7}^{s}(\mathrm{SO}/\mathrm{SO}(6))\to\pi_{1}^{s}(\Sigma^{\infty-6}\mathrm{SO}/\mathrm{SO}(6)_{+})\to\pi_{7}^{s}(S^{0})=\mathbb{Z}/240\\{\sigma\\}\to
0.$

###  6.3. Relation to $X_{d}$

The embedding $e$ gives a map of central extensions

${1}$${\Theta_{7}}$${\mathrm{Stab}_{\Gamma_{g,1}}(\ell^{\mathrm{hyp}}_{W_{g,1}})}$${\mathrm{Sp}_{2g}^{\text{q
or
a}}(\mathbb{Z})}$${1}$${1}$${\mathrm{K}_{d}}$${\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})}$${\mathrm{Aut}(\pi_{3}(X_{d}),\lambda,\mu)}$${1.}$$\scriptstyle{\Phi}$

As “finite residual” is covariantly functorial for group homomorphisms,
Corollary 6.3 (ii) shows that $\mathrm{Im}(\Phi)$ is contained in the finite
residual of $\mathrm{Stab}_{\mathrm{MCG}_{d}}(\ell^{\mathrm{hyp}}_{X_{d}})$.
Combined with Theorem A this proves Theorem B.

######  Remark 6.4.

Using this map of central extensions, Corollary 6.3 (i) implies that the
differential $d_{2}\mathrel{\mathop{\mathchar
58\relax}}H_{2}(\mathrm{Aut};\mathbb{Z})\to\mathrm{K}_{d}$ in (5.3) hits the
subgroup $\mathrm{Im}(\Phi)$; the rest of the discussion in Section 5 shows
that this is precisely what it hits.

######  Remark 6.5.

It follows from the calculation in Section 6.2 that if
$[f]\in\mathrm{MCG}_{d}^{\mathrm{hyp}}$ can be supported on $e(W_{g,1})\subset
X_{d}$, then its image under

$\mathrm{MCG}_{d}^{\mathrm{hyp}}\longrightarrow
H_{1}(\mathrm{MCG}_{d}^{\mathrm{hyp}};\mathbb{Z})\overset{\sim}{\longrightarrow}\pi_{1}^{s}(MT{\theta}^{\mathrm{hyp}}(6))\longrightarrow\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$

lies in the image of $\iota_{*}\mathrel{\mathop{\mathchar
58\relax}}\pi_{7}^{s}(S^{0})\to\pi_{7}^{s}(MT\bar{\theta}^{\mathrm{hyp}})$.
This gives another point of view on the proof of Lemma 5.8.

## References

  * [Ada63] J. F. Adams, _On the groups $J(X)$. I_, Topology 2 (1963), 181–195.
  * [AW65] J. F. Adams and G. Walker, _On complex Stiefel manifolds_ , Proc. Cambridge Philos. Soc. 61 (1965), 81–103.
  * [Bea86] A. Beauville, _Le groupe de monodromie des familles universelles d’hypersurfaces et d’intersections complètes_ , Complex analysis and algebraic geometry (Göttingen, 1985), Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, pp. 8–18.
  * [Bro79] W. Browder, _Complete intersections and the Kervaire invariant_ , Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 88–108.
  * [Bru71] G. Brumfiel, _Homotopy equivalences of almost smooth manifolds_ , Comment. Math. Helv. 46 (1971), 381–407.
  * [Fri17] N. Friedrich, _Homological stability of automorphism groups of quadratic modules and manifolds_ , Doc. Math. 22 (2017), 1729–1774.
  * [Gil67] M. E. Gilmore, _Complex Stiefel manifolds, some homotopy groups and vector fields_ , Bull. Amer. Math. Soc. 73 (1967), 630–633.
  * [GRW14] S. Galatius and O. Randal-Williams, _Stable moduli spaces of high-dimensional manifolds_ , Acta Math. 212 (2014), no. 2, 257–377.
  * [GRW16] by same author, _Abelian quotients of mapping class groups of highly connected manifolds_ , Math. Ann. 365 (2016), no. 1-2, 857–879.
  * [GRW17] by same author, _Homological stability for moduli spaces of high dimensional manifolds. II_ , Ann. of Math. (2) 186 (2017), no. 1, 127–204.
  * [GRW18] by same author, _Homological stability for moduli spaces of high dimensional manifolds. I_ , J. Amer. Math. Soc. 31 (2018), no. 1, 215–264.
  * [Hai23] R. Hain, _Mapping class groups of simply connected Kähler manifolds_ , https://arxiv.org/abs/2304.01410, 2023.
  * [Ker60] M. A. Kervaire, _Some nonstable homotopy groups of Lie groups_ , Illinois J. Math. 4 (1960), 161–169.
  * [Kra20] M. Krannich, _Mapping class groups of highly connected $(4k+2)$-manifolds_, Selecta Math. (N.S.) 26 (2020), no. 5, Paper No. 81, 49.
  * [Kre79] M. Kreck, _Isotopy classes of diffeomorphisms of $(k-1)$-connected almost-parallelizable $2k$-manifolds_, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 643–663.
  * [Kre99] by same author, _Surgery and duality_ , Ann. of Math. (2) 149 (1999), no. 3, 707–754.
  * [KRW20] M. Krannich and O. Randal-Williams, _Mapping class groups of simply connected high-dimensional manifolds need not be arithmetic_ , C. R. Math. Acad. Sci. Paris 358 (2020), no. 4, 469–473.
  * [KRW21] A. Kupers and O. Randal-Williams, _Framings of $W_{g,1}$_, Q. J. Math. 72 (2021), no. 3, 1029–1053.
  * [Kry03] N. A. Krylov, _On the Jacobi group and the mapping class group of $S^{3}\times S^{3}$_, Trans. Amer. Math. Soc. 355 (2003), no. 1, 99–117.
  * [KS20] M. Kreck and Y. Su, _Mapping class group of manifolds which look like $3$-dimensional complete intersections_, https://arxiv.org/abs/2009.08054v2, 2020.
  * [L0̈9] M. Lönne, _Fundamental groups of projective discriminant complements_ , Duke Math. J. 150 (2009), no. 2, 357–405.
  * [Loo10] E. Looijenga, _Fermat varieties and the periods of some hypersurfaces_ , Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), Adv. Stud. Pure Math., vol. 58, Math. Soc. Japan, Tokyo, 2010, pp. 47–67.
  * [Mat63] H. Matsunaga, _Applications of functional cohomology operations to the calculus of $\pi_{2n+i}\,(U(n))$ for $i=6$ and $7$, $n\geq 4$_, Mem. Fac. Sci. Kyushu Univ. Ser. A 17 (1963), 29–62.
  * [Mat62] by same author, _The homotopy groups $\pi_{2n+i}(U(n))$ for $i=3,\,4$ and $5$_, Mem. Fac. Sci. Kyushu Univ. Ser. A 15 (1961/62), 72–81.
  * [Mos69] R. E. Mosher, _Some homotopy of stunted complex projective space_ , Illinois J. Math. 13 (1969), 192–197.
  * [Pae56] G. F. Paechter, _The groups $\pi_{r}(V_{n,\,m})$. I_, Quart. J. Math. Oxford Ser. (2) 7 (1956), 249–268.
  * [Pha65] F. Pham, _Formules de Picard-Lefschetz généralisées et ramification des intégrales_ , Bull. Soc. Math. France 93 (1965), 333–367.
  * [Sul77] D. Sullivan, _Infinitesimal computations in topology_ , Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 269–331 (1978).
  * [Wal66] C. T. C. Wall, _Classification problems in differential topology. V. On certain $6$-manifolds_, Invent. Math. 1 (1966), 355–374; corrigendum, ibid. 2 (1966), 306.
  * [Woo75] J. W. Wood, _Removing handles from non-singular algebraic hypersurfaces in $CP_{n+1}$_, Invent. Math. 31 (1975), no. 1, 1–6.