$\displaystyle\Phi^{\mathsf{emb}}:\mathsf{QT}^{\mathsf{emb}}_{d,d}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta)\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}[(Y,\partial
Y),(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)]$

and a surjection

(3.27)
$\displaystyle\Phi^{\mathsf{imm}}:\mathsf{QT}^{\mathsf{imm}}_{d,d}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta)\longrightarrow[(Y,\partial
Y),(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)].\quad\diamondsuit$

######  Corollary 3.8.

Let $\Theta\subset\Lambda\subset\mathbf{\Omega}_{\langle d]}$ be a pair of
closed sub-posets. There exists a group isomorphism (abelian for $n>1$)

(3.28)
$\displaystyle\mathbf{\Phi}^{\mathsf{emb}}:\mathbf{QT}^{\mathsf{emb}}_{d,d}(D^{n},\partial
D^{n};\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta)\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\pi_{n}(\mathcal{P}_{d}^{\mathbf{c}\Theta},\mathcal{P}_{d}^{\mathbf{c}\Lambda}).$

Assuming the $\Lambda$-condition (3.3), we get a group isomorphism

(3.29)
$\displaystyle\Phi^{\mathsf{emb}}:\mathsf{OG}^{\mathsf{emb}}_{d,d}(n;\mathbf{c}\Theta;\mathbf{c}\Theta)\approx\mathsf{G}^{\mathsf{emb}}_{d,d}(n;\mathbf{c}\Theta;\mathbf{c}\Theta)\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\pi_{n}(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt).$

###### Proof.

The claim follows instantly from Theorem 3.2 and Proposition 3.2. ∎

The next proposition is an attempt “to reverse the direction” of the maps in
(• ‣ 3.3), (• ‣ 3.4).

######  Proposition 3.5.

Let $\Theta^{\prime}\subset\Theta\subset\Lambda\subset\mathbf{\Omega}$ be
closed subposets, and $Y$ be a smooth compact manifold. Assume that
$d^{\prime}\geq d$ and $d^{\prime}\equiv d\mod 2$.

Then the obvious map

(3.30)
$\displaystyle\mathcal{A}:\;\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})\to\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$

is injective. That is, if two embeddings $\beta_{1}:(M_{1},\partial
M_{1})\hookrightarrow(\mathbb{R}\times Y,\mathbb{R}\times\partial Y)$ and
$\beta_{2}:(M_{2},\partial M_{2})\hookrightarrow(\mathbb{R}\times
Y,\,\mathbb{R}\times\partial Y)$ are quaitopic via immersions, then they are
quaitopic via embeddings.

Moreover, there exists a surjective “resolution map”

(3.31)
$\displaystyle\mathcal{R}:\;\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})\to\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$

that serves as the right inverse of $\mathcal{A}$. In other words, there
exists a canonical resolution of any such immersion into an embedding, well-
defined within the relevant quasitopy classes.

If $Y$ is orientable, similar claims hold for the oriented quasitopies and the
analogues of maps $\mathcal{A}$, $\mathcal{R}$ that are available for them.

###### Proof.

In the proof, to simplify the notations further, put
$\mathcal{QT}^{\mathsf{imm}}(Y)=_{\mathsf{def}}\;\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$,
$\mathcal{QT}^{\mathsf{emb}}(Y)=_{\mathsf{def}}\;\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$.

Using the map (• ‣ 3.3), we see that the map $\Phi^{\mathsf{emb}}(Y)$ is a
composition of the map $\mathcal{A}$ and the map

$\Phi^{\mathsf{imm}}(Y):\;\mathcal{QT}^{\mathsf{imm}}(Y)\to\big{[}\big{[}(Y,\partial
Y),\;\epsilon_{d,d^{\prime}}:(\mathcal{P}_{d}^{\mathbf{c}\Theta},\mathcal{P}_{d}^{\mathbf{c}\Lambda})\to(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}},\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Lambda})\big{]}\big{]}.$

However, by formula (3.23) from Theorem 3.2, we have a bijective map

$\Phi^{\mathsf{emb}}(Y):\mathcal{QT}^{\mathsf{emb}}(Y)\approx\big{[}\big{[}(Y,\partial
Y),\;\epsilon_{d,d^{\prime}}:(\mathcal{P}_{d}^{\mathbf{c}\Theta},\mathcal{P}_{d}^{\mathbf{c}\Lambda})\to(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}},\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Lambda})\big{]}\big{]}.$

Hence we define the desired surjective map $\mathcal{R}$ by the formula
$\mathcal{R}=\Phi^{\mathsf{emb}}(Y)^{-1}\circ\,\Phi^{\mathsf{imm}}(Y)$. It is
on the level of definitions to check that
$\mathcal{R}\circ\mathcal{A}=\mathsf{id}$.

If $Y$ is orientable, the same arguments hold for the oriented quasitopies
since
$\mathcal{OQT}^{\mathsf{emb}}(Y)\hfill\break\approx\mathcal{QT}^{\mathsf{emb}}(Y)$
by Corollary 3.5. ∎

######  Remark 3.7.

_Of course, applying the resolution $\mathcal{R}$ to an immersion
$\beta:M\to\mathbb{R}\times Y$ produces an embedding
$\beta^{\prime}:M^{\prime}\to\mathbb{R}\times Y$, where the topology of
$M^{\prime}$ is quite different from the topology of $M$. However, the _sets_
$\beta^{\prime}(M^{\prime})$ and $\beta(M)$ can be assumed to be arbitrary
close in the Hausdorff distance in $\mathbb{R}\times Y$. _ $\diamondsuit$

######  Corollary 3.9.

Each fiber of the resolution map $\mathcal{R}$ consists of the elements that
share the same $\Phi^{\mathsf{imm}}$-value in $[[(Y,\partial
Y),\;\epsilon_{d,d^{\prime}}:(\mathcal{P}_{d}^{\mathbf{c}\Theta},\mathcal{P}_{d}^{\mathbf{c}\Lambda})\to(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}},\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Lambda})]].$

In particular, for $Y=D^{n}$, we get a splittable groups’ extension

(3.32) $\displaystyle
1\to\mathsf{K}\to\mathsf{QT}^{\mathsf{imm}}_{d,d^{\prime}}(n;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\stackrel{{\scriptstyle\mathcal{R}}}{{\rightarrow}}\mathsf{QT}^{\mathsf{emb}}_{d,d^{\prime}}(n;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\to
1,$

whose kernel
$\mathsf{K}\subset\mathsf{QT}^{\mathsf{imm}}_{d,d^{\prime}}(n;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
consists of quasitopies of immersions that belong to the kernel of the map in
(3.4) with $Y=D^{n}$. Via the operation $\uplus$, the subgroup $\mathsf{K}$
acts on the fibers of the map $\mathcal{R}$ from (3.31).

###### Proof.

We use the notations from the previous proof. By Proposition 3.5, each fiber
of $\mathcal{R}$ consists of the elements that share the same
$\Phi^{\mathsf{imm}}$-value in $[[(Y,\partial
Y),\;\epsilon_{d,d^{\prime}}:(\mathcal{P}_{d}^{\mathbf{c}\Theta},\mathcal{P}_{d}^{\mathbf{c}\Lambda})\to(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}},\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Lambda})]].$
For $Y=D^{n}$, this implies that the group
$\mathcal{QT}^{\mathsf{imm}}(D^{n})$ is a semidirect product
$\mathsf{K}\bowtie\mathcal{QT}^{\mathsf{emb}}(D^{n})$ of the groups
$\mathsf{K}$ and $\mathcal{QT}^{\mathsf{emb}}(D^{n})$. Again, $\mathsf{K}$ is
characterized by the property of having trivial $\Phi^{\mathsf{imm}}$-values.

Since, for $\beta_{1}\in\mathcal{QT}^{\mathsf{imm}}(Y)$ and
$\beta_{2}\in\mathcal{QT}^{\mathsf{imm}}(D^{n})$, we have
$\Phi^{\mathsf{imm}}(\beta_{1}\uplus\beta_{2})=\Phi^{\mathsf{imm}}(\beta_{1})\uplus\Phi^{\mathsf{imm}}(\beta_{2})$,
we conclude that
$\Phi^{\mathsf{imm}}(\beta_{1}\uplus\mathsf{K})=\Phi^{\mathsf{imm}}(\beta_{1})\uplus
0=\Phi^{\mathsf{imm}}(\beta_{1})$. Therefore $\mathsf{K}$ acts on fibers of
the map $\mathcal{R}$. ∎

Since $\beta\in\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$ and
$\mathcal{R}(\beta)\in\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$ share the
same invariants
$\Phi^{\mathsf{imm}}(\beta)=\Phi^{\mathsf{emb}}(\mathcal{R}(\beta))\in\big{[}\big{[}(Y,\partial
Y),\;\epsilon_{d,d^{\prime}}:(\mathcal{P}_{d}^{\mathbf{c}\Theta},\mathcal{P}_{d}^{\mathbf{c}\Lambda})\to(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}},\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Lambda})\big{]}\big{]}$,
we should be looking for new invariants that distinguish between $\beta$ and
$\mathcal{R}(\beta)$.

Relying on Proposition 3.1, we may use the bordism classes
$[\Sigma_{k}^{\beta}]\in\mathbf{B}_{n-k+1}(Y,\partial Y)$ (or
$[\Sigma_{k}^{\beta}]\in\mathbf{OB}_{n-k+1}(Y,\partial Y)$ when $Y$ is
oriented) of the the $k$-self-intersection manifolds
$\\{\Sigma_{k}^{\beta}\\}_{k\geq 2}$ of $\beta$ to produce the desired
distinguishing invariants. In a sense, these quasitopy invariants “ignore” the
foliations $\mathcal{L},\mathcal{L}^{\bullet}$. Evidently, they vanish on
$\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$.

Therefore, for $\dim(Y)=n$, assuming that
$\min\\{n+1,\,\hat{m}(\mathbf{c}\Theta_{\langle d]})\\}\geq 2$ (see Definition
3.8), the correspondence $\beta\leadsto\\{\pi\circ\beta\circ
p_{1}:(\Sigma_{k}^{\beta},\partial\Sigma_{k}^{\beta})\to(Y,\partial Y)\\}_{k}$
produces a map

(3.33)
$\displaystyle\mathcal{BB}:\,\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})\big{/}\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})\to$
$\displaystyle\to\bigoplus_{k\in[2,\,\min\\{n+1,\,\hat{m}(\mathbf{c}\Theta_{\langle
d]})\\}]}\mathbf{B}_{n-k+1}(Y,\partial Y),$

which has a potential to discriminate between
$\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$ and
$\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y,\partial
Y;\hfill\break\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$.
Lemma 3.8 provides the simplest example where it does.

Next, we “mix” the tangency patterns of immersions $\beta$ to $\mathcal{L}$
with the self-intersections of $\beta$ (which also influence the
$\mathcal{L}$-tangency patterns). For simplicity, we assume the
$\Lambda$-condition and that $Y$ is oriented.

Let $\beta:(M,\partial M)\to(\mathbb{R}\times Y,\mathbb{R}\times\partial Y)$
be an immersion whose quasitopy class belongs to
$\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Theta^{\prime})$. Let $\mathsf{A}$ be an abelian
group. We pick a cohomology class $\theta\in
H^{n-k+1}(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt;\mathsf{A})$ and evaluate its
pull-back $(\Phi^{\beta}\circ\pi\circ\beta\circ p_{1})^{\ast}(\theta)$ on the
relative fundamental class $[\Sigma_{k}^{\beta},\partial\Sigma_{k}^{\beta}]$.
This construction leads to the following proposition.

######  Proposition 3.6.

Let $\Theta^{\prime}\subset\Theta\subset\mathbf{\Omega}$ be closed posets. We
assume the $\Lambda$-condition. Let $Y$ be an oriented smooth compact
manifold. Pick a positive integer
$k\leq\min\\{\hat{m}(\mathbf{c}\Theta_{\langle d]},n+1\\}$.

For any cohomology class $\theta\in
H^{n-k+1}(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt;\mathsf{A})$, the evaluation

$\chi_{k}(\theta,\beta):=\big{\langle}(\Phi^{\beta}\circ\pi\circ\beta\circ
p_{1})^{\ast}(\theta),\;[\Sigma_{k}^{\beta},\partial\Sigma_{k}^{\beta}]\big{\rangle}\in\mathsf{A}$

is an invariant of the quasitopy class
$[\beta]\in\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Theta^{\prime})$.

If $\beta$ is an embedding, then $\chi_{k}(\theta,\beta)=0$ for $k\geq 2$.

###### Proof.

By Corollary 3.1, the bordism class of $\pi\circ\beta\circ
p_{1}:(\Sigma_{k}^{\beta},\partial\Sigma_{k}^{\beta})\to(Y,\partial Y)$ is
invariant under a change of $\beta$ within its quasitopy class. By Theorem
3.1, the cocycle $(\Phi^{\beta})^{\ast}(\theta)$ is an invariant of the
quasitopy class of $\beta$. By the topological Stokes’ Theorem, the cocycle
$(\Phi^{\beta})^{\ast}(\theta)$, being evaluated on the cycle
$(\pi\circ\beta\circ p_{1})(\Sigma_{k}^{\beta},\partial\Sigma_{k}^{\beta})$ is
an invariant of
$[\beta]\in\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Theta^{\prime})$. ∎

Here is the simplest example of an invariant
$\rho_{\beta}:=\frac{1}{n+1}\big{[}(\pi\circ\beta\circ
p_{1}):\Sigma_{n+1}^{\beta}\to Y\big{]}$, delivered by $\mathcal{BB}$ from
(3.33), which does discriminate between the quasitopies of embeddings and
immersions.

######  Lemma 3.8.

Let $\dim M=n$ and let $\rho_{\beta}$ be the number of points in $Y$ where
exactly $n+1$ branches of $\beta(M)$ meet transversally in $\mathbb{R}\times
Y$. If $\rho_{\beta}\equiv 1\mod 2$, then
$\beta\in\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$ is
nontrivial, provided $\mathbf{c}\Lambda=(\emptyset),(1)$.

###### Proof.

Let $F_{\beta}$ denote the finite set of points, where exactly $n+1$ branches
of $\beta(M)$ meet transversally in $\mathbb{R}\times Y$. Assume that $\beta$
be quasitopic to trivial immersion with the help of a cobordism
$B:N\to\mathbb{R}\times Y\times[0,1]$. Then, in general, $B(N)$ may have
points where exactly $n+2$ branches of $B(N)$ meet transversally. Let $E_{B}$
denote their set and let $\eta_{B}:=\\#(E_{B})$. The points, where exactly
$n+1$ branches of $B(N)$ meet transversally, form a graph
$\Gamma_{B}\subset\mathbb{R}\times Y\times[0,1]$. If
$\mathbf{c}\Lambda=(\emptyset),(1)$, then $\Gamma_{B}$ has $\rho_{\beta}$
univalent vertices and $\eta_{B}$ vertices of valency $n+2$. Every edge of
$\Gamma_{B}$ that does not terminate at a valency one vertex from $F_{\beta}$
is attached to $E_{B}$. Thus counting the edges of $\Gamma_{B}$ we get
$2(n+2)\eta_{B}-\rho_{\beta}\equiv 0\mod 2$. Therefore, when
$\rho_{\beta}\equiv 1\mod 2$, we get a contradiction with the assumption that
$\beta$ is quasitopic to trivial immersion. As a result, any immersion $\beta$
with an odd number $\rho_{\beta}$ is nontrivial in
$\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$, provided
$\mathbf{c}\Lambda=(\emptyset),(1)$. ∎

Figure 4. Six generators $a,b,c,d,e,f$ of the free group
$\mathsf{QT}^{\mathsf{emb}}_{6,6}(1;\mathbf{c}\Theta;\mathbf{c}\Theta)$, where
$\Theta\subset\mathbf{\Omega}_{\langle 6]}$ is the closed poset generated by
the elements $\omega$ with two $2$’s or one $3$. The vertical lines mark
crossing the walls of the $6$-dimensional chambers in
$\mathcal{P}_{6}^{\mathbf{c}\Theta}$.

######  Example 3.1.

_Pick $n=1$, $d^{\prime}=d=6$. Let
$\mathbf{c}\Theta^{\prime}=\mathbf{c}\Theta$ consist of
$\omega\in\mathbf{\Omega}_{\langle 6]}$ such that all their entries are $1$’s
and $2$’s and a single $2$ is present at most. Put $Y=D^{1}$. Then
$\mathsf{QT}^{\mathsf{emb}}_{6,6}(1;\mathbf{c}\Theta;\mathbf{c}\Theta)\approx\pi_{1}(\mathcal{P}_{6}^{\mathbf{c}\Theta},pt)$.
By [KSW1], Theorem 2.4, the latter group is a free group $\mathsf{F}_{6}$ in
$6$ generators (see Fig. 3). The group extension in (3.32) reduces to_

$1\to\mathsf{K}\to\mathsf{QT}^{\mathsf{imm}}_{6,6}(1;\mathbf{c}\Theta;\mathbf{c}\Theta)\stackrel{{\scriptstyle\mathcal{R}}}{{\rightarrow}}\mathsf{F}_{6}\to
1.$

_The figure $\infty$, placed “horizontally” in $\mathbb{R}\times D^{1}$
($D^{1}$ being the horizontal direction) represents an immersion
$\beta:S^{1}\to\mathbb{R}\times D^{1}$. We claim that $\beta$ belongs to the
kernel $\mathsf{K}$ from (3.32). Indeed, consider a flip
$\tau:\mathbb{R}\times D^{1}\to\mathbb{R}\times D^{1}$ whose fixed point set
is the vertical line through the singularity of figure $\infty$. Due to the
symmetry of $\infty$ with respect to $\tau$, the path
$\Phi^{\mathsf{imm}}(D^{1},\partial
D^{1})\subset(\mathcal{P}_{6}^{\mathbf{c}\Theta},pt)$ is relatively
contractible. Thus $\infty$ is indeed in the kernel $\mathsf{K}$. By Lemma
3.8, $\beta$ is nontrivial in
$\mathsf{QT}^{\mathsf{imm}}_{6,6}(1;\mathbf{c}\Theta;\mathbf{c}\Theta)$. By
the same lemma, the quasitopy class of any collection of smooth curves with an
_odd_ number of crossings is nontrivial in
$\mathsf{QT}^{\mathsf{imm}}_{6,6}(1;\mathbf{c}\Theta;\mathbf{c}\Theta)$. At
the same time, the immersion $\infty\uplus\infty\subset\mathbb{R}\times D^{1}$
is trivial in
$\mathsf{QT}^{\mathsf{imm}}_{6,6}(1;\mathbf{c}\Theta;\mathbf{c}\Theta)$ since
it is the boundary of a horseshoe surface $N\approx\infty\times
I\subset\mathbb{R}\times D^{1}\times[0,1]$ with the
$\mathcal{L}^{\bullet}$-tangency patterns that belong to $\mathbf{c}\Theta$.
So the immersion $\beta$, such that $\beta(S^{1})=\infty$, is an element of
order two in the kernel $\mathsf{K}$. However, if we orient $S^{1}$, then
$\infty$ becomes an element of infinite order in
$\mathsf{ker}(\mathsf{OQT}^{\mathsf{imm}}_{6,6}(1;\mathbf{c}\Theta;\mathbf{c}\Theta)\to\mathsf{OQT}^{\mathsf{emb}}_{6,6}(1;\mathbf{c}\Theta;\mathbf{c}\Theta))$._

_In fact, any collection of curves $\beta(M)$ in $\mathbb{R}\times D^{1}$ with
the tangency patterns in $\mathbf{c}\Theta$, with an odd number of transversal
crossings, and which is symmetric under an involution $\tau:\mathbb{R}\times
D^{1}\to\mathbb{R}\times D^{1}$ which preserves the oriented foliation
$\mathcal{L}$, belongs to $\mathsf{K}$ and has order $2$ there. Based on a
sparse evidence, we conjecture that $\mathsf{K}\approx\mathbb{Z}_{2}$. If this
is true, then
$\mathsf{QT}^{\mathsf{imm}}_{6,6}(1;\mathbf{c}\Theta;\mathbf{c}\Theta)$ is a
semi-direct product $\mathbb{Z}_{2}\bowtie\mathsf{F}_{6}$. _ $\diamondsuit$

The next theorem describes one very general mechanism for generating
characteristic cohomology classes for immersions/embeddings
$\beta:M\to\mathbb{R}\times Y$ with $\Theta$-restricted tangency patterns to
$\mathcal{L}$.

######  Theorem 3.3.

For a compact manifold $Y$, $d^{\prime}=d$, and $\Theta^{\prime}=\Theta$,
assuming the $\Lambda$-condition, any immersion/embedding $\beta:(M,\partial
M)\to(\mathbb{R}\times Y,\mathbb{R}\times\partial Y)$ as in Proposition 3.3
induces a characteristic homomorphism $(\Phi^{\beta})^{\ast}$ from the
(co)homology of the differential complex
$\\{\partial^{\\#}:\mathbb{Z}[\Theta^{\\#}_{\langle
d]}]\to\mathbb{Z}[\Theta^{\\#}_{\langle d]}]\\}$ in (2.15) to the cohomology
$H^{\ast}(Y,\partial Y;\mathbb{Z})$.

Quasitopic immersions/embeddings induce the same characteristic homomorphisms.

###### Proof.

For any closed poset $\Theta\subset\mathbf{\Omega}_{\langle d]}$, by Corollary
2.6 and Theorem 2.2 ([KSW2]), the homology of the differential complex in
(2.15) is isomorphic to
$H^{\ast}(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt;\mathbb{Z})$. By Proposition
3.3, any immersion $\beta:(M,\partial M)\to(\mathbb{R}\times
Y,\,\mathbb{R}\times\partial Y)$ whose tangency patterns belong to
$\mathbf{c}\Theta$ and the ones of $\beta|_{\partial M}$ to
$\mathbf{c}\Lambda=(\emptyset)\text{ or }(1)$, produces a map
$\Phi^{\beta}:(Y,\partial Y)\to(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)$ whose
homotopy class is determined by $\beta$. This $\beta$ induces a natural map
$\beta^{\ast}:H^{\ast}(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt;\mathbb{Z})\to
H^{\ast}(Y,\partial Y;\mathbb{Z})$ in cohomology, and therefore, a
characteristic homomorphism
$(\tilde{\Phi}^{\beta})^{\ast}:H^{\ast}(\mathbb{Z}[\Theta^{\\#}],\partial^{\\#})\to
H^{\ast}(Y,\partial Y;\mathbb{Z})$.

By Proposition 3.3,
$(d,d;\mathbf{c}\Theta,(\emptyset);\mathbf{c}\Theta)$-quasitopic
immersions/embeddings $\beta$ induce homotopic maps $\tilde{\Phi}^{\beta}$ and
hence the same characteristic homomorphisms $(\tilde{\Phi}^{\beta})^{\ast}$. ∎

Thus various homomorphisms $(\tilde{\Phi}^{\beta})^{\ast}$ may distinguish
between different quasitopy classes of immersions $\beta$. We will soon
exhibit multiple examples, where they do.

###  3.4. Quasitopies of immersions/embeddings with special forbidden
combinatorics $\Theta$ and the stabilization by $d$

In this subsection, we take advantage of a few results from [V] and [KSW2], as
formulated in Section 2, to advance computations of quasitopies of embeddings
with special combinatorial patterns $\mathbf{c}\Theta$ and
$\mathbf{c}\Theta^{\prime}$ of tangency to the foliations $\mathcal{L}$ and
$\mathcal{L}^{\bullet}$, respectively.

We start with the Arnold-Vassiliev case of real polynomials with, so called,
moderate singularities [Ar], [V]. Let $\Theta_{\max\geq
k}\subset\mathbf{\Omega}_{\langle d]}$ be the closed poset consisting of
$\omega$’s with the maximal entry $\geq k$. For $k\geq 3$, the cohomology
$H^{j}(\mathcal{P}^{\mathbf{c}\Theta_{\max\geq k}},pt;\mathbb{Z})$ is
isomorphic to $\mathbb{Z}$ in each dimension $j$ of the form $(k-2)m$, where
the integer $m\leq d/k$, and vanishes otherwise [Ar].

Based on Vassiliev’s computation of the cohomology ring
$H^{\ast}(\mathcal{P}^{\mathbf{c}\Theta_{\max\geq k}},pt;\mathbb{Z})$ (see
[V], Theorem 1 on page 87), consider the graded ring $\mathcal{V}ass_{d,k}$,
multiplicatively generated over $\mathbb{Z}$ by the elements
$\\{e_{m}\\}_{m\leq d/k}$ of the degrees $\deg(e_{m})=m(k-2)$, subject to the
relations

(3.34) $\displaystyle e_{l}\cdot e_{m}=\frac{(l+m)!}{l!\cdot
m!}\;e_{l+m}\text{\; for }k\equiv 0\mod 2,\text{ and the relations }$
$\displaystyle e_{1}\cdot e_{1}=0,\quad e_{1}\cdot e_{2m}=e_{2m+1},\ $ (3.35)
$\displaystyle e_{2l}\cdot e_{2m}=\frac{(l+m)!}{l!\cdot m!}\;e_{2l+2m}\text{\;
for }k\equiv 1\mod 2.$

Guided by Theorem 3.3 and employing Theorem 3.1, we get the following claim.

######  Proposition 3.7.

Let $k\geq 3$. Consider an immersion $\beta:(M,\partial M)\to(\mathbb{R}\times
Y,\mathbb{R}\times\partial Y)$ whose tangency patterns to the foliation
$\mathcal{L}$ belong to $\mathbf{c}\Theta_{\max\geq
k}\subset\mathbf{\Omega}_{\langle d]}$ and the ones of $\beta|_{\partial M}$
either form an empty set $(\emptyset)$ when $d\equiv 0\mod 2$, or the set
$(1)$ when $d\equiv 1\mod 2$.

Then $\beta$ generates a characteristic ring homomorphism
$(\Phi^{\beta})^{\ast}:\mathcal{V}ass_{d,k}\to H^{\ast}(Y,\partial
Y;\mathbb{Z})$. The homomorphism $(\Phi^{\beta})^{\ast}$ is an invariant of
the quasitopy class of $\beta$. In other words, we get a map
$\Phi^{\ast}_{d,k}:\mathsf{QT}^{\mathsf{imm}}_{d,d}(Y;\mathbf{c}\Theta_{\max\geq
k};\mathbf{c}\Theta_{\max\geq
k})\to\mathsf{Hom_{ring}}\big{(}\mathcal{V}ass_{d,k},\;H^{\ast}(Y,\partial
Y;\mathbb{Z})\big{)}.\hfill\diamondsuit$

######  Remark 3.8.

_Note that if some generator $e_{\ell}\in\mathcal{V}ass_{d,k}$ is mapped by
$(\Phi^{\beta})^{\ast}$ to zero in $H^{(k-2)\ell}(Y,\partial Y;\mathbb{Z})$,
then the images of all $e_{q}$ with $q>\ell$ must be torsion elements. Thus
some cohomology rings $H^{\ast}(Y,\partial Y;\mathbb{Z})$ may not be able to
accommodate nontrivial images of the Vassiliev ring $\mathcal{V}ass_{d,k}$ in
positive degrees._

_In contrast, here is a case when the accommodation seems possible: take
$Y=\mathbb{C}\mathbb{P}^{3}$, $k=4$, and $d=12$. Consider $9$-dimensional
variety $\mathcal{P}_{12}^{\Theta_{\max\geq 4}}$, its compliment
$\mathcal{P}_{12}^{\mathbf{c}\Theta_{\max\geq 4}}$, and the cohomology ring
$\mathcal{V}ass_{12,4}\approx
H^{\ast}(\mathcal{P}_{12}^{\mathbf{c}\Theta_{\max\geq 4}};\mathbb{Z})$. Then
the ring homomorphism $\Phi^{\ast}:\mathcal{V}ass_{12,4}\to
H^{\ast}(\mathbb{C}\mathbb{P}^{3};\mathbb{Z})\approx\mathbb{Z}[a]/\\{a^{4}=0\\}$
that sends $\Phi^{\ast}(e_{1}):=6a$, $\Phi^{\ast}(e_{2}):=18a^{2}$, and
$\Phi^{\ast}(e_{3}):=36a^{3}$ embeds $\mathcal{V}ass_{12,4}$ as a subring of
$\mathbb{Z}[a]/\\{a^{4}=0\\}$. We speculate that there is an embedding
$\Phi:\mathbb{C}\mathbb{P}^{3}\to\mathcal{P}_{12}^{\mathbf{c}\Theta_{\max\geq
4}}$ that induces that $\Phi^{\ast}$. _ $\diamondsuit$

Propositions 3.6, 3.7 lead to the following corollary.

######  Corollary 3.10.

Let $M,Y$ be compact oriented smooth $n$-manifolds. For any generator
$e_{\ell}\in\mathcal{V}ass_{d,k}$ of degree $\ell(k-2)$ and an immersion
$\beta:(M,\partial M)\to(\mathbb{R}\times Y,\mathbb{R}\times\partial Y)$ with
$k$-moderate tangency patterns to $\mathcal{L}$, the integer

$\chi_{k}(e_{\ell},\beta)=\big{\langle}(\Phi^{\beta}\circ\pi\circ\beta\circ
p_{1})^{\ast}(e_{\ell}),\,[\Sigma_{\ell(k-2)}^{\beta},\partial\Sigma_{\ell(k-2)}^{\beta}]\big{\rangle}$

is an invariant of the quasitopy class
$[\beta]\in\mathcal{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta_{\max\geq k},\mathbf{c}\Theta_{\max\geq k})$. The invariant
$\chi_{k}(e_{\ell},\beta)=0$ when $\beta$ is an embedding and $k>1$, or when
$H^{\ell(k-2)}(Y,\partial Y;\mathbb{Z})=0$.

$\diamondsuit$

The next proposition is a stabilization result by the increasing
$d^{\prime}\geq d$ for the embeddings with moderate tangencies to the
foliation $\mathcal{L}$ on $\mathbb{R}\times Y$.

######  Proposition 3.8.

Let $k\geq 4$. If $\dim Y\leq(k-2)(\lceil d/k\rceil+1)-2$, then the
classifying map

(3.36)
$\displaystyle\qquad\qquad\Phi^{\mathsf{emb}}:\mathsf{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta_{\max\geq
k};\mathbf{c}\Theta_{\max\geq
k})\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\big{[}(Y,\partial
Y),(\mathcal{P}_{d}^{\mathbf{c}\Theta_{\max\geq k}},pt)\big{]}$

is a bijection, and the classifying map

(3.37)
$\displaystyle\qquad\qquad\Phi^{\mathsf{imm}}:\mathsf{QT}^{\mathsf{imm}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta_{\max\geq
k};\mathbf{c}\Theta_{\max\geq
k})\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\big{[}(Y,\partial
Y),(\mathcal{P}_{d}^{\mathbf{c}\Theta_{\max\geq k}},pt)\big{]}$

is a surjection for any $d^{\prime}\geq d$, $d^{\prime}\equiv d\mod 2$.

In particular, for a given $Y$,
$\mathsf{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta_{\max\geq
k};\mathbf{c}\Theta_{\max\geq k})$ stabilizes for all $d^{\prime}\geq
d\geq\frac{k}{k-2}(\dim Y+4-k)$, a linear function in $\dim Y$.

###### Proof.

Let $q:=(k-2)(\lceil d/k\rceil+1)-2$. By [V], Theorem 3 on page 88, the
$\epsilon_{d,d^{\prime}}$-induced homomorhpism
$(\epsilon_{d,d^{\prime}})_{\ast}:\pi_{i}(\mathcal{P}_{d}^{\mathbf{c}\Theta_{\max\geq
k}})\to\pi_{i}(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta_{\max\geq k}})$ of
homotopy groups is an isomorphism for all $i\leq q$. Thus the the two spaces
$q$-connected. Therefore, if $\dim Y\leq q$, then no element of
$\big{[}(Y,\partial Y),(\mathcal{P}_{d}^{\mathbf{c}\Theta_{\max\geq
k}},pt)\big{]}$ becomes, under the composition with $\epsilon_{d,d^{\prime}}$,
nill-homotopic. Hence,

$\big{[}\big{[}(Y,\partial
Y),\;\epsilon_{d,d^{\prime}}:\mathcal{P}_{d}^{\mathbf{c}\Theta_{\max\geq
k}}\to\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta_{\max\geq
k}}\big{]}\big{]}=\big{[}(Y,\partial
Y),(\mathcal{P}_{d}^{\mathbf{c}\Theta_{\max\geq k}},pt)\big{]}.$

In particular, if $\dim Y\leq q$, then $\big{[}(Y,\partial
Y),(\mathcal{P}_{d}^{\mathbf{c}\Theta_{\max\geq k}},pt)\big{]}$ is stable in
$d$, provided $d\geq\frac{k}{k-2}(\dim Y+4-k)$. Therefore, by Theorem 3.2, all
the claims of the proposition are validated. ∎

######  Example 3.2.

Let $k=4$. By Proposition 3.8 and formula (3.36), for any compact connected
surface $Y$, we get bijections:

$\displaystyle\mathsf{QT}^{\mathsf{emb}}_{4,4}(Y;\mathbf{c}\Theta_{\max\geq
4};\mathbf{c}\Theta_{\max\geq
4})\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\pi^{2}(Y/\partial Y),$
$\displaystyle\mathsf{QT}^{\mathsf{emb}}_{4,6}(Y;\mathbf{c}\Theta_{\max\geq
4};\mathbf{c}\Theta_{\max\geq
4})\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\pi^{2}(Y/\partial Y),$
$\displaystyle\mathsf{QT}^{\mathsf{emb}}_{6,6}(Y;\mathbf{c}\Theta_{\max\geq
4};\mathbf{c}\Theta_{\max\geq
4})\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\pi^{2}(Y/\partial Y),$

whose target is the second cohomotopy group $\pi^{2}(Y/\partial
Y)\approx\mathbb{Z}$, the latter isomorphism being generated by the degree of
maps to $S^{2}$.

Let $Z$ be a simply-connected $CW$-complex whose cohomology ring, truncated in
dimensions $\geq 5$, is isomorphic to
$\mathbb{Z}[e_{1},e_{2}]/\\{e_{1}^{2}-2e_{2}\\}$, where $\deg e_{1}=2,\deg
e_{2}=4$. Then by Proposition 3.8, for any compact smooth manifold $Y$ of
dimension $\leq 4$, we get bijections:

$\displaystyle\mathsf{QT}^{\mathsf{emb}}_{8,8}(Y;\mathbf{c}\Theta_{\max\geq
4};\mathbf{c}\Theta_{\max\geq 4})$
$\displaystyle\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}$
$\displaystyle[Y/\partial Y,Z],$
$\displaystyle\mathsf{QT}^{\mathsf{emb}}_{8,10}(Y;\mathbf{c}\Theta_{\max\geq
4};\mathbf{c}\Theta_{\max\geq 4})$
$\displaystyle\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}$
$\displaystyle[Y/\partial Y,Z],$
$\displaystyle\mathsf{QT}^{\mathsf{emb}}_{10,10}(Y;\mathbf{c}\Theta_{\max\geq
4};\mathbf{c}\Theta_{\max\geq 4})$
$\displaystyle\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}$
$\displaystyle[Y/\partial Y,Z].\;\diamondsuit$

######  Corollary 3.11.

Let $\Theta_{\max\geq k}\subset\mathbf{\Omega}_{\langle d]}$ be the closed
poset of $\omega$’s with at least one entry $\geq k$. Assume that $k\leq d\leq
2k-1$.

Then there is a group isomorphism

$\mathsf{OG}^{\mathsf{emb}}_{d,d}(n;\mathbf{c}\Theta_{\max\geq
k};\mathbf{c}\Theta_{\max\geq
k})\approx\mathsf{G}^{\mathsf{emb}}_{d,d}(n;\mathbf{c}\Theta_{\max\geq
k};\mathbf{c}\Theta_{\max\geq k})\approx\pi_{n}(S^{k-2}).$

In other words, the quasitopy group of embeddings
$\beta:M\hookrightarrow\mathbb{R}\times D^{n}$ into the cylinder
$\mathbb{R}\times D^{n}$ with no vertical tangencies of $\beta(M)$ to
$\mathcal{L}$ of the order $\geq k$ and with the total multiplicities
$\\{\mu_{\beta}(y)\leq d\\}_{y\in D^{n}}$ is isomorphic to the group
$\pi_{n}(S^{k-2})$, provided that $d\in[k,2k-1]$.

As a result, in this range of $d$’s and for $n>1$, all the groups
$\mathsf{G}^{\mathsf{emb}}_{d,d}(n;\mathbf{c}\Theta_{\max\geq
k};\mathbf{c}\Theta_{\max\geq k})$ are _finite_ abelian, except for $n=k-2$
and $n=2k-5$, where they are infinite of rank one.

###### Proof.

By Arnold’s theorem [Ar], for $d\in[k,2k+1]$,
$\pi_{n}(\mathcal{P}_{d}^{\mathbf{c}\Theta_{\max\geq
k}})\approx\pi_{n}(S^{k-2})$. Applying Corollary 3.8, the claim follows by
examining the tables of homotopy groups of spheres. ∎

######  Example 3.3.

_For $k=4$, $d\in[4,9]$, and all $n$, we get a group isomorphism_

$\mathsf{G}^{\mathsf{emb}}_{d,d}(n;\mathbf{c}\Theta_{\max\geq
4};\mathbf{c}\Theta_{\max\geq 4})\approx\pi_{n}(S^{2}).$

_In particular, for $d\in[4,9]$, visiting the table of homotopy groups of
spheres, we get:_

$\displaystyle\mathsf{G}^{\mathsf{emb}}_{d,d}(3;\mathbf{c}\Theta_{\max\geq
4};\mathbf{c}\Theta_{\max\geq 4})\approx\mathbb{Z},\quad$
$\displaystyle\mathsf{G}^{\mathsf{emb}}_{d,d}(4;\mathbf{c}\Theta_{\max\geq
4};\mathbf{c}\Theta_{\max\geq 4})\approx\mathbb{Z}_{2},$
$\displaystyle\mathsf{G}^{\mathsf{emb}}_{d,d}(14;\mathbf{c}\Theta_{\max\geq
4};\mathbf{c}\Theta_{\max\geq
4})\approx\mathbb{Z}_{84}\times\mathbb{Z}_{2}^{2}.\quad\quad$

_It would be very interesting to understand and describe, in the sprit of the
Arnold’s pink “kidneys” in Fig. 3 (see[Ar]), _the shapes_ of embeddings
$\beta:M\hookrightarrow\mathbb{R}\times D^{n}$ that produce such mysterious
periods… In principle, Theorem 3.2 contains the instructions for such
attempts. Perhaps, at least for $n=1,2$, one has a fighting chance…
$\diamondsuit$ _

Let $H,G$ be two groups, and $\mathsf{Hom}(H,G)$ their group of homomorphisms.
Then $G$ acts on $\mathsf{Hom}(H,G)$ by the conjugation: for any $\phi:H\to
G$, $h\in H$, and $g\in G$, we define $(Ad_{g}\phi)(h)$ by the formula
$g^{-1}\phi(h)g$. We denote by $\mathsf{Hom}^{\bullet}(H,G)$ the quotient
$\mathsf{Hom}(H,G)/Ad_{G}$.

The next corollary deals with special $\Theta$’s for which
$\mathcal{P}_{d}^{\mathbf{c}\Theta}$ is a $K(\pi,1)$-space [KSW1].

######  Corollary 3.12.

Let $\mathbf{c}\Theta$ consist of all $\omega$’s with entries $1$ and $2$ only
and no more than a single entry $2$. Put $\kappa(d):=\frac{d(d-2)}{4}$ for
$d\equiv 0\mod 2$, and $\kappa(d):=\frac{(d-1)^{2}}{4}$ for $d\equiv 1\mod 2$.
Let $\mathsf{F}_{\kappa(d)}$ be the _free_ group in $\kappa(d)$ generators.

Assume that either $Y$ is a closed manifold, or $\partial Y\neq\emptyset$ and
the $\Lambda$-condition is in place.

If $\partial Y\neq\emptyset$, then there is a bijection

$\Phi^{\mathsf{emb}}:\mathsf{QT}^{\mathsf{emb}}_{d,d}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta)\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\mathsf{Hom}(\pi_{1}(Y),\mathsf{F}_{\kappa(d)})$

and a surjection

$\Phi^{\mathsf{imm}}:\mathsf{QT}^{\mathsf{imm}}_{d,d}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta)\longrightarrow\mathsf{Hom}(\pi_{1}(Y),\mathsf{F}_{\kappa(d)}).$

When $Y$ is closed, then similar claims hold with the targets of
$\Phi^{\mathsf{emb}}$ and $\Phi^{\mathsf{imm}}$ being replaced by the set
$\mathsf{Hom}^{\bullet}(\pi_{1}(Y),\mathsf{F}_{\kappa(d)})$.

In particular,
$\Phi^{\mathsf{emb}}:\mathsf{QT}^{\mathsf{emb}}_{d,d}(S^{1};\mathbf{c}\Theta;\mathbf{c}\Theta)\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\mathsf{F}_{\kappa(d)}/Ad_{\,\mathsf{F}_{\kappa(d)}}$,
the free group of cyclic words in $\kappa(d)$ letters (see Fig. 3).

Thus, if $\pi_{1}(Y)$ has no nontrivial free images, then the group
$\mathsf{QT}^{\mathsf{emb}}_{d,d}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta)$ is
trivial.

###### Proof.

By Theorem 2.4, [KSW1], for such a $\mathbf{c}\Theta$, the space
$\mathcal{P}_{d}^{\mathbf{c}\Theta}$ is of the homotopy type of
$K(\mathsf{F}_{\kappa(d)},1)=\bigvee_{\sigma=1}^{\kappa(d)}S^{1}_{\sigma}$.
Thus, when $Y$ is closed, by the obstruction theory,
$[Y,\bigvee_{\sigma=1}^{\kappa(d)}S^{1}_{\sigma}]\approx\mathsf{Hom}^{\bullet}(\pi_{1}(Y),\mathsf{F}_{\kappa(d)})$
(see [Hu], Section VI, F). When $\partial Y\neq\emptyset$, the boundary of $Y$
is mapped by $\Phi^{\beta}$ to the preferred point
$pt\in\mathcal{P}_{d}^{\mathbf{c}\Theta}$. This makes it possible to consider
the based loops in $Y$, the base point $b$ being chosen in $\partial Y$, and
the based loops in $\bigvee_{\sigma=1}^{\kappa(d)}S^{1}_{\sigma}$, the base
point $pt$ being the apex $\star$ of the bouquet. Again, by the standard
obstruction theory, we get $[(Y,\partial
Y),\,(\bigvee_{\sigma=1}^{\kappa(d)}S^{1}_{\sigma},\star)]\approx\mathsf{Hom}(\pi_{1}(Y,b),\,\mathsf{F}_{\kappa(d)})$.

Now, by Corollary 3.8, the claim follows. In particular, if $\pi_{1}(Y)$ has
no nontrivial free images (say, $\pi_{1}(Y)$ is a finite group), the group
$\mathsf{QT}^{\mathsf{emb}}_{d,d}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta)$ is
trivial.

See Fig. 3 and [K4], [KSW1] for more illustrations of the cases when
$\mathcal{P}_{d}^{\mathbf{c}\Theta}$ is a $K(\pi,1)$-space. ∎

######  Proposition 3.9.

Let $\Theta^{\prime}=\Theta=_{\mathsf{def}}\\{\omega:\;|\omega|^{\prime}\geq
k\\}$, and $d^{\prime}\geq d>k>2$, $d^{\prime}\equiv d\mod 2$.

Then, assuming the $\Lambda$-condition (3.3), we get a group homomorphism

$\displaystyle\Phi_{d,d^{\prime}}^{\mathsf{i/e}}(n;\mathbf{c}\Theta;\mathbf{c}\Theta):\;\mathsf{G}^{\mathsf{i/e}}_{d,d^{\prime}}(n;\mathbf{c}\Theta;\mathbf{c}\Theta)\longrightarrow\pi_{n}(\mathcal{P}_{d}^{\mathbf{c}\Theta})\big{/}\ker((\epsilon_{d,d^{\prime}})_{\ast})\approx$
(3.38)
$\displaystyle\approx\;\pi_{n}\big{(}\bigvee_{i=1}^{A(d,\,k)}S^{k-1}_{i}\big{)}\big{/}\ker\Big{\\{}\pi_{n}\big{(}\bigvee_{i=1}^{A(d,\,k)}S^{k-1}_{i}\big{)}\stackrel{{\scriptstyle(\tilde{\epsilon}_{d,d^{\prime}})_{\ast}}}{{\longrightarrow}}\pi_{n}\big{(}\bigvee_{j=1}^{A(d^{\prime},\,k)}S^{k-1}_{j}\big{)}\Big{\\}},$

which is an isomorphism for the embeddings and an epimorphism for immersions.
The homomorphism $(\tilde{\epsilon}_{d,d^{\prime}})_{\ast}$ in (3.9) is
induced by the embedding
$\epsilon_{d,d^{\prime}}:\mathcal{P}_{d}^{\mathbf{c}\Theta}\to\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta}$.
The number $A(d,k):=A(d,k,0)$ has been introduced in (2.21).

For $n<k-1$, the group
$\mathsf{G}^{\mathsf{emb}}_{d,d^{\prime}}(n;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta)$
is trivial.

###### Proof.

By formula (• ‣ 3.4) from Theorem 3.4, we produce the map
$\Phi_{d,d^{\prime}}^{\mathsf{i/e}}(n;\mathbf{c}\Theta;\mathbf{c}\Theta)$. By
Corollary 3.8, it is a group epimomorphism for immersions and an isomorphism
for the embeddings.

By Proposition 2.1 and the Alexander duality, the spaces
$\mathcal{P}_{d}^{\mathbf{c}\Theta}$ and
$\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta}$ have the homology types of
bouquets of $(k-1)$-spheres. Recall that a simply-connected $CW$-complex whose
torsion-free $\mathbb{Z}$-homology is concentrated in a single dimension $q$
is homotopy equivalent to a bouquet of $q$-spheres (see [Ha], Theorem 4C.1).
Therefore the spaces $\mathcal{P}_{d}^{\mathbf{c}\Theta}$ and
$\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta}$ have the homotopy types of
bouquets of $(k-1)$-spheres, provided $k>2$. With the help of these homotopy
equivalences, the homomorphism
$(\tilde{\epsilon}_{d,d^{\prime}})_{\ast}:\pi_{n}\big{(}\bigvee_{i=1}^{A(d,\,k)}S^{k-1}_{i}\big{)}\to\pi_{n}\big{(}\bigvee_{j=1}^{A(d^{\prime},\,k)}S^{k-1}_{j}\big{)}$
is induced by the embedding
$\epsilon_{d,d^{\prime}}:\mathcal{P}_{d}^{\mathbf{c}\Theta}\to\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta}$.
This validates the first claim.

The last claim follows from the cellular approximation of a continuous map in
its homotopy class. ∎

For $d^{\prime}=d$, we are able to compute the quasitopy classes of $k$-flat
(see Definition 3.6) embeddings into $\mathbb{R}\times Y$. They correspond to
the case $q=0$ in the following proposition.

######  Proposition 3.10.

For $k\in[3,d-1]$ and $q\in[0,d]$, consider the closed subposet
$\Theta_{|\sim|^{\prime}\geq k}^{(q)}:=\big{\\{}\omega:\;|\omega|^{\prime}\geq
k,\,|\omega|\in[q,d]\big{\\}}$. Assume the $\Lambda$-condition (3.3).

Then, for a compact smooth $n$-manifold $Y$ with boundary, the quasitopy set
of embeddings is described by the formula

$\mathsf{QT}^{\mathsf{emb}}_{d,d}\big{(}Y;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)}\big{)}\;\approx\;\big{[}(Y,\partial
Y),\;(\bigvee_{i=1}^{A(d,\,k,\,q)}S^{k-1}_{i},\;\star)\big{]},$

where $\star$ is the apex of the bouquet, and the integer $A(d,\,k,\,q)$ is
defined by (2.21).

In particular, the quasitopic classes of embeddings in the cylinder
$\mathbb{R}\times D^{n}$ with the combinatorial patterns $\omega$ such that
$|\omega|^{\prime}<k$ and $|\omega|<q$ are described by the group isomorphism

$\mathsf{QT}^{\mathsf{emb}}_{d,d}\big{(}D^{n};\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)}\big{)}\;\approx\;\pi_{n}\big{(}\bigvee_{i=1}^{A(d,\,k,\,q)}S^{k-1}_{i},\,\star\big{)}.$

###### Proof.

The arguments are similar to the ones used in the proof of Proposition 3.9. To
simplify notations, temporarily, put $\Theta=\Theta_{|\sim|^{\prime}\geq
k}^{(q)}$.

By Theorem 2.1, $\bar{\mathcal{P}}^{\Theta}_{d}$ has the integral homology
type of a bouquet of $A(d,k,q)$ many $(d-k)$-dimensional spheres ([KSW2]).
Since the space $\bar{\mathcal{P}}^{\Theta}_{d}$ is $(d-k)$-dimensional
$CW$-complex [KSW1], for $k>2$, the space $\mathcal{P}^{\mathbf{c}\Theta}_{d}$
is simply-connected. By the Alexander duality, its integral homology is
concentrated in the dimension $k-1$ and is isomorphic to
$\mathbb{Z}^{A(d,k,q)}$. By Proposition 4C.1 from [Ha],
$\mathcal{P}^{\mathbf{c}\Theta}_{d}$ has the homotopy type of the bouquet
$\bigvee_{i=1}^{A(d,k,q)}S^{k-1}_{i}$. Now, by Theorem 3.2, the claim follows.
∎

######  Theorem 3.4.

Let $Y$ be either a closed smooth $n$-manifold or a manifold with boundary, in
which case, we assume the $\Lambda$-condition. We denote by
$\langle\omega\rangle$ the closed subposet of $\mathbf{\Omega}_{\langle d]}$,
consisting of elements $\omega^{\prime}\preceq\omega$.

Then for all $d\leq 13$ and all $\omega$ such that $|\omega|^{\prime}>2$, the
quasitopy
$\mathsf{QT}^{\mathsf{emb}}_{d,d}(Y;\mathbf{c}\langle\omega\rangle;\mathbf{c}\langle\omega\rangle)$
either is trivial (consists of single element), or is isomorphic to the
cohomotopy set $\pi^{k}(Y)$, where
$k=k(\omega)\in[|\omega|^{\prime}-1,\,d-1]$.

For $d\leq 13$, the table (4.1) in Appendix lists all $\omega$ and the
corresponding $k=k(\omega)$ for which
$\mathcal{P}_{d}^{\mathbf{c}\langle\omega\rangle}$ is non-contractible (in
fact, a homology $k$-sphere).

###### Proof.

The claim follows by combining Theorem 3.2 with the table (4.1) in Appendix. ∎

######  Example 3.4.

_Let $d=8$ and $\omega=(4)$. Then, for any closed $Y$, using the list (4.1) in
Appendix, we get a bijection
$\mathsf{QT}^{\mathsf{emb}}_{8,8}(Y;\mathbf{c}\langle(4)\rangle;\mathbf{c}\langle(4)\rangle)\approx\pi^{4}(Y)$,
the $4$-cohomotopy set of $Y$. In particular, the we get the following group
isomorphism:_

$\mathsf{QT}^{\mathsf{emb}}_{8,8}(S^{7};\mathbf{c}\langle(4)\rangle;\mathbf{c}\langle(4)\rangle)\approx\pi_{7}(S^{4})\approx\mathbb{Z}\times\mathbb{Z}_{12}.$

_Let $d=12$ and $\omega=(11213)$. Then, for any closed $Y$, by (4.1), we get a
bijection_

$\mathsf{QT}^{\mathsf{emb}}_{12,12}(Y;\mathbf{c}\langle(11213)\rangle;\mathbf{c}\langle(11213)\rangle)\approx\pi^{6}(Y).$

_In particular, we get an isomorphism
$\mathsf{QT}^{\mathsf{emb}}_{12,12}(S^{9};\mathbf{c}\langle(11213)\rangle;\mathbf{c}\langle(11213)\rangle)\approx\pi_{9}(S^{6})\approx\mathbb{Z}_{24}$._

$\diamondsuit$

###  3.5. Relation between quasitopies and inner framed cobordisms

For a compact smooth $Y$ of dimension $n\geq k-1$, Proposition 3.10 below
points to a peculiar connection between the quasitopies of embeddings
$\mathsf{QT}^{\mathsf{emb}}_{d,d}\big{(}Y;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}\big{)}$ and the inner
framed cobordisms $\mathcal{FB}^{k-1}_{\Xi}(Y)$ of $Y$ (see the definition
below).

######  Definition 3.12.

The cobordisms $\mathcal{FB}^{k-1}_{\Xi}(Y)$ are based on codimension $k-1$
smooth closed submanifolds $Z$ of $Y$ of the form
$Z=\coprod_{\sigma=1}^{A(d,k,q)}Z_{\sigma}$. The normal bundle $\nu(Z,Y)$ is
framed. The disjoint “components” $\\{Z_{\sigma}\\}_{\sigma}$ of $Z$ are
marked with different colors $\sigma$ from a pallet $\Xi$ of cardinality
$A(k,d,q)$.

Two such submanifolds $Z_{0},Z_{1}\subset Y$ are said to be inner cobordant if
there exist codimension $k-1$ submanifold
$W=\coprod_{\sigma=1}^{A(d,k,q)}W_{\sigma}$ of $Y\times[0,1]$, such that
$\partial W=(Z_{0}\times\\{0\\})\cup(Z_{1}\times\\{1\\})$; moreover, $\partial
W_{\sigma}=\big{(}(Z_{0})_{\sigma}\times\\{0\\}\big{)}\coprod\big{(}(Z_{1})_{\sigma}\times\\{1\\}\big{)}$
for each color $\sigma\in\Xi$. The normal bundle $\nu:=\nu(W,Y\times[0,1])$ is
framed, the restrictions $\nu|_{Y\times\\{0\\}}\approx\nu(Z_{0},Y)$,
$\nu|_{Y\times\\{1\\}}\approx\nu(Z_{1},Y)$, and the framing of $\nu$ extends
the framings of $\nu(Z_{0},Y)$ and $\nu(Z_{1},Y)$. $\diamondsuit$

As with quasitopies of immersions, for any pair $Y_{1}$, $Y_{2}$ of compact
smooth manifolds with boundary and any choice of
$\kappa_{1}\in\pi_{0}(\partial Y_{1})$, $\kappa_{2}\in\pi_{0}(\partial
Y_{2})$, an operation

(3.39)
$\displaystyle\uplus_{\mathcal{F}}:\mathcal{FB}^{k-1}_{\Xi}(Y_{1})\times\mathcal{FB}^{k-1}_{\Xi}(Y_{2})\to\mathcal{FB}^{k-1}_{\Xi}(Y_{1}\\#_{\partial}Y_{2})$

may be defined in the obvious way as $Z_{1}\coprod Z_{2}\subset
Y_{1}\\#_{\partial}Y_{2}$. This operation $\uplus_{\mathcal{F}}$ converts the
inner framed cobordisms $\mathcal{FB}^{k-1}_{\Xi}(D^{n})$ into a group.
Choosing $\kappa\in\pi_{0}(\partial Y)$, we get a group action

(3.40)
$\displaystyle\uplus_{\mathcal{F}}:\mathcal{FB}^{k-1}_{\Xi}(D^{n})\times\mathcal{FB}^{k-1}_{\Xi}(Y)\to\mathcal{FB}^{k-1}_{\Xi}(Y).$

######  Proposition 3.11.

Let a pallet $\Xi$ be of the cardinality $A(d,k,q)$. For a compact smooth
manifold $Y$ of dimension $n\geq k-1$, where $k\in[3,d-1]$, there is a
bijection

$Th:\mathcal{FB}^{k-1}_{\Xi}(Y)\approx\mathsf{QT}^{\mathsf{emb}}_{d,d}\big{(}Y;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}\big{)},$

delivered by the Thom construction on the trivialized normal bundle
$\nu(Z,Y)$, where $[Z\hookrightarrow Y]\in\mathcal{FB}^{k-1}_{\Xi}(Y)$. The
bijection $Th$ respects the operations $\uplus_{\mathcal{F}}$ and $\uplus$ in
framed inner bordisms and in the quasitopies:

(3.41) $\displaystyle Th(Z_{1}\uplus_{\mathcal{F}}Z_{2})=Th(Z_{1})\uplus
Th(Z_{2}).$

###### Proof.

Given a framed embedding
$Z:=\coprod_{\sigma=1}^{A(d,k,q)}Z_{\sigma}\hookrightarrow Y$ of a closed
smooth $Z$ of codimension $k-1$, the Thom construction on $\nu(Z,Y)$ produces
a continuous map $T_{Z}:Y\to\bigvee_{\sigma=1}^{A(d,k,q)}S^{k-1}_{\sigma}$ to
the bouquet. Since $Z$ is closed, $\partial Y$ is mapped to the base point
$\star$ of the bouquet. The homotopy class of $T_{Z}$ depends only on the
inner cobordism class of $Z\subset Y$ in $\mathcal{FB}^{k-1}_{\Xi}(Y)$.

Let us fix a homotopy equivalence
$\tau:(\mathcal{P}^{\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)}},pt)\sim(\bigvee_{\sigma=1}^{A(d,k,q)}S^{k-1}_{\sigma},\star).$ By
Corollary 3.7, the homotopy class of the map $\tilde{T}_{Z}:=\tau^{-1}\circ
T_{Z}$ determines the quasitopy class of some embedding
$\beta_{\tilde{T}_{Z}}:M\subset\mathbb{R}\times Y$, whose combinatorial
tangency patterns to $\mathcal{L}$ belong to the poset
$\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}$ and the tangency patterns of
$\beta_{\tilde{T}_{Z}}:\partial M\to\mathbb{R}\times\partial Y$ to the poset
$\mathbf{c}\Lambda=(\emptyset),(1)$. Thus the map
$Th:Z\leadsto\beta_{\tilde{T}_{Z}}$ is well-defined.

To show that $Th$ is onto, we take any embedding
$\beta:M\subset\mathbb{R}\times Y$ of a $n$-manifold $M$, whose tangency
patterns reside in $\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}$ and such
that the tangency patterns of $\beta:\partial M\subset\mathbb{R}\times\partial
Y$ reside in $\mathbf{c}\Lambda$ as above. We use $\beta$ to produce a map
$\Phi^{\beta}:(Y,\partial
Y)\to(\mathcal{P}^{\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}},pt)$.
Composing $\Phi^{\beta}$ with the homotopy equivalence $\tau$, we get a map
$\tilde{\Phi}^{\beta}:=\tau\circ\Phi^{\beta}$ from $Y$ to the bouquet such
that $\partial Y$ is mapped to the base point $\star$ of the bouquet. For each
sphere $S^{k-1}_{\sigma}$ from the bouquet, we fix a point $b_{\sigma}\in
S^{k-1}_{\sigma}$, different from $\star$, and a $(k-1)$-frame at
$b_{\sigma}$. Perturbing $\tilde{\Phi}^{\beta}$ within its homotopy class, we
may assume that $\tilde{\Phi}^{\beta}$ is transversal to each $b_{\sigma}$.
Then the closed framed submanifold
$Z_{\beta}:=(\tilde{\Phi}^{\beta})^{-1}(\coprod_{\sigma\in\Xi}b_{\sigma})\subset
Y$ is an element of $\mathcal{FB}^{k-1}_{\Xi}(Y)$. Moreover, for a given
$\beta$, the map $\tau^{-1}\circ Th(Z_{\beta})$ is in the homotopy class of
$\Phi^{\beta}$.

The injectivity of $Th$ has a similar validation, using the
$(n+1)$-dimensional cobordism $B:N\to\mathbb{R}\times Y\times[0,1]$ that
bounds an embedding $\beta:M\to\mathbb{R}\times Y$ which is quasitopic to the
trivial embedding. Again, we perturb
$\tau\circ\Phi^{B}:Y\times[0,1]\to\bigvee_{\sigma=1}^{A(d,k,q)}S^{k-1}_{\sigma}$
to make it transversal to $\coprod_{\sigma\in\Xi}b_{\sigma}$, thus producing a
framed cobordism $W$ in $Y\times[0,1]$ which bounds $Z_{\beta}$.

Finally, the validation of the property
$Th(Z_{1}\uplus_{\mathcal{F}}Z_{2})=Th(Z_{1})\uplus Th(Z_{2})$ is on the level
of definitions. ∎

For $k>2$ and an oriented $n$-dimensional $Y$, where $\partial
Y\neq\emptyset$, Proposition 3.11 gives a limited insight into the actions of
the (abelian when $n\geq 2$) group
$\mathsf{G}^{\mathsf{emb}}_{d,d}(n;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)})$ on the set
$\mathsf{QT}^{\mathsf{emb}}_{d,d}\big{(}Y;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}\big{)}$.

Here is one useful observation: if $\beta:M\hookrightarrow\mathbb{R}\times Y$
is represented, via $Th^{-1}$, by a closed normally framed submanifold
$Z_{\beta}=\coprod_{\sigma\in\Xi}Z_{\beta,\sigma}\subset Y$, then the
fundamental homology classes $\\{[Z_{\beta,\sigma}]\in
H_{n-k+1}(Y;\mathbb{Z})\\}_{\sigma\in\Xi}$ are _constant_ along the
$\mathsf{G}^{\mathsf{emb}}_{d,d}(n;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)})$-orbit through
$Th(Z_{\beta})$. Indeed, using the property
$Th(Z_{1}\uplus_{\mathcal{F}}Z_{2})=Th(Z_{1})\uplus Th(Z_{2})$, all the
changes of $Z_{\beta}$, produced by the $\uplus_{\mathcal{F}}$-action, are
incapsulated in a ball $B^{n}\subset Y$, and thus their fundamental classes
are null-homologous in $Y$.

For a pallet $\Xi$ of cardinality $A(d,k,q)$, consider the subset
$H_{n-k+1}^{[\Xi\sim fr]}(Y;\mathbb{Z})$ of the group
$\big{(}H_{n-k+1}(Y;\mathbb{Z})\big{)}^{A(d,k,q)}$ that is realized by
$A(d,k,q)$ _disjoint_ distinctly colored and normally framed closed
submanifolds $\\{Z_{\sigma}\\}_{\sigma\in\Xi}$ of $Y$, $\dim
Z_{\sigma}=n-k+1$. For $2k-2>n$, by a general position argument,
$H_{n-k+1}^{[\Xi\sim fr]}(Y;\mathbb{Z})$ is a _subgroup_ of
$\big{(}H_{n-k+1}(Y;\mathbb{Z})\big{)}^{A(d,k,q)}$.

Using Proposition 3.11 and tracing the correspondence $[\beta]\leadsto
Th^{-1}([\beta])\leadsto\\{[Z_{\beta,\sigma}]\\}_{\sigma}$, validates
instantly to the following claim.

######  Proposition 3.12.

Let $Y$ be a compact smooth oriented $n$-manifold $Y$, where $n\geq k-1$ and
$k\in[3,d-1]$. Assume that $\partial Y\neq\emptyset$. Then the orbit-space of
the $\mathsf{G}^{\mathsf{emb}}_{d,d}(n;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)})$-action on the set
$\mathsf{QT}^{\mathsf{emb}}_{d,d}\big{(}Y;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}\big{)}$ admits a
surjection on the set $H_{n-k+1}^{[\Xi\sim fr]}(Y;\mathbb{Z})$.

$\diamondsuit$

######  Example 3.5.

_Take $k=3$, $d=6$, $q=0$, and $n=3$. Then $A(6,3,0)=10$ and $\\#\Xi=10$. Thus
we get a homotopy equivalence
$\tau:\mathcal{P}_{6}^{\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3}}\sim\bigvee_{\sigma=1}^{10}S^{2}_{\sigma}$. For a compact oriented
$3$-manifold $Y$, any normally framed submanifold $Z$ acquires an orientation.
Using the isomorphism_

$Th:\mathcal{FB}^{2}_{\Xi}(Y)\approx\mathsf{QT}^{\mathsf{emb}}_{6,6}\big{(}Y;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3};\mathbf{c}\Theta_{|\sim|^{\prime}\geq 3}\big{)}$

_from Proposition 3.11, any collection of at most ten disjoint framed closed
$1$-dimensional submanifolds $Z_{1},\ldots Z_{10}\subset Y$ (that is, any
_framed link_ $Z\subset Y$, colored with $10$ distinct colors at most)
produces a quasitopy class of an embedding $\beta:M^{3}\to\mathbb{R}\times Y$,
where $M$ is closed. Its tangency patterns $\omega$, with the exception of
$(\emptyset)$, have only entries from the list $\\{1,2,3\\}$ so that no more
than one $3$ and no more than two $2$’s are present in $\omega$, while
$|\omega|\leq 6$._

_The orbit-space of the
$\mathsf{G}^{\mathsf{emb}}_{6,6}(3;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3};\mathbf{c}\Theta_{|\sim|^{\prime}\geq 3})$-action on
$\mathsf{QT}^{\mathsf{emb}}_{6,6}\big{(}Y;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3};\mathbf{c}\Theta_{|\sim|^{\prime}\geq 3}\big{)}$ admits a surjection onto
the group $(H_{1}(Y;\mathbb{Z}))^{10}$. For example, for
$Y=T^{3}\setminus\textup{int}(D^{3})$, where $T^{3}$ is the $3$-torus, the
orbit-space is mapped onto the lattice $\mathbb{Z}^{30}$. $\diamondsuit$_

######  Corollary 3.13.

We adopt the hypotheses of Proposition 3.10.

  * •

For $\dim Y=k-1$ and any choice of $\kappa\in\pi_{0}(\partial Y)$, the group
$\mathsf{G}^{\mathsf{emb}}_{d,d}(k-1;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\hfill\break\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)})$ acts
freely and transitively on the set
$\mathsf{QT}^{\mathsf{emb}}_{d,d}\big{(}Y;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}\big{)}$. Thus both
sets are in a $1$-to-$1$ correspondence.

  * •

For a simply-connected $Y$, $\dim Y=k>4$, the group
$\mathsf{G}^{\mathsf{emb}}_{d,d}\big{(}k;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}\big{)}$ acts freely
and transitively on the set
$\mathsf{QT}^{\mathsf{emb}}_{d,d}\big{(}Y;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}\big{)}$. Again, both
sets are in a $1$-to-$1$ correspondence.

###### Proof.

By Proposition 3.12, there exists a bijection

$Th:\mathcal{FB}^{k-1}_{\Xi}(Y)\to\mathsf{QT}^{\mathsf{emb}}_{d,d}\big{(}Y;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\hfill\break\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}\big{)}.$

When $\dim Y=k-1$, the elements of $\mathcal{FB}^{k-1}_{\Xi}(Y)$ are
represented by finite sets $Z$ of framed oriented singletons, marked with
$\\#\Xi$ colors at most. (We may assume that no two singletons of the same
color have opposite orientations.) Evidently, any such finite set $Z$ can be
incapsulated into a standard smooth ball $D^{k-1}\subset Y$ such that
$\partial D^{k-1}\cap\partial Y=D^{k-2}$, a standard $(k-2)$-ball. Therefore
$Z$ belongs to the $\mathcal{FB}^{k-1}_{\Xi}(D^{k-1})$-orbit of the empty
collection $Z_{\emptyset}$. By (3.41), $Th(Z)$ belongs to the
$\mathsf{G}^{\mathsf{emb}}_{d,d}(k-1;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)})$-orbit of the trivial
embedding (the empty one for $d\equiv 0\mod 2$ and the embedding
$\beta_{\star}:Y\to\\{0\\}\times Y\subset\mathbb{R}\times Y$ for $d\equiv
1\mod 2$). This validates the first claim.

For a simply-connected $Y$, $\dim Y=k>4$, again by Proposition 3.12, any
element of
$\mathsf{QT}^{\mathsf{emb}}_{d,d}\big{(}Y;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)}\big{)}$ is represented
by a framed collection $Z$ of disjoint loops in $Y$, colored with $\\#\Xi$
colors at most. Using that $\pi_{1}(Y)=1$ and $\dim Y=k>4$, we can bound each
loop $Z_{\sigma}$ by a regularly embedded smooth $2$-disk
$W_{\sigma}\subset\textup{int}(Y)$ so that all $W_{\sigma}$’s are disjoint.
Therefore each loop $Z_{\sigma}$ is contained in a standard ball
$D^{k}_{\sigma}$, the regular neighborhood of $W_{\sigma}$. By performing
$1$-surgeries on $\coprod_{\sigma}D^{k}_{\sigma}$, we construct a $k$-ball
$\tilde{B}^{k}$ that contains all $Z_{\sigma}$’s. Adding a $1$-handle
$H\subset Y\setminus\tilde{B}^{k}$ that connects $\partial\tilde{B}^{k}$ with
$\partial Y$, we exhibit a ball $B^{k}:=\tilde{B}^{k}\cup H$ which contains
all the loops and such that $\partial B^{k}\cap\partial Y$ is a
$(k-1)$-dimensional ball. Therefore $Z$ is, once more, in the orbit of the
empty collection $Z_{\emptyset}$ of loops. By (3.41), $Th(Z)$ belongs to the
$\mathsf{G}^{\mathsf{emb}}_{d,d}(k;\mathbf{c}\Theta_{|\sim|^{\prime}\geq
k}^{(q)};\mathbf{c}\Theta_{|\sim|^{\prime}\geq k}^{(q)})$-orbit of the trivial
embedding.

In fact, this conclusion is also valid for $3$-manifolds $Y$ with a _spherical
boundary_ , but its validation is not as direct as for the case $k>4$. If
$\pi_{1}(Y)=1$, by the Perelman’s solution of the Poincaré Conjecture
[P1]-[P3], we conclude that $Y$ is the standard ball $D^{3}$. Thus any
$\Xi$-colored and framed link $Z\subset Y$ is obviously contained in $D^{3}$.

As usual, the case $k=4$ seems to be challenging. ∎

###  3.6. More about the stabilization of
$\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,\mathbf{c}\Lambda;\mathbf{c}\Theta^{\prime})$ by the
degrees $d,d^{\prime}$

We start with the embeddings of $1$-manifolds in the product $\mathbb{R}\times
D^{1}$ and the forbidden poset $\Theta$ that consists of $\omega$’s with the
reduced norms $\geq 2$.

######  Proposition 3.13.

Let $d^{\prime}\geq d+2$, $d^{\prime}\equiv d\mod 2$. For a closed subposet
$\Theta\subset\mathbf{\Omega}_{\langle d],\,|\sim|^{\prime}\geq 2}$, let
$\hat{\Theta}_{\\{d^{\prime}\\}}$ be the smallest subposet in
$\mathbf{\Omega}_{\langle d^{\prime}]}$ that contains $\Theta$. Then we get a
stabilization by $d^{\prime}$:

$\mathsf{QT}^{\mathsf{emb}}_{d+2,\,d^{\prime}}(D^{1};\mathbf{c}\hat{\Theta}_{\\{d+2\\}};\mathbf{c}\hat{\Theta}_{\\{d^{\prime}\\}})\approx\pi_{1}(\mathcal{P}_{d+2}^{\mathbf{c}\hat{\Theta}_{\\{d+2\\}}},pt).$

###### Proof.

The claim follows from Theorem 3.2 by combining it with Theorem 2.2 (see also
Theorem 2.15 from [KSW1]), which claims that
$(\epsilon_{d+2,d^{\prime}})_{\ast}:\pi_{1}(\mathcal{P}_{d+2}^{\mathbf{c}\hat{\Theta}_{\\{d+2\\}}})\approx\pi_{1}(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta_{\\{d^{\prime}\\}}})$
is an isomorphism, induced by the embedding $\epsilon_{d+2,d^{\prime}}$ of the
two spaces. See [KSW1] for an explicit description of presentations of such
fundamental groups
$\pi_{1}(\mathcal{P}_{d+2}^{\mathbf{c}\hat{\Theta}_{\\{d+2\\}}})$. ∎

######  Theorem 3.5.

Let $\Theta$ be a closed poset such that $|\omega|^{\prime}\geq 3$ for any
$\omega\in\Theta$. Let $d^{\prime}\geq d$ and $d^{\prime}\equiv d\mod 2$.
Assume the $\Lambda$-condition (3.3).

For any compact smooth $Y$ such that $\dim Y\leq d+2-\psi_{\Theta}(d+2)$
(see(2.18)) we get the classifying map

(3.42)
$\displaystyle\Phi_{d,d+2}^{\mathsf{i/e}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta):\;\mathsf{QT}^{\mathsf{i/e}}_{d,d+2}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta)\to[(Y,\partial
Y),(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)],$

which is bijective for embeddings and surjective for immersions.

For any profinite (in the sense of Definition 2.2) poset $\Theta$ and any $Y$,
the set
$\mathsf{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\hfill\break\mathbf{c}\Theta)$
stabilizes for all sufficiently big $d$.

###### Proof.

The argument is a bit similar to the one we used in the proof of Proposition
3.9.

Under the hypotheses, by Theorem 2.2, the homomorphism
$(\epsilon_{d,d^{\prime}})_{\ast}:H_{j}(\mathcal{P}_{d}^{\mathbf{c}\Theta};\mathbb{Z})\to
H_{j}(\mathcal{P}_{d+2}^{\mathbf{c}\Theta};\mathbb{Z})$ is an isomorphism for
all $j<d+2-\psi_{\Theta}(d+2)$. Since $|\omega|^{\prime}\geq 3$ for any
$\omega\in\Theta$, all the spaces
$\\{\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta}\\}_{d^{\prime}\geq d}$ are
simply connected. Hence, $\epsilon_{d,d+2}$ induces an isomorphism in the
integral $j$-homology of $\mathcal{P}_{d}^{\mathbf{c}\Theta}$ and of
$\mathcal{P}_{d+2}^{\mathbf{c}\Theta}$ for all $j<d+2-\psi_{\Theta}(d+2)$. By
the Whitehead Theorem (the inverse Hurewicz Theorem) (see Theorem (7.13) in
[Wh]), $\epsilon_{d,d+2}$ induces isomorphisms of the homotopy groups of
$\mathcal{P}_{d}^{\mathbf{c}\Theta}$ and of
$\mathcal{P}_{d+2}^{\mathbf{c}\Theta}$ in dimensions $\leq
d+2-\psi_{\Theta}(d+2)$. By standard application of the obstruction theory, if
$\dim Y\leq d+2-\psi_{\Theta}(d+2)$, no nontrivial element of $[(Y,\partial
Y),(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)]$ becomes, via $\epsilon_{d,d+2}$,
null-homotopic in $[(Y,\partial
Y),(\mathcal{P}_{d+2}^{\mathbf{c}\Theta},pt^{\prime})]$. Thus, for such $Y$,
we get

$[[(Y,\partial
Y),\;\epsilon_{d,d+2}:(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)\to(\mathcal{P}_{d+2}^{\mathbf{c}\Theta^{\prime}},pt^{\prime})]]=[(Y,\partial
Y),(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)].$

By Lemma 4.3 from [KSW2], for any closed profinite $\Theta$,
$\lim_{d^{\prime}\to\infty}d^{\prime}-\psi_{\Theta}(d^{\prime})=+\infty$.
Hence, repeating the previous arguments, we get a similar conclusion for all
$d^{\prime}\geq d\gg\dim Y$.

Therefore, by Theorem 3.2, the all the claims of Theorem 3.5 follow. ∎

##  4\. Appendix: Topology of
$\mathcal{P}_{d}^{\mathbf{c}\langle\omega\rangle}$ for small $d$

In conclusion, let us state somewhat surprising results of one computer-
assisted computation from [KSW2], Figures 6-8. The tables below are produced
from these three figures with the help of the Alexander duality. For a given
$\omega\in\mathbf{\Omega}$, we denote by $\langle\omega\rangle$ the minimal
closed poset that contains $\omega$. In (4.1), for $d\leq 13$, we list all
compositions $\omega$ for which the space
$\mathcal{P}_{d}^{\mathbf{c}\langle\omega\rangle}$ is homologically
nontrivial. In fact, for $d\leq 13$, every homologically nontrivial
$\mathcal{P}_{d}^{\mathbf{c}\langle\omega\rangle}$ is a _homology_ sphere!
Moreover, at least for $\omega$ with $|\omega|^{\prime}>2$, all such spaces
$\mathcal{P}_{d}^{\mathbf{c}\langle\omega\rangle}$ are _homotopy spheres_.
Unfortunately, the reason for such phenomena is mysterious to us…

In the list below, we denote by $``\mathbf{S^{k}}"$ the homology $k$-sphers.
We use the notation $``\mathbf{S^{k}_{\star}}"$ for the homotopy $k$-spheres.
To make the writing of $\omega$’s more compact, we adopt the following
notation: for example, we abbreviate $\omega=(1132555)$ as $(1^{2}325^{3})$.

(4.1) $\displaystyle\text{\bf{List of nontrivial $\mathbb{Z}$-homology types
of }}\mathcal{P}_{d}^{\mathbf{c}\langle\omega\rangle}\text{\bf{ for }}4\leq
d\leq 13:$

$\underline{\mathsf{d=4}}$.

$|\omega|^{\prime}=0$: $\omega=(1^{2})\;\Rightarrow\mathbf{S^{0}}$

$|\omega|^{\prime}=1$: $\omega=(2)\;\Rightarrow\mathbf{S^{0}}$

$|\omega|^{\prime}=3$: $\omega=(4)\;\Rightarrow\mathbf{S^{2}_{\star}}$

$\underline{\mathsf{d=5}}$.

$|\omega|^{\prime}=0$: $\omega=(1^{3})\;\Rightarrow\mathbf{S^{0}}$

$|\omega|^{\prime}=4$: $\omega=(5)\;\Rightarrow\mathbf{S^{3}_{\star}}$

$\underline{\mathsf{d=6}}$.

$|\omega|^{\prime}=0$: $\omega=(1^{4}),(1^{2})\;\Rightarrow\mathbf{S^{0}}$

$|\omega|^{\prime}=1$: $\omega=(2)\;\Rightarrow\mathbf{S^{0}}$

$|\omega|^{\prime}=5$: $\omega=(6)\;\Rightarrow\mathbf{S^{4}_{\star}}$

$\underline{\mathsf{d=7}}$.

$|\omega|^{\prime}=0$: $\omega=(1^{5}),(1^{3})\;\Rightarrow\mathbf{S^{0}}$

$|\omega|^{\prime}=6$: $\omega=(7)\;\Rightarrow\mathbf{S^{5}_{\star}}$

$\underline{\mathsf{d=8}}$.

$|\omega|^{\prime}=0$:
$\omega=(1^{6}),(1^{4}),(1^{2})\;\Rightarrow\mathbf{S^{0}}$

$|\omega|^{\prime}=1$: $\omega=(1,2,1)\;\Rightarrow\mathbf{S^{4}}$
$\omega=(2)\;\Rightarrow\mathbf{S^{0}}$

$|\omega|^{\prime}=2$: $\omega=(1,3),(3,1)\;\Rightarrow\mathbf{S^{4}}$

$|\omega|^{\prime}=3$: $\omega=(4)\;\Rightarrow\mathbf{S^{4}_{\star}}$

$|\omega|^{\prime}=7$: $\omega=(8)\;\Rightarrow\mathbf{S^{6}_{\star}}$.

$\underline{\mathsf{d=9}}$.

$|\omega|^{\prime}=0$:
$\omega=(1^{7}),(1^{5}),(1^{3})\;\Rightarrow\mathbf{S^{0}}$;

$|\omega|^{\prime}=1$:
$\omega=(1^{2},2,1),(1,2,1^{2}),\;\Rightarrow\mathbf{S^{4}}$;

$|\omega|^{\prime}=2$:
$\omega=(1^{2},3),(1,3,1),(3,1^{2})\;\Rightarrow\mathbf{S^{4}}$;

$|\omega|^{\prime}=8$: $\omega=(9)\;\Rightarrow\mathbf{S^{7}_{\star}}$.

$\underline{\mathsf{d=10}}$.

$|\omega|^{\prime}=0$:
$\omega=(1^{8}),(1^{6}),(1^{4}),(1^{2})\;\Rightarrow\mathbf{S^{0}}$;

$|\omega|^{\prime}=1$:
$\omega=(1^{3},2,1),(1^{2},2,1^{2}),(1,2,1^{3}),\;\Rightarrow\mathbf{S^{4}}$;
$\omega=(2)\;\Rightarrow\mathbf{S^{0}}$;

$|\omega|^{\prime}=2$:
$\omega=(1^{3},3),(1^{2},3,1),(1,3,1^{2}),(3,1^{3})\;\Rightarrow\mathbf{S^{4}}$;

$|\omega|^{\prime}=9$: $\omega=(10)\;\Rightarrow\mathbf{S^{8}_{\star}}$.

$\underline{\mathsf{d=11}}$.

$|\omega|^{\prime}=0$:
$\omega=(1^{9}),(1^{7}),(1^{5}),(1^{3})\;\Rightarrow\mathbf{S^{0}}$;

$|\omega|^{\prime}=1$:
$\omega=(1^{4},2,1),(1^{3},2,1^{2}),(1^{2},2,1^{3}),(1,2,1^{4})\;\Rightarrow\mathbf{S^{4}}$;

$|\omega|^{\prime}=2$:
$\omega=(1^{4},3),(1^{3},3,1),(1^{2},3,1^{2}),(1,3,1^{3}),(3,1^{4})\;\Rightarrow\mathbf{S^{4}}$;

$\omega=(1,2,1,2,1)\;\Rightarrow\mathbf{S^{6}}$;

$|\omega|^{\prime}=3$:
$\omega=(1,2,1,3),(3,1,2,1)\;\Rightarrow\mathbf{S^{6}_{\star}}$;

$|\omega|^{\prime}=4$: $\omega=(3,1,3)\;\Rightarrow\mathbf{S^{6}_{\star}}$;

$|\omega|^{\prime}=10$: $\omega=(11)\;\Rightarrow\mathbf{S^{9}_{\star}}$.

$\underline{\mathsf{d=12}}$.

$|\omega|^{\prime}=0$:
$\omega=(1^{10}),(1^{8}),(1^{6}),(1^{4}),(1^{2})\;\Rightarrow\mathbf{S^{0}}$;

$|\omega|^{\prime}=1$:
$\omega=(1^{5},2,1),(1^{4},2,1^{2}),(1^{3},2,1^{3}),(1^{2},2,1^{4}),(1,2,1^{5})\;\Rightarrow\mathbf{S^{4}}$;

$\omega=(1,2,1)\;\Rightarrow\mathbf{S^{6}}$;

$\omega=(2)\;\Rightarrow\mathbf{S^{0}}$;

$|\omega|^{\prime}=2$:
$\omega=(1^{5},3),(1^{4},3,1),(1^{3},3,1^{2}),(1^{2},3,1^{3}),(1,3,1^{4}),(3,1^{5})\;\Rightarrow\mathbf{S^{4}}$;

$\omega=(1^{2},2,1,2,1),(1,2,1^{2},2,1),(1,2,1,2,1^{2}),(1,3),(3,1)\;\Rightarrow\mathbf{S^{6}}$;

$\omega=(1,2^{2},1)\;\Rightarrow\mathbf{S^{8}}$;

$|\omega|^{\prime}=3$:
$\omega=(1^{2},2,1,3),(1,2,1^{2},3),(1,2,1,3,1),(1,3,1,2,1),(3,1^{2},2,1),$

$(3,1,2,1^{2}),(4)\;\Rightarrow\mathbf{S^{6}_{\star}}$;

$\omega=(1,2,3),(3,2,1),(1,4,1)\;\Rightarrow\mathbf{S^{8}_{\star}}$;

$|\omega|^{\prime}=4$:
$\omega=(1,3,1,3),(3,1,3,1)\;\Rightarrow\mathbf{S^{6}_{\star}}$;

$\omega=(1,5),(5,1),(3^{2})\;\Rightarrow\mathbf{S^{8}_{\star}}$;

$|\omega|^{\prime}=5$: $\omega=(6)\;\Rightarrow\mathbf{S^{8}_{\star}}$;

$|\omega|^{\prime}=11$: $\omega=(12)\;\Rightarrow\mathbf{S^{10}_{\star}}$.

$\underline{\mathsf{d=13}}$.

$|\omega|^{\prime}=0$:
$\omega=(1^{11}),(1^{9}),(1^{7}),(1^{5}),(1^{3})\;\Rightarrow\mathbf{S^{0}}$;

$|\omega|^{\prime}=1$:
$\omega=(1^{6},2,1),(1^{5},2,1^{2}),(1^{4},2,1^{3}),(1^{3},2,1^{4}),(1^{2},2,1^{5}),(1,2,1^{6})\;\Rightarrow\mathbf{S^{4}}$;

$\omega=(1^{2},2,1),(1,2,1^{2})\;\Rightarrow\mathbf{S^{6}}$;

$|\omega|^{\prime}=2$:
$\omega=(1^{6},3),(1^{5},3,1),(1^{4},3,1^{2}),(1^{2},1,3,1^{3}),(1^{2},3,1^{4}),(1,3,1^{5}),(3,1^{6})\;\Rightarrow\mathbf{S^{4}}$;

$\omega=(1^{2},3),(1,3,1),(3,1^{2}),(1^{3},2,1,2,1),(1^{2},2,1^{2},2,1),(1^{2},2,1,2,1^{2}),(1,2,1^{3},2,1),\hfill\break(1,2,1^{2},2,1^{2}),(1,2,1,2,1^{3})\;\Rightarrow\mathbf{S^{6}}$;

$\omega=(1^{2},2^{2},1),(1,2^{2},1^{2})\;\Rightarrow\mathbf{S^{8}}$;

$|\omega|^{\prime}=3$:
$\omega=(1^{3},2,1,3),(1^{2},2,1^{2},3),(1^{2},2,1,3,1),(1^{2},3,1,2,1),(1,2,1^{2},3,1),\hfill\break(1,2,1,3,1^{2}),(1,3,1^{2},2,1),(1,3,1,2,1^{2}),(3,1^{2},2,1^{2}),(3,1,2,1^{3})\;\Rightarrow\mathbf{S^{6}_{\star}}$;

$\omega=(1^{2},2,3),(1,2,3,1),(1,3,2,1),(3,2,1^{2}),(1^{2},4,1),(1,4,1^{2})\;\Rightarrow\mathbf{S^{8}_{\star}}$;

$|\omega|^{\prime}=4$:
$\omega=(1^{2},3,1,3),(1,3,1,3,1),(3,1,3,1^{2}),\;\Rightarrow\mathbf{S^{6}_{\star}}$;

$\omega=(1,3^{2}),(3^{2},1),(1^{2},5),(1,5,1),(5,1^{2})\;\Rightarrow\mathbf{S^{8}_{\star}}$;

$|\omega|^{\prime}=12$: $\omega=(13)\;\Rightarrow\mathbf{S^{11}_{\star}}$.
$\diamondsuit$

Acknowledgment: The author is grateful to the Department of Mathematics of
Massachusetts Institute of Technology for many years of hospitality.

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