$\mathbb{CP}^2$-stable classification of $4$-manifolds with finite fundamental group
Daniel Kasprowski Peter Teichner
math.GT 57N13


^2-stable classification of 4-manifolds with finite π_1]^2-stable classification of 4-manifolds with finite fundamental group


D. Kasprowski]Daniel Kasprowski
Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut,Endenicher Allee 60, 53115 Bonn, Germany
kasprowski@uni-bonn.de

P. Teichner]Peter Teichner
Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
teichner@mpim-bonn.mpg.de


[2010]57N13




We show that two closed, connected 4-manifolds with finite fundamental groups are ^2-stably homeomorphic if and only if their quadratic 2-types are stably isomorphic and their Kirby-Siebenmann invariants agree.

[
    [
    December 30, 2023
=====================



§ INTRODUCTION

Two 4-manifolds are ^2-stably homeomorphic if they become homeomorphic after taking connected sum with finitely many copies of ^2, where we allow different numbers of copies for
the two manifolds. If the manifolds are orientable, we allow connected sums with both orientations of ^2. We will show that the ^2-stable classification for closed connected 4-manifolds with finite fundamental group is determined by their (stable) quadratic 2-type. 

The quadratic 2-type Q_M consists of the Postnikov 2-type of the 4-manifold M, determined by π_1M, π_2M (as a π_1M-module) and the k-invariant in H^3(π_1M;π_2M), together with the orientation character w_1M: π_1M→{± 1} and the equivariant intersection form λ_M: π_2M ×π_2M [π_1M]:

    Q_M:=(π_1M,w_1M,π_2M,k_M,λ_M).

We say that the quadratic 2-types of two manifolds are stably isomorphic if they become isomorphic after finitely many
stabilizations as follows:

    Q_M ↦ Q_M#±^2≅ (π_1M,w_1M,π_2M ⊕[π_1M],(k_M,0), λ_M ⊥⟨± 1 ⟩ )

For a manifold M with an isomorphism π_1M ≅π, we pick a continuous map c M→ Bπ inducing the isomorphism on π_1. Then c is uniquely determined up to based homotopy by π_1(c).

	
Let M_1, M_2 be closed, connected 4-manifolds with finite fundamental group π, orientation character w:π→{± 1} and equal Kirby-Siebenmann invariants. Then the following are equivalent:
	
		
  *  c_*[M_1]=c_*[M_2]∈ H_4(π;^w)/±(π);
		
  *  M_1 and M_2 are ^2-stably homeomorphic;
		
  *  the quadratic 2-types Q_M_1 and Q_M_2 are stably isomorphic.
	


	Since stably homeomorphic smooth 4-manifolds are stably diffeomorphic, <ref> remains true in the smooth category (where Kirby-Siebenmann vanishes). In the topological case, one could also allow taking connected sums with Freedman's manifold *^2. Then the Kirby-Siebenmann invariant can be changed and <ref> has a version where one does not have to control it. 

Since ^2#^2#(-^2)≅^2#(S^2× S^2), two manifolds are ^2-stably homemorphic if they are even stably homeomorphic, i.e. homeomorphic after taking connected sum with finitely many copies of S^2× S^2.

If follows from Kreck's modified surgery <cit.> that (<ref>) and (<ref>) are equivalent for arbitrary fundamental groups, see also <cit.>. Moreover, (<ref>) implies (<ref>) by definition and hence the interesting implication for us is (<ref>)⇒(<ref>). 

In <cit.> we showed that already the quadruple (π_1,π_2,k_,w_1) determines the ^2-stable classification, for groups that have one end or are torsion-free. 

We also described two 4-manifolds of the form lens space times circle which both have  fundamental group /n× and π_2=0 but are not ^2-stably homeomorphic <cit.>. Hence for infinite groups (with two ends and torsion), the implication (<ref>)⇒(<ref>) is false in general.

The following question remains open, compare <cit.>, even though we show there that the answer is “no” if H_4(π; ^w) is annihilated by 4 or 6.


Are there finite groups such that the equivariant intersection form λ is needed in <ref> .



Our proof is based on the following results on finite Poincaré complexes. Let X be a finite connected Poincaré 4-complex with finite fundamental group π, orientation character w and chosen fundamental class [X]∈ H_4(X;^w)≅. If B is a 3-coconnected CW-complex then X is B-polarized if it is equipped with a 3-equivalence f X→ B. The set of 4-dimensional B-polarized Poincaré complexes with orientation character w:π_1B→{± 1}, up to homotopy equivalence over B, is denoted by _4^(B,w). 

Assume that _4^(B,w) is non-empty and that π:=π_1B is finite. Then there is an exact sequence

    0→(^w⊗_πΓ(π_2(B)))→_4^(B,w)→/|π| ×(π_2(B)).


Hambleton-Kreck <cit.> prove this result in the oriented case w=0 but we need the above generalization to the non-orientable case. The general case as well as <ref> below were obtained by the second author in his PhD-thesis <cit.> but have not yet been published, so we'll review them in this paper.

For a hermitian form λ on π_2 B, let _4^(B,w,λ)
be the subset of _4^(B,w) of B-polarized Poincaré complexes (X,f) such that λ is mapped to the intersection form λ_X via f^*.


	
	Assume that _4^(B,w,λ) is non-empty. Then there  is a bijection

    _4^(B,w,λ)⟷(^w⊗_πΓ(π_2B)).




	If (^w⊗_πΓ(π_2X))=0, the quadratic 2-type Q_X of X determines the homotopy type of X. 


In the orientable case, the above torsion group vanishes if π has cyclic or quaternion 2-Sylow-subgroups <cit.> whereas in the non-orientable case, the following is a simple non-vanishing result:


	
	Let (X,w) be a Poincaré 4-complex with fundamental group π. If π_2 (X) splits off a free module and there exists an element g∈π with g^2=1 and w(g)= -1, then (^w⊗_πΓ(π_2X))≠ 0. 


	This applies to simple 4-manifolds such as X=^4#k^2 and X=(^2× S^2)#k^2 for k≥ 1. In these cases the quadratic 2-type still determines the homotopy type by <cit.> when we restrict to manifolds instead of Poincaré complexes. 
	
	Considering X=^4#^2 we have π_2(X)≅[/2] and by the proof of <ref> we have (^w⊗_[/2]Γ([/2]))≅/2. We will show in <ref> that there is a Poincaré complex with the same quadratic 2-type as X which is not homotopy equivalent to X. By the above this implies that it cannot be homotopy equivalent to a manifold. 


For X=^2× S^2 we have that π_2(X)≅⊕^w and thus Γ(π_2(X))≅⊕⊕^w and ^w⊗Γ(π_2(X))≅/2⊕/2⊕. Hence up to homotopy equivalence there are at most four Poincaré complexes with the same quadratic 2-type as X. For manifolds this case was studied by Kim, Kojima and Raymond <cit.> and Hambleton and Kreck together with the second author <cit.>. Among the four homotopy types of Poincaré complexes there are three that are realized by manifolds. These are distinguished by their  intersection form on /2-homology and either a /4-valued quadratic refinement of the equivariant intersection form <cit.> or a /8-valued Arf invariant <cit.>. Hambleton and Milgram <cit.> showed that there is a Poincaré complex with this quadratic 2-type which is not homotopy equivalent to a manifold. Its Spivak normal fibration admits no reduction to a vector bundle <cit.>. Hence up to homotopy equivalence there are precisely four Poincaré with the quadratic 2-type of ^2× S^2.



We will recall the necessary background on the equivariant intersection form and its connection to Whitehead's quadratic functor in <ref>. In <ref>, we will then prove the statements from the introduction. 



Acknowledgements: The authors thank Matthias Kreck for insightful conversations and Mark Powell for helpful comments on a previous version - this all happened at the Max Planck Institute for Mathematics in Bonn. The first author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - GZ 2047/1, Projekt-ID 390685813.



§ BACKGROUND
 
Let X be a finite connected Poincaré 4-complex with finite fundamental group π, orientation character w:π_1X→{± 1} and chosen fundamental class [X]∈ H_4(X;^w)≅. Here the coefficient group ^w is  viewed as a π-module with action gz=w(g)z

We have an involution on π given by g↦g̅:=w(g)g^-1 and we use this involution to consider a right π-module as a left π-module if necessary. We fix a point x_0∈ X and a lift x_0∈ X. We abbreviate π_2(X,x_0) by π_2(X) and write H_*(-) for H_*(-;). Using the isomorphism H^2_cs( X)H_2( X)π_2( X)π_2(X), we can view the equivariant intersection form as a pairing on π_2X, or, as we prefer in this paper, as a pairing H^2_cs( X) × H^2_cs( X)→π.
It is given by 

    λ_X(α,β)=λ_X(β)(α)=∑_g∈π(gα∪β)∩[X])g^-1=∑_g∈πgα∩ ((β∩[X]))g^-1.

The form λ_X is hermitian with respect to our involution: For h_i∈π we have

    λ_X(h_1α,h_2β)   =∑_g∈π((gh_1α∪ h_2β)∩[X])g^-1
       =∑_g∈π((h_2^-1gh_1α∪β)∩ w(h_2)[X])g^-1
       =∑_g∈π((gα∪β)∩ w(h_2)[X])h_1g^-1h_2^-1
       =h_1λ(α,β)w(h_2)h_2^-1 = h_1λ(α,β)h̅_2

and λ_X(α,β)=λ_X(β,α).

The 2-type B of X is the 2nd stage of a Postnikov tower for X. This means that there is a 3-equivalence f X→ B and B fibers over K(π,1) with fiber K(π_2(X),2). Such fibrations are classified by the unique obstruction k_X for finding a section.

    k_X∈ H^3(K(π,1);π_2(K(π_2(X),2)))=H^3(π;π_2(X))

is by definition the k-invariant of X.
The quadratic 2-type of X is

    Q_X:=(π,w,π_2(X),k_X, λ_X).



	Let A be an abelian group. Then Γ(A) is the abelian group with generators v(a)  for all a∈ A and the following relations:
	 
    {v(-a)-v(a)| a∈ A}   and

	 
    {v(a+b+c)-v(a+b)-v(b+c)-v(c+a)+v(a)+v(b)+v(c)| a,b,c∈ A}.




	
	If A is free abelian with basis B, then Γ(A) is free abelian with basis {v(b), v(b+b')-v(b)-v(b')| b,b'∈ B}. Another important case is Γ(/2)≅/4. 


Let Y be a simply-connected CW-complex and let η S^3→ S^2 be the Hopf map. It induces a map ηΓ(π_2(Y))→π_3(Y) given by v(α)↦α∘η.


	For every simply-connected CW-complex Y we have Whitehead's exact sequence
	
    …π_4(Y)→ H_4(Y)Γ(π_2(Y))π_3(Y)→ H_3(Y;Z)→ 0.



We will now explain how the equivariant intersection form λ_X for a finite Poincaré 4-complex with finite fundamental group can be viewed as an element of Γ(π_2(X)). This is well-known, but for convenience and to fix notation we sketch the argument here.

Let A free abelian. An element ϕ∈_(_(A,),),A) is symmetric if for all f,g∈_(A,) we have f(ϕ(g))=g(ϕ(f)). And we denote the subgroup of symmetric homomorphisms by _(_(A,),A). Let Y be a finite simply-connected CW-complex with H_2(Y) free abelian. We define _(H^2(Y),H_2(Y)) as _(_(H_2(Y),),H_2(Y)) using the canonical isomorphism H^2(Y)→_(H_2(Y),).


	
	Let Y be a finite simply-connected CW-complex with H_2(Y) free abelian. There is an isomorphism
	
    ζ_(H^2(Y),H_2(Y))→Γ(π_2(Y))

	such that for every class z∈ H_4(Y) we have ζ(-∩ z)=b_4(z).


Let X be a finite Poincaré 4-complex with finite fundamental group π and orientation character w. Note that by Poincaré duality
H_2( X)≅_(H_2( X),)
and thus H_2( X) is free abelian. Let (H^2( X)) denote the hermitian forms on H^2( X). We have a map

    (H^2( X))→_(H^2( X),_(H^2( X), )

given by

    λ↦ (β↦(α↦_0λ(α,β)) ),

where _0π→ takes the coefficient at the identity element. This map is injective with image the π-linear homomorphisms, where we view _(H^2( X), ) as a left π-module with the involution on π coming from w.
After identifying _(H^2( X), ) with H_2( X), the intersection form λ_X∈(H^2( X)) maps to -∩ [ X]. Therefore, we consider the intersection form as an element of Γ(π_2(X)) as follows.


	
	Let X be a finite Poincaré 4-complex with finite fundamental group π and orientation character w. Then we can consider the element 
    _X:=ζ(-∩ [ X])=b_4([ X])∈Γ(π_2(X)).




	
	Let X be a finite Poincaré 4-complex with finite fundamental group and 2-type B. Let f X→ B be a 3-equivalence. Then there is an isomorphism Γ(π_2(X))≅ H_4( B,) mapping _X to f_*[ X].


	Since B is the 2-type of X, we have π_3(B)=π_4(B)=0. Consider the diagram:
	
    π_4(X)[r][d]   H_4( X)[r][d]^ f_*   Γ(π_2(X,x_0))[r][d]^≅   π_3(X)[r][d]   0
    
    		0[r]   H_4( B)[r]^-≅   Γ(π_2(B,f(x_0)))[r]   0

	From the commutativity of the diagram and <ref> it follows that _X is mapped to f_*[ X] under the composition of the two isomorphisms.



	
Let X=K∪_α D^4 be a finite Poincaré complex with finite fundamental group, with orientation character w and with 3-skeleton K. Under the map Γ(π_2X)=Γ(π_2K)→π_3(K) the element _X is mapped to N^wα (up to a unit), where N^w=∑_g∈πw(g)g∈π.


We can assume that π_2X is nontrivial since otherwise X is homotopy equivalent to either a sphere S^4 or a projective plane ^4.
The Whitehead sequences for X and K fit into the following commutative diagram:


    H_4( X)[r]^-b_4   Γ(π_2X)[r]^-η   π_3(X)[r]   H_3( X)[r]   0
    	
    	0=H_4( K)[r][u]   Γ(π_2K)@=[u][r]^-η   π_3(K)[r][u]   H_3( K)[u][r]   0
          π_4(X,K)[u][r]^-≅   H_4( X, K)≅π[u]   
          π_4(X)[u][r]   H_4( X)≅[u]^N^w

There is a choice of isomorphism π≅π_4(X,K) such that 1 is mapped to α under π_4(X,K)→π_3(K). Since λ_X comes from H_4( X), the image η(λ_X)∈π_3(K) lies in the kernel of the maps to π_3(X) and H_3( K). Therefore, by a quick diagram chase, η(_X) is a multiple of N^wα and since (-∩ [ X])∈_(H^2( X),H_2( X,)) is an isomorphism and hence primitive, it is mapped to N^wα up to a unit.	


In <ref> we will need the following results about the transfer map.

Let U≤π have finite index. Let X be a CW-complex with fundamental group π and let X denote the finite covering of X with fundamental group U. Then X and X have the same universal covering and let C_* denote its cellular π-chain complex. We can consider it as a right π-module using the involution on π. If M is a π-module we can restrict the action to U and consider the projection

    C_*⊗_ UMC_*⊗_πM

inducing

    p_* H_*(X;M)→ H_*(X;M).

On the chain level we obtain a map in the other direction by

    C_*⊗_πM→ C_*⊗_ UM,   c⊗ m↦∑_Ug∈ U\πcg^-1⊗ gm.

This map induces a map

    _* H_*(X;M)→ H_*(X;M)

and similarly one constructs a map

    ^* H^*(X;M)→ H^*(X;M).



These transfer maps have the following properties:

	
  * p_*∘_* and ^*∘ p^* are multiplication by the index [U:π].
	
  * If U≤π is normal, then π/U acts on X and also on H_*(X;M) and H^*(X;M). In this case, tr_*∘ p_* and p^*∘ tr^* are multiplication by ∑_g∈π/Ug.
	
  * If π acts trivially on a commutative ring R, then tr^* is an H^*(X;R)-module homomorphism, i.e.
	
    tr^*(p^*(x)∪ y)=x∪ tr^*(y)

	for all x∈ H^*(X;R) and y∈ H^*(X;R).
	
  * The transfer commutes with Kronecker-products and Steenrod-squares:
	
    ⟨ tr^*(y),b⟩=⟨ y,tr_*b⟩∈ R

	and
	
    Sq^i(tr^*(y))=tr^*(Sq^i(y))∈ H^*(X;/2).




(1) and (2) follow directly from the definitions. For (3) and (4) one uses that for trivial modules tr_* and tr^* are induced by a stable map trΣ^∞ X→Σ^∞X of suspension spectra, see <cit.>.











§ THE CP2-STABLE CLASSIFICATION


In this section we will recall the proof of <cit.> and then prove <ref>. At the end of the section we will prove <ref> and <ref> which was mentioned in the introduction.


Let X, Y be finite Poincaré 4-complexes with finite fundamental group π. Assume that X and Y have isomorphic quadratic 2-types:

    Q_X=(π, w,π_2(X),k,λ_X)≅ Q_Y.

In the following we will always assume that π_2(X)≠ 0 since otherwise X is homotopy equivalent to either a sphere S^4 or a projective plane ^4.
Let B be the 2-type of X and let f X→ B be a 3-equivalence.
From the results of <ref>, in particular <ref>, it follows that there is a 3-equivalence g Y→ B such that g_*[ Y]= f_*[ X]. 

If in this situation there is a map h X→ Y making the diagram

    X[rr]^h[rd]_f      Y[ld]^g
       B

homotopy commutative, then h is an isomorphism on π_1, π_2. By <ref> there is an isomorphism Γ(π_2(Y))≅ H_4( B) mapping λ_Y to g_*[ Y] and by assumption, g_*[ Y]= f_* [ X]= g_* h_* [ X]. Since H_4( Y)≅, H_4( B) is torsion-free and g_*[ Y] is non-trivial, this implies h_*[ X]=[ Y]. Thus by Poincaré duality h is an isomorphism on homology of universal coverings in all degrees and hence h is a homotopy equivalence.  


	
	A map h as above exists if and only if f_*[X]=g_*[Y]∈ H_4(B;^w).


	This result is proved by Hambleton and Kreck <cit.> in the oriented case. Since the proof in the non-oriented case works the same we will not give any details here. However, we want to point out the following two things.
	
	Firstly, there is a minor mistake at the beginning of the proof of <cit.>, where the authors construct B by attaching cells of dimension 4 and higher to Y, i.e. g Y→ B is the inclusion. Then they say that they want to extend
	
    X^(3)B^(3)=Y^(3)⊆ Y

	over the 4-cell of X.
	
	In fact, using the vanishing of obstructions in cohomology they can only extend the map X^(2)→ Y, because they might have to change the given map on the 3-cells in order to extend over the 4-cell. But also such an extension h suffices for the proof because the given (trivial) homotopy between f and g∘ h on X^(2)× I extends to X× I using π_3(B)=π_4(B)=0.
	
	Secondly, to get the non-oriented case, note that Hambleton and Kreck show for w≡ 1 that the obstruction in
	
    H^4(X;π_3Y)≅ H_0(X;π_3Yπ⊗^w)≅π_3Yπ⊗^w≅π_4(B,Y)π⊗^w≅ H_4(B,Y;^w)

	is given by the image of f_*[X] under H_4(B;^w)→ H_4(B,Y;^w) and therefore vanishes if f_*[X]=g_*[Y]. All these isomorphisms continue to hold for non-trivial w.



Let X be a finite Poincaré complex with orientation character w, finite fundamental group π, intersection form λ_X and with a fixed 3-equivalence f X→ B, where B is 3-coconnected.  

Let λ be a hermitian form on π_2(B).
Recall that _4^(B,w,λ) denotes the set of 4-dimensional B-polarized Poincaré complexes (Y,g) with orientation character w, such that λ maps to λ_Y, up to homotopy equivalence over B. Assume (X,f)∈_4^(B,w,λ). 
By <ref>, sending (Y,g) to g_*[Y]-f_*[X] defines an injection

    _4^(B,w,λ)→ H_4^w(B):=H_4(B;^w).


We want to compute the image of this map. Under the isomorphism H_4( B)≅Γπ_2(B), the element f_*[ X] is determined by λ=f^*λ_X by <ref>. Furthermore, recall that _*[X]=[ X]. Therefore |π|(g_*[Y]-f_*[X])=p_*_*(g_*[Y]-f_*[X])=0, i.e. the above image lies in the torsion subgroup of H_4^w(B).

Write X as K∪_α D^4, where K is a 3-complex. Since K is a 3-dimensional complex, H_3^w(K) is a subgroup of a free abelian group and thus torsion-free. Furthermore, H^1(X)≅ H^1(π)≅_(π,)=0. Using this, from the exact sequence

    0[r]    H_4^w(X)@=[d][r]    H_4^w(X,K)@=[d][r]   H_3^w(K)[r]   H_3^w(X)[d]^≅_^-1
                H^1(X)

we obtain H_3^w(K)=0. Therefore, H_4^w(B)→ H_4^w(B,K) is an isomorphism. Since B is 3-coconnected, π_4(B,K)→π_3(K) is an isomorphism. Combining these, we obtain the very useful isomorphism

    H_4^w(B)H_4^w(B,K)π_4(B,K)⊗_π^wπ_3(K)⊗_π^w.

Consider the commutative diagram

    H^w_4(X)[r][d]^f_*   H^w_4(X,K)[d]^f_*   π_4(X,K)⊗_π^w[l][d]^f_*[dr]^∂   
    
    	H^w_4(B)[r]^≅   H^w(B,K)   π_4(B,K)⊗_π^w[l]_≅[r]^≅_∂   π_3(K)⊗_π^w

and view the top cell of X as an element in π_4(X,K). Then it has boundary α and its image in H^w_4(X,K) is also the image of [X]∈ H_4^w(X). Hence the composition of the horizontal isomorphisms (<ref>) maps f_*[X] to [α]⊗ 1.

Finally, we use Whitehead sequences for X and K:

    π_4( X)[r]^-0   H_4( X)[r]^-b_4   Γ(π_2(X))[r]^-η   π_3 (X)[r]   H_3( X)@=[r]   0
       0[u][r]   Γ(π_2(K))@=[u][r]^-η   π_3 (K)[u][r]   H_3( K)[u][r]   0
             π_4( X, K)[u][r]^≅   H_4( X, K)[r]^-≅[u]   π
             π_4( X)[u]^0[r]^0   H_4( X)[u]

The image of the composition H_4( X)Γ(π_2 (X))=Γ(π_2(K))π_3(K) lies in the kernel of the map to π_3(X) and thus inside π_4( X, K)≅ H_4( X, K)≅π. We see that it fits into a short exact sequence

    0→ H_4( X)→ H_4( X, K)→ H_3( K)→ 0

and we obtain the following commutative diagram of short exact sequences:

    0[r]   H_4( X)[r][d]^b_4   π[r][d]   H_3( K)[r]@=[d]   0
    
    	0[r]   Γ(π_2(K))[r]   π_3(K)[r]   H_3( K)[r]   0

The fundamental class [ X] maps downward to λ_X∈Γ(π_2X)=Γ(π_2K) and, by <ref>, to the right to ± N^w:=∑_g∈πw(g)g. Note that H_4( X)≅^w. By the upper short exact sequence above, H_3( K)≅π/N^w.
Tensoring the diagram with ^w over π we obtain the following diagram of the long exact Tor-sequences:
    @C=.7pc@R=.8pc      @=[d][r]^·|π|   @=[d]      
    
    	0[r]   _1^π(π/N^w,^w)[r]@=[d]   ^w⊗_π^w[d][r]^-N^w⊗𝕀   π⊗_π^w[r][d]   π/N^w⊗_π^w[r]@=[d]   0
    
    	0[r]   _1^π(π/N^w,^w)[r]   Γ(π_2(K))⊗_π^w[r]   π_3(K)⊗_π^w[r]   π/N^w⊗_π^w[r]   0

The top row shows that _1^π(π/N^w,^w)=0 and π/N^w⊗_π^w=/|π|. Using that π_3K⊗_π^w≅ H_4^w(B) and π_2(B)≅π_2(K), the bottom row can be identified with the short exact sequence:

    0→Γ(π_2(B))⊗_π^w H_4^w(B)/|π|→ 0

with i(λ⊗ 1)=|π|f_*[X] and q(f_*[X])=1+|π|, exactly as in the oriented case <cit.>. From here <ref> follows as in the proof of <cit.>.

Now we are nearly ready to prove the following theorem which completes the proof of <ref>. Recall that (Y,g)↦ g_*[Y]-f_*[X] defines an injection _4^(B,w,f_*λ_X)→ H_4^w(B) with image inside the torsion subgroup.

	
	For all (Y,g)∈_4^(B,w,λ) we have q(g_*[Y]-f_*[X])=0 and (Y,g)↦ i^-1(g_*[Y]-f_*[X]) defines a bijection
	
    _4^(B,w,λ)↔(Γ(π_2B)⊗_π^w).


Before proving <ref> we will first show how it implies <ref>.

It remains to show that for a given closed, connected 4-manifold M with fundamental group π and orientation character w the element c_*[M]∈ H_4(π;^w) is determined by the stable quadratic 2-type of M. Let B denote the Postnikov 2-type of M and consider the sequence M B Bπ. 
Considering the following diagram and following the construction of (<ref>), we see that the map i in (<ref>) can be identified with the canonical map H_4( B)⊗_π^w→ H_4^w(B).

    0   0   
       0[r]   Γ(π_2( K))[r]^-η   π_3 ( K)[u][r]   H_3( K)[u][r]   0
             π_4( B, K)[u]_≅[r]^≅   H_4( B, K)[u]   
             0[u][r]   H_4( B)[u]   
                0[u]

Now given another closed, connected 4-manifold N with orientation character w and a 3-equivalence g N→ B such that g^*λ_N=f^*λ_M, then f_*[M]-g_*[N] is contained in (Γ(π_2(B))⊗_π^w)≅(H_4( B)⊗_π^w) by <ref>. Since the composition B→ B Bπ is trivial, we thus have c_*f_*[M]=c_*g_*[N]∈ H_4^w(Bπ) as claimed.

For the proof of <ref> we need the following lemma.

	
	If w≡ 1 or if the 2-Sylow subgroup of π has order bigger than 2, there exists a homomorphism
	
    κΓ(π_2B)⊗_π^w→ with λ⊗ 1↦ 1.


We will first prove <ref> assuming <ref>, and then we will prove <ref>.

	We will first show that q(g_*[Y]-f_*[X])=0.
	
	In the case that w≡ 1 or the 2-Sylow subgroup of π has order more than 2, we will show that i((^w⊗_πΓ(π_2B)))=(H_4^w(B)). This proves q(g_*[Y]-f_*[X])=0 in this case. The image of (^w⊗_πΓ(π_2B)) is obviously contained in (H_4^w(B)) and we have to show the other inclusion.
	
	Let x∈(H_4^w(B)) be given. Because H_4(B)≅Γ(π_2B) is torsion free, we see that |π|x=p_*_*(x)=0. From (<ref>) and q(f_*[X])=1∈/|π| we see that there exists k∈ and y∈Γ(π_2 B)⊗_π^w such that x+kf_*[X]=i(y). It follows that
	
    |π|i(y)=|π|x+k|π|f_*[X]=ki(λ⊗ 1)

	Thus, |π|y=k(λ⊗ 1) and by <ref> we have |π|κ(y)=kκ(λ⊗ 1)=k. Therefore, |π| divides k and
	
    x=i(y-k/|π|(λ⊗ 1))∈(i).

	
	We now consider the case π=Q⋊/2 with |Q| odd and w=p_2 the projection onto /2≅{± 1}. Since f_*[X]∈ H_4^w(B) maps to a generator of /|π|≅/|Q|⊕/2 under q there is x∈/|Q| with q(f_*[X])=(x,1+2)∈/|Q|⊕/2. For any (Y,g)∈_4^(B,w,λ) there is in the same way y∈/|Q| with q(g_*[Y])=(y,1+2) and hence q(g_*[Y]-f_*[X])=(y-x,0).
	
	Taking double coverings with respect to the subgroup Q≤π we get a commutative square
	
    X[r]^f[d]   B[d]^p
    
    		X[r]^f   B

	where p_*f_*[X]=2f_*[X]. So we may conclude from the following commutative diagram that q(g_*[Y])=y.
		
    0[r]   Γ(π_2B)⊗_ Q[r]^-i[d]   H_4(B)[d]^p_*[r]^-q   /|Q|[r][d]^· 2   0
    
    			0[r]   Γ(π_2B)⊗_π^w[r]^-i   H_4^w(B)[r]^-q   /|π|[r]   0

	Since X is an orientable Poincaré complex we already know that <ref> holds for X. Therefore, 0=q(g_*[Y]-f_*[X])=y-x∈/|Q| and thus also q(g_*[Y]-f_*[X])=0.
	
	By the above, the map _4^(B,w,λ)→(Γ(π_2B)⊗_π^w) is well defined and we already now that it is injective. Showing surjectivity is easy since we can use <cit.> to reattach the top cell of X as follows. Recall that X=K∪_α D^4. For a given element β∈(Γ(π_2B)⊗_π^w) pick a preimage β∈Γ(π_2B)⊆π_3(K) and attach a 4-cell to K along a representative of β+α. The resulting complex X_β has the same intersection form as X and also H_3(X_β)=0 because β maps trivially to H_3( K). As in <cit.>, the map f|_K extends to a 3-equivalence f_β X_β→ B such that f_β*[X_β]=f_*[X]+β.

For the proof of <ref> we need the following constructions of maps from Γ(π_2(X))=Γ(π_2(B)) to ^w.
Consider the isomorphism

    ζ^-1Γ(π_2(X))→_(H^2( X),H_2( X))

from <ref>.
Precomposing with the inverse of H^2( X) H_2( X), we get a map

    κ'Γ(π_2(X))→_(H_2( X),H_2( X)).

The construction of κ' and the following two lemmas below are due to Stefan Bauer <cit.>.

	If we let π act on _(H_2( X),H_2( X)) by gϕ=w(g)(g_*∘ϕ∘ g_*^-1), then κ' is equivariant. In particular,
	
    ∘κ'Γ(π_2(X))→^w

	is equivariant.


	If we let π act on (H^2( X),H_2( X)) by gψ=g_*∘ψ∘ g^*, then by naturality ζ is equivariant.
	We have
	
    g_*(g^*α∩ [ X])=α∩ g_*[ X]=w(g)(α∩[ X])

	and thus g^*∘^-1=w(g)(^-1∘ g^-1_*). This proves the lemma since κ'(x)=ζ(x)∘^-1.


	
	When we view λ_X as an element of Γ(π_2(X)) as in <ref>, then
	
    κ'(λ_X)=𝕀_H_2( X).

	

	Recall that λ_X is the image of -∩ [ X] under ζ. Thus, precomposing with the inverse of =-∩[ X] gives the identity on H_2( X).


A central involution in a group G is an element of order 2 in the center of G.

	
	Let G be a 2-group, w G→/2. If w≡ 1 or |G|>2, there exists a central involution τ∈ G with w(τ)=1∈{± 1}.


	Since G is a 2-group there exists a central involution. We are done if w(τ)=1. Otherwise the map w splits and since τ is central, we have G=⟨τ⟩×(w). Since |G|>2 there again exists a central involution τ'∈ (w) and this is still central in G=⟨τ⟩×(w).


Let τ_*_Z(H_2( X),H_2( X))→_Z(H_2( X),H_2( X)) be given by precomposition with the action of τ on X.

	
	The map
	
    ∘τ_*∘κ'Γ(π_2(X))→^w

	is equivariant and the element λ_X is mapped to -2. 


	Since κ' and the trace are equivariant, we have to show that τ_* is equivariant. This is the case, because the group element τ is central.
	Since κ'(λ_X)=𝕀_H_2( X) by <ref>, we have τ_*(κ'(λ_X))=τ H_2( X)→ H_2( X; ). Since w(τ)=1 and τ acts fixed point free on X, the Lefschetz formula yields
	
    0=(τ_H_0( X))+(τ_H_4( X))+(τ_H_2( X))=2+(τ_*(κ'(λ_X)))

	and therefore (τ_*(κ'(λ_X)))=-2.

We want to show that the image of ∘τ_*∘κ' is always even. For this we need the following lemma. This is well-known and for example follows from <cit.>. For the reader's convenience we give a proof.

	
	Let τ∈π have order 2. Then the symmetric bilinear form
	
    H^2(X;/2)⊗ H^2(X;/2)→/2,   x⊗ y↦⟨τ x∪ y,[X]⟩

	is even, i.e. ⟨τ x, x⟩=0 for all x∈ H^2( X;/2).


	Let qX→X:=X/τ be the double covering. Then q^*∘^*(x)=x+τ x and
	
    ⟨ q^*∘^*(x)∪ x,[ X]⟩   =⟨^*(q^*∘^*(x)∪ x),[ X]⟩
       =⟨^*(x)∪^*(x),[X]⟩
       =⟨ Sq^2(^*(x)),[X]⟩
       =⟨ Sq^2(x),_*[X]⟩
       =⟨ x∪ x,[ X]⟩.

	Here we used the identities from <ref> (with p=q).
	It follows that 
    ⟨ x∪τ x,[ X]⟩=⟨ x∪ (q^*∘^*(x)+x),[ X]⟩=2⟨ x∪ x,[ X]⟩=0

	for all x∈ H^2( X;/2).


	
	The image of ∘τ_*∘κ' is contained in 2^w.


	The trace fits into the following commutative diagram:
	
    _(H_2( X),H_2( X))[rr]^      
       H^2( X)⊗ H_2( X)[lu]^ϕ_1[ru]^ϕ_2

	where ϕ_1 is the slant map and ϕ_2 the evaluation, i.e. ϕ_1(α⊗ x):=(y↦ (α∩ y)x) and ϕ_2(α⊗ x):=α∩ x.
	
	Let {β_i} be a -basis of H_2( X). Every element in H^2( X)⊗ H_2( X) can be written as ∑_i,jλ_i,jβ_i⊗(β_j∩[ X]) for some λ_i,j∈. The map
	
    Ψ:=ϕ_1(∑_i,jλ_i,jβ_i⊗(β_j∩[ X]))∘ H^2( X)→ H_2( X)

	is given by Ψ(α)=∑_i,jλ_i,j((α∪β_i)∩[ X])(β_j∩[ X]). Recall that it is symmetric if α'∩Ψ(α)=α∩Ψ(α'). Let {β_i^*} be the basis dual to {β_i} in the sense that (β_i^*∪β_j)∩[ X]=δ_ij. We see that if Ψ is symmetric, then
	
    λ_i',j'   =∑_i,jλ_i,j((β_i'^*∪β_i)∩[ X])((β^*_j'∪β_j)∩[ X])
       =β_j'^*∩Ψ(β^*_i')
       =β_i'^*∩Ψ(β^*_j')
       =∑_i,jλ_i,j((β_j'^*∪β_i)∩[ X])((β^*_i'∪β_j)∩[ X])
       =λ_j',i'

	One easily computes
	
    τ_*(ϕ_1(∑_i,jλ_i,jβ_i⊗(β_j∩[ X])))=ϕ_1(∑_i,jλ_i,jτβ_i⊗(β_j∩[ X]))

	and using that τ has order two and w(τ)=1, we get
	
    (τβ_i∪β_j)∩[ X]=(τ^2β_i∪τβ_j)∩ w(τ)[ X]=(β_i∪τβ_j)∩[ X].

	Let Ψ be symmetric. By <ref> we then get
	
    ϕ_2(∑_i,jλ_i,jτβ_i⊗(β_j∩[ X]))   =∑_i,jλ_i,j(τβ_i∪β_j)∩[ X]
       =∑_i(τβ_i∪β_i)∩[ X]+2∑_i<jλ_i,j(τβ_i∪β_j)∩[ X]
       =2∑_i(β_i∪β_i)∩[ X]+2∑_i<jλ_i,j(τβ_i∪β_j)∩[ X]

	This implies the lemma since by definition of κ' every element in its image is coming from _(H^2( X),H_2( X)).



	Since ^w⊗_π^w=, f_*π_2(X)→π_2(B) is an isomorphism and f^*λ=λ_X, it suffices to show that there exists a π-homomorphism
	
    κΓ(π_2(X))→^w

	with κ(λ_X)=1.
	
	We have rank(H_2( X))=χ(X)-2=|π|χ(X)-2, where χ denotes the Euler characteristic. Thus it suffices to show that there exist homomorphisms κ_iΓ(π_2(X))→^w such that:
	
		
  * κ_1(λ_X)=|π|,
		
  * κ_2(λ_X)=rank(H_2( X)) and
		
  * κ_3(λ_X) is odd,
	
	because if κ_3(λ_X)=2n+1, then for κ=κ_3+nκ_2-nχ(X)κ_1 we have κ(λ_X)=1.
	(1):
	For a π-module M let M^π_w denote the submodule consisting of those objects m with gm=w(g)m for all g∈π. We have λ_X∈Γ(π_2(X))^π_w. Any -homomorphism Γ(π_2(X))^π_w→^w is already a π-homomorphism, since π acts on both sides via w. Since X is a finite 4-dimensional Poincaré complex with finite fundamental group, π_2(X) is free abelian. By <ref>, Γ(π_2(X)) is therefore free abelian. Capping with [ X] gives an isomorphism H^2( X)→ H_2( X) and hence λ_X∈Γ(π_2(X))^π_w is primitive. Therefore, there exists a - and thus π-linear map
	
    dΓ(π_2(X))^π_w→^w

	with d(λ_X)=1. We define κ_1 as the composition
	
    κ_1Γ(π_2(X))(Γ(π_2(X))^w)^π^w.

	(2):
	Define κ_2:=∘κ'. By <ref>, 
    κ_2(λ_X)=(_H_2( X))=rank(H_2( X)).

	(3):
	Let G≤π be a 2-Sylow subgroup and p X'→ X the corresponding covering. Then the map
	
    tr_G^πΓ(π_2X)⊗_π^w→Γ(π_2X)⊗_ G^p^*w

	has the property that 
	
    tr_G^π(λ_X⊗ 1)=∑_k∈ G\π(k(λ_X)⊗ k· 1)=[π:G](λ_X⊗ 1),

	where the last equation follows from k(λ_X) = w(k)·λ_X ∈Γ(π_2X) for all k∈π.
	Since [π:G] is odd and λ_X=λ_X', we can therefore assume that π is a 2-group by composing κ_3 for G with tr_G^π.
	Note that p^* H^1(X;/2)→ H^1(X';/2) is injective and thus p^*w is non trivial if and only if w is non trivial.
	
	Let κ' be the homomorphism from (<ref>). By <ref>, the image of ∘τ_*∘κ' is always even and we can define κ_3:=-1/2(∘τ_*∘κ'). By <ref>, κ_3(λ_X)=1.



	By <cit.> we have
	
    Γ(π)≅ F⊕⊕_g∈π∖{1}, g^2=1π/π(1-g),

	for some free module F. It is an easy calculation that this implies
	
    (^w⊗_πΓ(π))≅ (/2)^r,

	where r is the number of g∈π∖{1} with g^2=1 and w(g)=-1.
	Since
	
    Γ(π_2 X)≅Γ(π)⊕Γ(M)⊕π⊗_ M,

	(^w⊗_πΓ(π)) is a summand of (^w⊗_πΓ(π_2X)).


	
	There is a finite Poincaré complex with the same quadratic 2-type as ^4#^2 which is not homotopy equivalent to ^4#^2.


	
	Let B be the Postnikov 2-type of ^4#^2. By <ref>, B-polarized Poincaré complexes up to homotopy equivalence over B are classified by the image of their fundamental class in H_4(B;^w). Hence to understand them up to homotopy equivalence, we have to study the action of all self-homotopy equivalences of B on H_4(B;^w). As π_2(B) is free, the k-invariant of B is trivial and the map B→ B/2 admits a splitting s. Using the Serre spectral sequence and the proof of <ref>, one computes	
	
	
    H_4(B;^w)   ≅^w⊗_[/2]H_4( B;)⊕ H_4(B/2;^w)
       ≅^w⊗_[/2]Γ([/2])⊕/2≅⊕/2⊕/2.

	
	By obstruction theory, for every self-homotopy equivalence ϕ of B we have ϕ∘ s≃ s. Hence ϕ acts diagonally on H_4(B;^w) with the identity on H_4(B/2;^w) and on ^w⊗_[/2]H_4( B;) by the map on π_2(B) induced by ϕ. The automorphisms of π_2(B)≅[/2] are given by multiplicaton with {± 1,± g} where g∈/2 is the generator. It follows that the action of ϕ on H_4(B;^w) is given by ± 1 and thus is the identity on the torsion subgroup. In particular, it is the identity on the torsion subgroup /2 of ^w⊗_[/2]Γ([/2])). It thus follows from <ref> that there is precisely one Poincaré complex which has the same quadratic 2-type as ^4#^2 but is not homotopy equivalent to it.

amsalpha

