On smooth manifolds with the homotopy type of a homology sphere
In this paper we study , the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of , via a map from into the quotient of by the action of the group of homotopy classes of simple self equivalences of . The map describes which bundles over can occur as normal bundles of manifolds in . We determine the image of when belongs to a certain class of homology spheres. In particular, we find conditions on elements of that guarantee they are pullbacks of normal bundles of manifolds in .
Key words and phrases:Homology sphere; Poincaré duality; -theory ; Cobordism; Spectral sequence
2010 Mathematics Subject Classification:57Q20, 55Q45
Unless otherwise stated, by a manifold we mean a smooth, oriented, closed manifold with dimension greater than or equal to . Given a simple Poincaré complex with formal dimension , a classical problem in topology is to understand the set of diffeomorphism classes of smooth manifolds in the simple homotopy type of . For such an aim, a fundamental object to study is the smooth simple structure set (see  page 125-126 for notation and details). Elements of are equivalence classes of simple homotopy equivalences from an -dimensional manifold . Two such homotopy equivalences and are said to be equivalent if there is a diffeomorphism such that is homotopic to the composition . An element of is called a simple smooth manifold structure on . Note that composition of an element in with a simple self equivalence of gives another element in , although the manifold is still the same. Hence, we need to quotient out simple self equivalences of in order to get the set of diffeomorphism classes of smooth manifolds in the simple homotopy type of . Denote the group of homotopy classes of simple self equivalences of . Then acts on by composition. The set of diffeomorphism classes of smooth manifolds in the simple homotopy type of , , is defined as the set of orbits of under the action of , i.e. .
Let denote the group of homotopy classes of maps (here, we abandon the traditional notation for simplicity). Every simple homotopy equivalence induces an automorphism on . Let denote the subgroup of that consist of automorphisms induced by the simple self equivalences of . There is a canonical action of on again given by composition. We denote by the set of orbits of under the action of .
As pointed out in  computations of and are in general difficult, so does the computation of . On the other hand, computations of and are easier in most cases as is a (generalized) cohomology group (see ). In this paper, we compare with where belongs to a certain class of Poincaré complexes.
There is a map defined by where denotes the normal bundle of (see Proposition 2.2). For a prime , by a -homology -sphere we mean a simple Poincaré complex of formal dimension such that . For a general reference to Poincaré complexes we refer to  and . Our purpose is to determine the image of for certain such homology spheres. Here we also assume that such a homology sphere admits a degree one normal map (equivalently the Spivak normal fibration has a vector bundle reduction), since otherwise the problem is trivial.
Let be an odd number and be a subset of the set of primes between and . Denote by the set of orbits in that can be represented by elements such that for each in the first Wu class of satisfy the identity (see  or ). Note that for odd and . The main object of this paper is to prove the following:
Let be an odd number and be a subset of the set of primes between and . Let with a given map be a -homology -sphere, so that is a -homology isomorphism for every prime with . Assume further that is of odd order. Then, the image of consists of orbits in that are represented by elements in the kernel of . In particular, if , then the image of is . Furthermore, if , then is surjective.
Observe that as gets larger, the image of gets smaller. In particular, if , then we do not need to make the assumption of Theorem 1.1 on the Wu classes for the primes . On the other hand, for an odd prime , being a -homology sphere implies there is no -torsion in . Hence, primes in also affect the image of .
It is well-known, due to , that a degree one normal map can be surgered to a simple homotopy equivalence if and only if the associated surgery obstruction vanishes. The main result in  states that if is of odd order, then the odd dimensional surgery obstruction groups, , vanish, i.e. every degree one normal map can be surgered to a simple homotopy equivalence. An essential step in the proof of Theorem 1.1 is that, under the stated conditions, an element in admits a degree one normal map if an only if it is in the kernel of and it has the same Wu classes as the Spivak normal fibration of for each . In particular, if , then bundles admitting degree one normal maps are completely determined by their Wu classes for . In the case when and every stable vector bundle over admits a degree one normal map. The rest is to determine the action of on , which is given by the restriction of the canonical action of .
Some examples of such come from the smooth spherical space forms. Some applications of our theorem are discussed in Section 4. Smith theory may also provide examples, although we do not mention any such example in this note.
2. Notation and Preliminaries
Let be a simple Poincaré complex with formal dimension . Given a stable vector bundle , a -manifold is a manifold whose stable normal bundles lifts to through , we refer  for more details (in  such objects are called -manifolds). We denote by the cobordism group of -dimensional -manifolds. An element of is often denoted by , where is a -dimensional manifold and is a lifting of its stable normal bundle to through , and brackets denote the homotopy class of such liftings (see  Proposition 2 for this notation). Such a map is called a normal map, and if degree of is equal to , i.e. , then it is called a degree one normal map (see  Definition 3.46.). Due to the Pontrjagin-Thom construction, the group is isomorphic to -th homotopy group of , the Thom spectra associated to .
Our primary tool is the James spectral sequence, which is a variant of the Atiyah-Hirzebruch spectral sequence (see , Section II). Let be a generalized homology theory represented by a connective spectrum, be an -orientable fibration with fiber and be a stable vector bundle. The James spectral sequence for , and has -page and converges to . In the case when is the stable homotopy, the edge homomorphism of this spectral sequence coming from the base line is as follows:
Proposition 2.1 (see  Proposition 2).
The edge homomorphism of the James spectral sequence for stable homotopy, and is a homomorphism given by
for every element .
The Atiyah-Hirzebruch spectral sequences for is isomorphic to the James spectral sequence for stable homotopy, and . This follows from the fact that is the sphere spectrum. This isomorphism is given by the Thom isomorphism (see proof of Proposition 1 in ). In this paper, we will only use this edge homomorphism of the James spectral sequence for the stable homotopy, the identity map and a given stable vector bundle . In this case is the map given by for every element . The other edge homomorphism for this spectral sequence will be denoted by , where denotes the sphere spectrum.
Recall that denotes the quotient (see Section 1). Let be the quotient map. We define a map from to as follows: Let be a smooth manifold equipped with a simple homotopy equivalence and let be the stable normal bundle of . Let be the homotopy inverse of . Then the pullback bundle defines an element in . If is the diffeomorphism class of in , we define .
is well defined.
Let be another manifold in the orbit with normal bundle , with a diffeomorphism and with a simple homotopy equivalence . Since , we have . Since is the homotopy inverse of , we have . Hence, and differ by an automorphism in as the composition is homotopic to a simple self homotopy equivalence of . By definition, in they are the same. ∎
Let be an odd prime. For a vector bundle, or in general a spherical fibration, over , there exist cohomology classes in , known as Wu classes, introduced in . We write instead of if the prime we consider is clear from the context. These classes are defined by the identity . Here, denotes the Steenrod’s reduced -th power operation and denotes the Thom isomorphism. For more details on Wu classes we refer to , Ch.19.
For each prime , let denote the negative of the Wu class of , the Spivak normal fibration of (which exists since is a finite complex, see ). Given a set of primes, let denote the subset of that consist of elements such that for each in the first Wu class of satisfies the identity (or equivalently ). Since the class is a homotopy type invariant of (see  Ch. 19), the subset is invariant under the action of . We denote the quotient of this action by . In particular, if , then .
will denote the James spectral sequence for the stable homotopy as the generalized homology theory, identity fibration and the stable bundle . The abbreviations JSS and AHSS will be used for the James and Atiyah-Hirzebruch Spectral sequences respectively. For any finite spectrum , will denote the -nilpotent completion of at the prime (also called localization at , corresponding to localization at the Moore spectrum of ), see .
3. Main results
Let be a simple Poincaré complex with formal dimension . We impose the following condition on a stable vector bundle :
For each the differential in the James spectral sequence is zero.
Observe that the image of the edge homomorphism of in is the intersection of the kernels of all of the differentials with source , i.e. . Thus, Condition 3.1 implies that the group is equal to , i.e. edge homomorphism is surjective. For a given class in we have . Therefore, we can find a class in such that is a generator of with the preferred orientation. As a result, we get a degree one normal map , i.e. we have a surgery problem.
If Condition 3.1 does not hold for , i.e. we have a non-trivial differential for some , then the edge homomorphism can not be surjective. This means can not be a generator of , i.e. can not be a degree one map. Hence, there is not a degree one normal map that represents a class in . As a result, there is not a manifold simple homotopy equivalent to whose stable normal bundle lifts to through . Hence, we have the following lemma:
A stable vector bundle admits a degree one normal map if and only if Condition 3.1 holds for .
For the JSS for , , there is a corresponding (isomorphic) AHSS for the Thom spectrum , i.e. the AHSS whose -page is which converges to , with the isomorphism given by the Thom isomorphism. For a given prime , it is well known that the -primary part of is zero whenever (see ). We use finiteness of . On each torsion part, the first non-trivial differentials of the AHSS are given by the duals of the stable primary cohomology operations. Due to Wu formulas, when we pass to the JSS we need to know the action of Steenrod algebra on the Thom class. For the action of Steenrod squares on the Thom class is determined by the Stiefel-Whitney classes. In fact the Wu formula asserts that (see , p.91).
Let denote the homology theory given by . Let be a spectrum. Consider the AHSS for the homology theory , i.e. the coefficient groups will be . Due to naturality of the AHSS, the first non-zero differentials have to be stable primary cohomology operations independent of the generalized cohomology theory, see pp. 208 . For each with we have and . Thus, the first non-trivial differential in this AHSS appears at the -th page. This differential has to be a stable primary cohomology operation. The only operations in this range are and dual of the Steenrod operation . As in the proof of Lemma in [33, pp. 751], letting as a test case, one can see that is not always zero. The differential in is given by the dual of the map , see  Proposition 1. Let us write for , where is a fixed odd prime. We obtain a similar formula for the first non-zero differentials in acting on torsion part.
For each the differential on the part is equal to the dual of the map
defined as , composed with reduction.
Consider the AHSS for and . In this case the coefficient groups of the AHSS will be and it will converge to . From above remarks, the differential in the AHSS for and , is the dual of the Steenrod operation
By the Thom isomorphism theorem an element of is of the form where and is the Thom class. On the passage to the JSS, Cartan’s formula implies
hence in the James spectral sequence these differentials become duals of the maps composed with reduction. ∎
We have the following lemma for the differential with source :
Let be a prime and be an odd number. Let be a -homology sphere with a given -homology isomorphism . Then for any stable vector bundle that is in the kernel of , the image of the differential in has trivial -torsion.
Let be the stable vector bundle given by the composition . The map induces a map of spectra . Since is a -homology isomorphism, the induced map is also -homology isomorphism, due to Thom isomorphism. Both and are connective spectra and of finite type. The space is -good (see Definition I.5.1 in ) due to 5.5 of . Thus, the induced map is a homotopy equivalence (see  Proposition 2.5 and Theorem 3.1). We have in . Hence, is a trivial bundle. The Thom spectrum is then homotopy equivalent to the wedge of spectra , as it is the suspension spectrum of . Recall that is the JSS for the identity fibration and the trivial stable vector bundle. Hence, collapses on the second page. As a result, -torsion in survives to the . The result follows by comparing the -torsion in , via , with the -torsion in .∎
The following lemma is a partial converse of Lemma 3.4:
Suppose that and Condition 3.1 holds for . Let be the nontrivial element in . Then, by a comparison as in the proof of Lemma 3.4, Condition 3.1 holds for . Thus, there is a degree one normal map representing a class in . Since , see , every such map can be surgered to a simple homotopy equivalence, . However, it is well known that every homotopy sphere is stably parallelizable (see ). Hence, we get a contradiction, as is nontrivial in . ∎
Proof of Theorem 1.1.
Since both and are odd, due to Theorem 1 in , the surgery obstruction groups vanish. Hence, every degree one normal map can be surgered to a homotopy equivalence. We will show that elements in are the ones that admit a degree one normal map.
Let be an orbit in represented by . Consider the James spectral sequence, . Let be a prime with and . Then is a -homology sphere. By Lemma 3.4 the image of has trivial -torsion. Since does not contain -torsion when , image of the differentials have trivial -torsion as well. Hence, all of the differentials based at have trivial -torsion in their images.
Now, let . Then . It is well known that for the -torsion in vanishes except when (see for example , III Theorem 3.13, B). Since , has trivial -torsion, we have . Hence, the only differential whose image can contain mod- torsion appears at degree . By Lemma 3.3, the differential on the -torsion part is equal to the dual of the map
defined as , composed with reduction. Let be an element in . By Poincaré duality, there exists an element in such that (see  Section 2). By definition . Then is trivial on mod- torsion as is an element in , i.e. as .
Now, assume that . Then by Poincaré duality there exist an element such that is nonzero in , i.e. .
As a result, Condition 3.1 holds for if and only if . By Lemma 3.2 if and only if admits a degree one normal map. Hence, we can do surgery and obtain a smooth manifold with a simple homotopy equivalence that represent a class in if and only if . By Lemma 3.4, the image of consists of orbits in represented by elements in the kernel of .
In the case when we have by Bott periodicity, i.e. image of is . If , then , i.e. is a surjection. This completes the proof. ∎
The following corollary is essentially stronger than the main result.
Let and be as in Theorem 1.1. Suppose that and are -homology -spheres fitting into a zigzag , so that both and are -homology isomorphisms for every prime with . If is a free product of finitely many odd order groups, then the image of contains all orbits in which are represented by elements in , where and are the induced maps .
Assume with . For a bundle over , Condition 3.1 holds for if and only if for each . One can compare spectral sequences for and as in the proof of Lemma 3.4, and show that differential is trivial on whenever for each such prime . Thus, Condition 3.1 holds for , whenever . Repeating the arguments of Lemma 3.5, one can show that if , then Condition 3.1 does not hold for . The result follows from Theorem 5 in , together with Lemma 3.2 above.∎
It can be seen from the proof of Theorem 1.1 (resp. Corollary 3.6) that we do not need a single map (or ) which is simultaneously a -homology isomorphism for every prime with . It is enough that for every prime with there exist maps and (depending on ) which are -homology isomorphisms. In this case, we need to replace (or ) by intersection over of all (or ).
Let denote the group of homotopy -spheres. For any smooth -manifold , there is a subgroup of called the inertia group of , defined as , where here means diffeomorphic (see ). Two manifolds and are said to be almost diffeomorphic if there is a such that . It is known that almost diffeomorphic manifolds have isomorphic stable normal bundles, as homotopy spheres are stably parallelizable (see ). Thus, their images are the same under . In order to determine the set of manifolds that are almost diffeomorphic to , one needs to compute . Hence, to determine , it is necessary to know for every in . It is known that is not a homotopy type invariant, in fact it is not even a PL-homeomorphism type invariant, see for example . As a result, complete determination of may not be possible in this generality.
The following corollary says that framed manifolds do not bound in for some .
Under the assumptions of Theorem 1.1 together with and , the edge map is an inclusion for any stable vector bundle .
As in the proof of Theorem 1.1 for a prime the first differential in that acts non-trivially on -torsion appears in dimension . Since , is a -homology sphere for every prime with . Hence, for every . As in the proof of Lemma 3.4, by comparing with we get (as collapses at the second page, due to degree reasons). Therefore, the first nontrivial differential appears when . But then the target of should be zero. Hence, collapses at the second page, and we get that is an inclusion. ∎
Observe that the degree of (as in Theorem 1.1) plays the important role here, as it is co-prime to smaller primes. One can ask what the necessary and sufficient conditions are on the pair so that the natural map induced by the inclusion of point is injective. It is well known that such is not the case for classical Thom spectra like or (see ). In the case when is a trivial bundle, there are examples for which this is true. Another possible question is: For which spaces , this natural map is injective for every stable vector bundle . Corollary 3.8 provides just one such example.
Suppose that and in Lemma 3.4. In this case induces the identity on cohomology with coefficients . Bott periodicity theorem asserts that . The map induces a map on the Atiyah-Hirzebruch spectral sequences. At the second page we have , which is an isomorphism. The mod- class in survives to the infinity page of the Atiyah-Hirzebruch spectral sequence for . Hence, is a surjection on the infinity page by the naturality of the AHSS. It follows that is surjective (for the case when is a spherical space form, this follows from , Theorem 1-(b)). Hence, we have the following remark:
In the case when and in Lemma 3.4, we have .
Since in Theorem 1.1, we have as well. Thus, we can determine the image in the case when as well.
Let denote the quotient space of the free linear action of on . If , then is a -homology sphere with the covering projection being the -homology isomorphism. Let denote the orbit space of a free action of on where acts by homeomorphisms. Such are often called fake lens spaces (see  and ,  for more details on topological and ,  for smooth fake lens spaces). For any such action , one can always find a lens space homotopy equivalent to (see  P.456). If is a prime and is an integer with , Theorem 1.1 applies to any fake lens space . If , then and if , then (note that dimension of is ). In this case, if , then for any prime , is a homology sphere and does not have any element of order for odd. In general, if is a natural number not divisible by primes less than or equal to , is an action of on , and , then Theorem 1.1 applies to where the image of consists of orbits in that are represented by elements in , where is the covering projection. Again, for odd prime, the group does not have any -torsion. We refer to Theorem 2 in  for the -theory of a lens space and to  for calculation of the group of homotopy classes of self homotopy equivalences of a lens space. For the particular cases when and , we can get from Theorem 2.A in  that has only -elements, namely the identity and the automorphism mapping an element to its algebraic inverse. Hence, in these cases each orbit (except the orbit of ) in has exactly two elements.
Another class of examples can be obtained from spherical space forms. There is a vast literature on classification of spherical space forms, see for example ,  and . Let be a homotopy -sphere with . Let be a group that can act freely and smoothly on and let , so that is a principal -bundle. Let be the smallest prime dividing the order of . Then there is a map that classifies . The group of self equivalences of contains a normal subgroup isomorphic to all inner automorphism of (see  Corollary 1 and Theorem 1.4). Note that, an inner automorphism induces the identity on all (generalized) cohomology groups of , due to the commutativity in cohomology. Let in be a self equivalence. Consider the diagram
so that and induce the same map on . By an argument as in  Theorem 7.26, the diagram commutes up to homotopy. It is well-known that induces a surjection on (see for example ). Thereby, inner automorphisms of induce identity on as well. Denote by the group of outer self-equivalences of . Then, in order to determine , we only need to consider automorphisms induced by self equivalences belonging to a fixed set of representatives of cosets in .
For and , then part 1 of Theorem 1.1 applies to , so that is surjective. In this case . For and no other prime between and divides the order of , then the image of is determined by the first Wu class of the Spivak normal bundle of . In general, if is the set of primes between and which divide the order of , then the image of is . We refer to  for the computation of (for as above) and the results given in  (and although indirectly, in ) for the computation of . Of course we only need these computations when is of odd order. The action of on is given by the restriction of the usual canonical action of the automorphism group on , which can be understood once is known.
Let and be as in Theorem 1.1. Let and with given maps for be two -homology -spheres, so that both and are -homology isomorphisms for with , i.e. Theorem 1.1 applies to both and . Then, is also a -homology sphere for primes with (which follows easily from Mayer-Vietoris sequence) and the fundamental group of is the free product (which follows from a simple application of Van Kampen’s Theorem). Thus, Corollary 3.6 applies to the connected sum . If there exists a map satisfying the conditions of Theorem 1.1, then we can choose as the identity map. In this case, consists of all orbits in which are represented by elements in . If we can not find such a map , then we can apply Corollary 3.6 by using the zigzags , where ’s are the obvious collapse maps. In this case, contains all orbits in which are represented by elements in .
For a given finite -complex , denote by the subgroup of that consists of automorphisms induced by elements in (we simply write when ). Due to the naturality of suspension isomorphism and Bott periodicity, we can identify with . Hence, the natural map from to factors through the group of stable self equivalences of , which is equal to (see for example ,  and  for more details about the group of stable self equivalences). If is a -homology sphere for a prime , then all Betti numbers of are less than or equal to . Thus, due to Theorem 1.1-(a) in , the group of stable self equivalences of is finite, which implies that (which is a subgroup of ) is finite.
I wish to thank Özgün Ünlü and Matthew Gelvin for their valuable advices. I also thank the anonymous referee for valuable comments. This research was partially supported by TÜBİTAK-BİDEB-2214/A Programme.
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