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Signature Homology J. reine angew. Math. 592 (2006), 79—122 Journal fu¨r die reine und angewandte Mathematik ( Walter de Gruyter Berlin Á New York 2006 Signature homology By Augusto Minatta at Bochum Abstract. In this note we give a new construction of Signature homology, and we ex- plain how to associate to any oriented manifold M a characteristic class in SigÃðMÞ which is an integral analog of the L-class. A connection with the Novikov conjecture is explained. Further applications are in the construction of a 2-local characteristic class in the singular cohomology of a topological manifold as well as in the determination of the homotopy type of G=Top. Contents Introduction 1. Stratifolds and transversality 1.1. Stratifolds 1.2. Transversality 2. H-stratifolds 2.1. Perverse self-dual complexes of sheaves 2.2. H-Stratifolds 2.3. Product structures 2.4. Existence of a small subclass 2.5. Collared H-Stratifolds 3. Signature homology 3.1. The functor SigÃðÞ 3.2. The coe‰cients of SigÃðÞ 4. The Signature fundamental class of a manifold 4.1. Signature homology with rational coe‰cients 4.2. The L-class of an H-stratifold 4.3. An integral formulation of the Novikov conjecture 4.4. Signature homology at odd primes 4.5. Signature homology at the prime 2 4.6. The relation between SigÃðÞ and the classifying space for surgery References The author was partially supported by a scholarship of the Istituto Nazionale di Alta Matematica ‘‘F. Severi’’. 80 Minatta, Signature homology Introduction This paper is devoted to a new construction of a generalized homology theory denoted by SigÃðÞ which we call Signature homology (in earlier versions the name ‘‘Hirzebruch homology’’ was also used). This homology functor has been firstly considered by Dennis Sullivan in the proof of the Hauptvermutung (see [Su]) and defined by means of the so called Sullivan-Baas construction. More recently, Matthias Kreck has observed that Signa- ture homology also can be used to state an integral formulation of the Novikov conjecture. In fact, according to Kreck, one can associate to every closed oriented smooth manifold M a fundamental class in SigÃðMÞ and it can be shown easily that the rational reduction of this class coincides with the L-class. This consideration allows then to get an integral formula- tion of the Novikov conjecture just by requiring the homotopy invariance of the Signature fundamental class for all singular manifolds over Kðp; 1Þ. Unfortunately this construction is very artificial and cannot be extended to topological manifolds (this is particularly un- satisfactory if one remembers that Novikov has defined rational Pontrjagin classes also for topological manifolds). This paper has the twofold purpose of both providing a more natural construction of Signature homology and of extending the considerations above to the topological category. It is perhaps interesting to notice that, while doing so, we could also prove a generalization of Novikov’s theorem about the topological invariance of ratio- nal Pontrjagin classes. This paper is essentially the fruit of a re-elaboration of my PhD thesis written under Matthias Kreck at the Ruprecht-Karls Universita¨t Heidelberg. There are some new consid- erations, but most results are already contained in the old version (see [Mi]). The first sec- tion contains a short introduction to the theory of stratifolds and provides a proof of the transversality theorem. Section 2 is devoted to the definition of H-stratifolds which are the geometric cycles of Signature homology. Section 3 deals with the construction of the Signa- ture homology functor and with the computation of its coe‰cients. Finally the last section is devoted to the definition of the Signature fundamental class and to its interpretation in terms of other known invariants like Pontrjagin classes. Acknowledgments. I wish to thank my advisor Professor Matthias Kreck for his con- stant suggestions, ideas, and corrections. I also wish to acknowledge the Istituto Nazionale di Alta Matematica ‘‘Francesco Severi’’ di Roma, whose financial support has made this work possible. I gratefully recognize all my colleagues and friends for funny and helpful discussions. A special thank goes to Markus Banagl and Gerd Laures. Finally I am infi- nitely indebted to my parents, Silvia, and all my other friends. 1. Stratifolds and transversality In this section we explain some properties of topological stratifolds. The notion of a stratifold has been firstly introduced by Matthias Kreck in 1998. Through a series of mod- ifications, the term ‘‘stratifold’’ has in the meanwhile come to indicate a di¤erent class of spaces, while the original objects are now called p-stratifolds. For simplicity we will how- ever adopt the original terminology. A more detailed treatment of the geometrical proper- ties of stratifolds can be found in Anna Grinberg’s work (see [Gr]). Minatta, Signature homology 81 1.1. Stratifolds. Let ðW; qWÞ be a pair of spaces with qW closed in W, and denote by W˚ the open set W À qW.Ifd : qW !ð0; þyÞ is a continuous function, then we set À Á <d qW ½0; þyÞ :¼fðx; tÞ A qW ½0; þyÞjt < dðxÞg: À Á À Á ed <d Analogously one defines the sets qW ½0; þyÞ and qW ð0; þyÞ . A collar of qW isÀ a homeomorphismÁ c : V ! U where V is an open neighborhood of <d qW f0g of the form qW ½0; þyÞ and U is an open neighborhood of qW in W,so that for any x A qW it is cðx; 0Þ¼x. Two collars are called equivalent if they coincide on an open neighborhood of qW. An equivalence class of collars will be called a germ of collars. Definition 1.1. An n-dimensional c-manifold is a pair ðW; qWÞ where – W is a metrizable space; – W˚ and qW are (metrizable topological) manifolds of dimension respectively n and n À 1 together with a germ of collars ½c. The manifold qW is called the boundary of W. If c : V ! U is a representative of the germ of collars of a c-manifold W, then we denote by p the composition cÀ1 p U ! V !1 qW: If M is a manifold, then a continuous map f : W ! M is called a c-map if there is a representative of the germ of collars c : V ! U such that for all x A U it holds À Á f ðxÞ¼ f pðxÞ : Observe that every continuous map can be approximated by a c-map. Definition 1.2. Let W1 and W2 be two c-manifolds. A homeomorphism f : W1 ! W2 is called an isomorphism if there are representativesÀ of theÁ germs of collars c1 : V1 ! U1 and c2 : V2 ! U2 such that for all ðx; tÞ A V1 with f c1ðx; tÞ A U2 it holds À Á À Á f c1ðx; tÞ ¼ c2 f ðxÞ; t : If W1 and W2 are two c-manifolds, then by smoothing the corners one defines the structure of a c-manifold on the product W1 Â W2. 82 Minatta, Signature homology Let X be an arbitrary topological space. A k-dimensional strat of X is a pair ðW; f Þ where W is a k-dimensional c-manifold, and f is a proper continuous map from W to X ˚ ˚ such that f jW˚ : W ! f ðWÞ is a homeomorphism. Definition 1.3. An n-dimensional stratifold is a pair ðX; PÞ where X is a topological space X, and P is a sequence of strats fWi; figien which satisfy the following conditions: F ˚ – fiðWiÞ¼X; i – dim Wi ¼ i; S – fiðqWiÞ H fjðWjÞ; jeiÀ1 À1 – a subset U H X is open if and only if for all i the set fi ðUÞ is open in Wi. The sequence W ; f is called the parametrization of X, and the restrictions f are P ¼f i ig ijqWi called the attaching maps of X. For simplicity a stratifold ðX; PÞ will be generally denoted just by X. If X is a stratifold, then the subspaces Sk SkðXÞ :¼ fiðWiÞ H X and Xk :¼ SkðXÞSkÀ1ðXÞ i¼0 are called respectively the k-th skeleton and the k-th stratum of X. A stratifold is said to be purely n-dimensional if Xn is dense in X. The k-th stratum of a stratifold X is by construc- tion a (possibly empty) k-dimensional manifold. The depth of a stratifold X is by definition the di¤erence in dimension of the highest and lowest dimensional non-empty strata. Definition 1.4. An n-dimensional stratifold X is called oriented if XnÀ1 is empty and the top stratum Xn is oriented. A standard argument (see [Gr]) shows that the collars of the manifolds Wi define, for any k, a canonical germ of retractions pk : Vk ! Xk where Vk is an open neighborhood of Xk in X. Example 1.5. Let X and X 0 be two stratifolds. Then the following spaces inherit a natural structure as stratifolds. (1) Any open subset U H X. (2) The cone over X,ifX is compact. (3) The product X Â X 0. Minatta, Signature homology 83 Definition 1.6. If X and X 0 are two n-dimensional stratifolds with stratifications re- 0 0 0 0 spectively P ¼fWi; fig and P ¼fWi ; fi g, then an isomorphism from X to X is a homeo- morphism j : X ! X 0 0 together with a sequence of isomorphisms of c-manifolds ji : WiðXÞ!WiðX Þ which make the following diagram commutative for every i: j Xx ! Xx0 ? ? ? ? 0 fi? ?fi ji 0 WiðXÞ ! WiðX Þ: Now, let X be a topological space and let fUi j i A Jg be a family of open subsets of X with the property that every Ui is a stratifold with parametrization Pi.
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