Results 1  10
of
12
Secondary analytic indices
, 1999
"... We define realvalued characteristic classes of flat complex vector bundles and flat real vector bundles with a duality structure. We construct pushforwards of such vector bundles with vanishing characteristic classes. These pushforwards involve the analytic torsion form in the first case and the et ..."
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We define realvalued characteristic classes of flat complex vector bundles and flat real vector bundles with a duality structure. We construct pushforwards of such vector bundles with vanishing characteristic classes. These pushforwards involve the analytic torsion form in the first case and the etaform of the signature operator in the second case. We show that the pushforwards are independent of the geometric choices made in the constructions and hence are topological in nature. We give evidence that in the first case, the pushforwards are given topologically by the BeckerGottliebDold transfer. 1
Singularities, Double Points, Controlled Topology and Chain Duality
"... A manifold is a Poincar'e duality space without singularities. McCrory obtained a homological criterion of a global nature for deciding if a polyhedral Poincar'e duality space is a homology manifold, i.e. if the singularities are homologically inessential. A homeomorphism of manifolds is a ..."
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A manifold is a Poincar'e duality space without singularities. McCrory obtained a homological criterion of a global nature for deciding if a polyhedral Poincar'e duality space is a homology manifold, i.e. if the singularities are homologically inessential. A homeomorphism of manifolds is a degree 1 map without double points. In this paper combinatorially controlled topology and the chain complex methods of the algebraic theory of surgery are used to simplify McCrory's work, and to provide a homological criterion of a global nature for deciding if a degree 1 map of polyhedral homology manifolds has acyclic point inverses, i.e. if the double points are homologically inessential. Other applications include double points of arbitrary maps, Whitehead torsion, homology fibrations and knot theory. AMS Classification numbers Primary: 55N45, 57R67 Secondary: 55U35 Keywords: Singularities, double points, controlled topology, chain complex, duality, surgery. 1 Contents 1 Chain duality 8 2 Sim...
OBSTRUCTIONS TO FIBERING A MANIFOLD
, 2009
"... Given a map f: M → N of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S 1, these torsion obstructions are identified with the ones due to Farrell [5]. ..."
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Given a map f: M → N of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S 1, these torsion obstructions are identified with the ones due to Farrell [5].
Topological Classification of Multiaxial U(n)Actions (with an appendix by Jared Bass)
, 2012
"... Since early 1980s, great progress has been made on the classification of finite group actions on the sphere. Deep but indirect connections to representation theory were discovered. The indirectness is reflected by the existence of nonlinear similarities between some linearly inequivalent representa ..."
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Since early 1980s, great progress has been made on the classification of finite group actions on the sphere. Deep but indirect connections to representation theory were discovered. The indirectness is reflected by the existence of nonlinear similarities between some linearly inequivalent representations [3], via the equivariant signature operator [MR] (see
AND STRATIFIED MANIFOLDS
, 2004
"... Abstract. Cappell and Shaneson pointed out in 1978 interesting properties of Browder Livesay invariants which are similar to differentials in some spectral sequence. Such spectral sequence was constructed in 1991 by Hambleton and Kharshiladze. This spectral sequence is closely related to a problem ..."
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Abstract. Cappell and Shaneson pointed out in 1978 interesting properties of Browder Livesay invariants which are similar to differentials in some spectral sequence. Such spectral sequence was constructed in 1991 by Hambleton and Kharshiladze. This spectral sequence is closely related to a problem of realization of elements of Wall groups by normal maps of closed manifolds. The main step of construction of the spectral sequence is an infinite filtration of spectra in which only the first two, as is wellknown, have clear geometric sense. The first one is a spectrum L(π1(X)) for surgery obstruction groups of a manifold X and the second LP∗(F) is a spectrum for surgery on a BrowderLivesay manifold pair Y ⊂ X. The geometric sense of the third term of filtration was explained by Muranov, Repovˇs, and Spaggiari in 2002. In the present paper we give a geometric interpretation of all spectra of filtration in construction of Hambleton and Kharshiladze. We introduce groups of obstructions to surgery on a system of embeddedd manifolds and prove that spectra which realize these groups coincide with spectra in the filtration of Hambleton and Kharshiladze. We describe algebraic and geometric properties of introduced obstruction
THE SIGNATURE OF A FIBRE BUNDLE IS MULTIPLICATIVE MOD 4
, 2005
"... Abstract. We express the signature modulo 4 of a closed, oriented, 4kdimensional PL manifold as a linear combination of its Euler characteristic and the new absolute torsion invariant defined in Korzeniewski [11]. Let F → E → B be a PL fibre bundle, where F, E and B are closed, connected, and compat ..."
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Abstract. We express the signature modulo 4 of a closed, oriented, 4kdimensional PL manifold as a linear combination of its Euler characteristic and the new absolute torsion invariant defined in Korzeniewski [11]. Let F → E → B be a PL fibre bundle, where F, E and B are closed, connected, and compatibly oriented PL manifolds. We give a formula for the absolute torsion of the total space E in terms of the absolute torsion of the base and fibre, and then combine these two results to prove that the signature of E is congruent modulo 4 to the product of the signatures of F and B. The signature sign(M) ∈ Z of a closed, oriented nmanifold M n is the index of its cup product form when n ≡ 0 mod4, and zero otherwise. For an orientable differentiable fibre bundle, Chern, Hirzebruch and Serre [4] proved that sign(E) = sign(F) · sign(B) provided that the fundamental group π1(B) acts trivially on the real cohomology H ∗ (F; R) of the fibre. In general, the signature is not multiplicative for differentiable fibre bundles: the first examples were constructed by Kodaira [10], Atiyah [2], and Hirzebruch [7]. These examples
SURGERY SPECTRAL SEQUENCE AND STRATIFIED MANIFOLDS
, 2006
"... Abstract. Cappell and Shaneson pointed out in 1978 interesting properties of Browder Livesay invariants which are similar to differentials in some spectral sequence. Such spectral sequence was constructed in 1991 by Hambleton and Kharshiladze. This spectral sequence is closely related to a problem ..."
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Abstract. Cappell and Shaneson pointed out in 1978 interesting properties of Browder Livesay invariants which are similar to differentials in some spectral sequence. Such spectral sequence was constructed in 1991 by Hambleton and Kharshiladze. This spectral sequence is closely related to a problem of realization of elements of Wall groups by normal maps of closed manifolds. The main step of construction of the spectral sequence is an infinite filtration of spectra in which only the first two, as is wellknown, have clear geometric sense. The first one is a spectrum L(π1(X)) for surgery obstruction groups of a manifold X and the second LP∗(F) is a spectrum for surgery on a BrowderLivesay manifold pair Y ⊂ X. The geometric sense of the third term of filtration was explained by Muranov, Repovˇs, and Spaggiari in 2002. In the present paper we give a geometric interpretation of all spectra of filtration in construction of Hambleton and Kharshiladze. We introduce groups of obstructions to surgery on a system of embeddedd manifolds and prove that spectra which realize these groups coincide with spectra in the filtration of Hambleton and Kharshiladze. We describe algebraic and geometric properties of introduced obstruction
Doc. Math. J. DMV 1 Singularities, Double Points, Controlled Topology and Chain Duality
, 1998
"... Abstract. A manifold is a Poincaré duality space without singularities. McCrory obtained a homological criterion of a global nature for deciding if a polyhedral Poincaré duality space is a homology manifold, i.e. if the singularities are homologically inessential. A homeomorphism of manifolds is a d ..."
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Abstract. A manifold is a Poincaré duality space without singularities. McCrory obtained a homological criterion of a global nature for deciding if a polyhedral Poincaré duality space is a homology manifold, i.e. if the singularities are homologically inessential. A homeomorphism of manifolds is a degree 1 map without double points. In this paper combinatorially controlled topology and chain complex methods are used to provide a homological criterion of a global nature for deciding if a degree 1 map of polyhedral homology manifolds has acyclic point inverses, i.e. if the double points are homologically inessential.