Obstructions to stably fibering manifolds

Steimle, Wolfgang

doi:10.2140/gt.2012.16.1691

Cited by 2 publications

(2 citation statements)

References 33 publications

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“…given by a nullhomotopy of the ordinary A-theory characteristic. This is the premise of [Ste12b], in which the author studies the 'parametrized excisive characteristic'. One of the main technical results of their work is an additivity theorem for the parametrized excisive characteristic, which is related to the Poincaré-Hopf theorems in the present work when they are restricted to bundles admitting fiberwise Morse functions.…”

Section: 2mentioning

confidence: 99%

Duality and vanishing theorems for topologically trivial families of smooth h-cobordisms

Jain¹

2021

Preprint

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Using the work of Dwyer, Weiss, and Williams we associate an invariant to any topologically trivial family of smooth h-cobordisms. This invariant is called the smooth structure class, and is closely related to the higher Franz-Reidemeister torsion of Igusa and Klein. We compute the smooth structure class in terms of a fiberwise generalized Morse function using fiberwise Poincaré-Hopf theory. This computation gives rise to a duality theorem for the smooth structure class that generalizes Milnor's duality theorem for the Whitehead torsion. From this result we deduce a vanishing theorem that implies the Rigidity Conjecture of Goette and Igusa. This conjecture states that, after rationalizing, there are no stable exotic smoothings of manifold bundles with closed even dimensional fibers.

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Section: 2mentioning

confidence: 99%

Duality and vanishing theorems for topologically trivial families of smooth h-cobordisms

Jain¹

2021

Preprint

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“…In a sequel paper [16], the author will apply this result to the fibering problem which asks whether a given map between manifolds is homotopic to the projection map of a fiber bundle. Theorem 1.2 together with the "converse Riemann-Roch theorem" of [6] allow to set up a complete obstruction theory to the stable fibering problem.…”

Section: Introductionmentioning

confidence: 99%

Higher Whitehead torsion and the geometric assembly map

Steimle

2012

Journal of Topology

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We construct a higher Whitehead torsion map, using algebraic K-theory of spaces, and show that it satisfies the usual properties of the classical Whitehead torsion. This is used to describe a 'geometric assembly map' defined on stabilized structure spaces in purely homotopy-theoretic terms. IntroductionGiven a space X, the structure space S n (X) is the space of all homotopy equivalences M → X, where M is an n-dimensional manifold (of some given type, such as compact or closed, differentiable, piecewise-linear, or topological). Obviously, X has the homotopy type of such a manifold if and only if the structure space of X is non-empty. In this case, π 0 S n (X) is the central object of interest for the classification of manifolds in the homotopy type of X; the higher homotopy type of S n (X) is closely related to the study of automorphisms of these and to the classification of families of manifolds homotopy equivalent to X [23].If p : E → B is a given fibration, then the above construction can be generalized as follows: A point in the structure space S n (p) of p is given by a bundle of n-dimensional manifolds E → B over B together with a fibre homotopy equivalence E → E. (So the structure space of the space X is the structure space of the trivial fibration X → * .) Pull-back defines a pairing S n (B) × S k (p) −→ S n+k (E), thus if B comes with a given structure, then evaluation of this pairing defines a map α : S k (p) −→ S n+k (E).Geometrically, this map assembles all the manifold structures on the individual fibres to one manifold structure on E, so we call it the geometric assembly map. It is relevant, for instance, in the study of fibring questions.Algebraic K-theory of spaces [19] is a key tool in understanding families of manifolds. The connection to the topology of manifolds is established by the stable parametrized h-cobordism theorem [20], which classifies families of h-cobordisms in terms of K-theory, in a stable range. Recently, Hoehn [7] has used the stable parametrized h-cobordism theorem to describe the homotopy type of the stabilized structure space S ∞ (p) := colim S n (p)

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Obstructions to stably fibering manifolds

Cited by 2 publications

References 33 publications

Duality and vanishing theorems for topologically trivial families of smooth h-cobordisms

Duality and vanishing theorems for topologically trivial families of smooth h-cobordisms

Higher Whitehead torsion and the geometric assembly map

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