Umkehr maps

Cohen, Ralph L.; Klein, John R.

doi:10.4310/hha.2009.v11.n1.a2

Cited by 19 publications

(21 citation statements)

References 14 publications

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“….a 1˝a2 /i D h ev ˇ; a 1˝a2 i D h ev Cˇ; a 1˝a2 i where the first equality holds by duality of coproduct to product and the second by (14). By Proposition 4.14, givenˇas a sum of products of !…”

Section: Lemma 415mentioning

confidence: 99%

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A homotopy-theoretic view of Bott–Taubes integrals and knot spaces

Koytcheff

2009

Algebr. Geom. Topol.

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We construct cohomology classes in the space of knots by considering a bundle over this space and "integrating along the fiber" classes coming from the cohomology of configuration spaces using a Pontrjagin-Thom construction. The bundle we consider is essentially the one considered by Bott and Taubes [8], who integrated differential forms along the fiber to get knot invariants. By doing this "integration" homotopy-theoretically, we are able to produce integral cohomology classes. Inspired by results of Budney and Cohen [11], we study how this integration is compatible with homology operations on the space of long knots. In particular we derive a product formula for evaluations of cohomology classes on homology classes, with respect to connect-sum of knots.

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Section: Lemma 415mentioning

confidence: 99%

“…References for these maps in general include Cohen and Klein [14], Becker and Gottlieb [7], and at a more elementary level, Bredon's book [9]. It will be natural to consider not just a map of spaces, but a map of spectra.…”

Section: The Pontrjagin-thom Mapmentioning

confidence: 99%

A homotopy-theoretic view of Bott–Taubes integrals and knot spaces

Koytcheff

2009

Algebr. Geom. Topol.

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“…where BJ = |N J| is the classifying space of J. If A is instead almost constant, then we get the following proposition (see [15,17]). Proposition 6.9.…”

Section: Lemma 63 (1) Each Adjunction Of Categories Induces a Homotmentioning

confidence: 90%

“…From numerous sources (e.g. [Cho07], [CK09], [MS06]) we expect such cohomology theories to be represented by spaces or spectra.…”

Section: Composition Of Inclusions Of Faces Then Even When F Is Not mentioning

confidence: 99%

“…The last step is the application of Poincaré duality (see [MS06], [CK09]) to the noncompact manifold C(M ; n) with twisted coefficients given by the bundle of spectra (ι * E ∧ n ) Σn . Since the manifold in question is noncompact, Poincaré duality gives an equivalence between cohomology with compact supports and homology desuspended by the tangent bundle of C(M ; n).…”

Section: In Addition Our Decoration (−)mentioning

confidence: 99%

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A tower connecting gauge groups to string topology

Malkiewich

2015

Journal of Topology

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Abstract. We develop a variant of calculus of functors, and use it to relate the gauge group G(P) of a principal bundle P over M to the Thom ring spectrum (P Ad ) −T M . If P has contractible total space, the resulting Thom ring spectrum is LM −T M , which plays a central role in string topology. Cohen and Jones have recently observed that, in a certain sense, (P Ad ) −T M is the linear approximation of G(P). We prove an extension of that relationship by demonstrating the existence of higher-order approximations and calculating them explicitly. This also generalizes calculations done by Arone in [Aro99].

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String topology on Gorenstein spaces

Félix

Thomas

2009

Math. Ann.

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The purpose of this paper is to describe a general and simple setting for defining (g, p + q)-string operations on a Poincaré duality space and more generally on a Gorenstein space. Gorenstein spaces include Poincaré duality spaces as well as classifying spaces or homotopy quotients of connected Lie groups. Our presentation implies directly the homotopy invariance of each (g, p + q)-string operation as well as it leads to explicit computations. IntroductionShriek maps play a central role in string topology and its generalizations. Following Dold, various presentations have been given [1,3,5,10,30]. Usually shriek maps are defined in (co)homology for maps f : N → M from a closed oriented n-manifold to a closed oriented m-manifold. Here we will consider shriek maps at the cochain level and in a more general setting.More precisely we work in the category of (left or right) differential graded modules over a differential graded lk-algebra (R, d) (lk is a fixed field), that we call for sake of simplicity the category of (R, d)-modules. Its associated derived category [20] is obtained by formally inverting quasi-isomorphisms, i.e. the homomorphisms of (R, d)-modules that induce isomorphisms in homology and hereafter denoted by in the diagrams. In the derived category the vector space of Y. Félix

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Umkehr maps

Cited by 19 publications

References 14 publications

A homotopy-theoretic view of Bott–Taubes integrals and knot spaces

A homotopy-theoretic view of Bott–Taubes integrals and knot spaces

A tower connecting gauge groups to string topology

String topology on Gorenstein spaces

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