On cobordism of manifolds with corners

Laures, Gerd

doi:10.1090/s0002-9947-00-02676-3

Cited by 80 publications

(114 citation statements)

References 32 publications

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“…The eventual complicated interactions between the moduli spaces in a framed flow category mean that allowing an unrestricted framed cobordism in the sense of [8] becomes intractable. Instead we will work with the following: …”

Section: Framed Cobordism Of Manifolds With Corners Rel Boundarymentioning

confidence: 99%

Framed cobordism and flow category moves

Lobb

Orson²,

Schütz³

2018

Algebr. Geom. Topol.

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. We give moves that change a framed flow category without changing the associated stable homotopy type. These are inspired by moves that can be performed in the Morse-Smale case without altering the underlying smooth manifold. We posit that if two framed flow categories represent the same stable homotopy type then a finite sequence of these moves is sufficient to connect the two categories. This is directed towards the goal of reducing the study of framed flow categories to a combinatorial calculus.We provide examples of calculations performed with these moves (related to the Khovanov framed flow category), and prove some general results about the simplification of framed flow categories via these moves.37D15, 55P42, 57M27

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Section: Framed Cobordism Of Manifolds With Corners Rel Boundarymentioning

confidence: 99%

Framed cobordism and flow category moves

Lobb

Orson²,

Schütz³

2018

Algebr. Geom. Topol.

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“…Equipped with its left-invariant one, Sp(2) represents a generator of π s 10 S 0 (3) , i.e. the element β at the prime 3 [38,42]. Note that there is also a bounding one, as stated previously.…”

Section: )mentioning

confidence: 99%

“…Hence f takes values in D Γ 6 ⊗ Q/Z, and thus is a natural generalization of the e-invariant, which takes values in Q/Z [42].…”

Section: Chromatic Level 2: Heterotic Corners and The F -Invariantmentioning

confidence: 99%

“…In the generalization of the e-invariant, which takes values in Q/Z, to modular forms one notes the following: Since Q/Z is not a ring then it does not make sense to consider "modular forms with coefficients mod Z". Hence, one has to consider modular forms with values in an appropriate ring, which turns out to be the ring of divided congruences D [42]. More precisely, unlike the 1-line in the Adams-Novikov spectral sequence (ANSS) the 2-line is not cyclic in each dimension, hence one needs more copies of Q/Z, and a good way to do so is via D ⊗ Q/Z.…”

Section: Introductionmentioning

confidence: 99%

“…This results in a constraint on degree twelve Chern numbers that generalize the constraint on degree ten Chern numbers in [22], and gives rise to a cup product composite Chern-Simons theory. We illustrate how these conditions affect the corner by highlighting the example of the ten-dimensional Lie group Sp (2) in [42] which turns out to be the physically important one (see [25]).…”

Section: Introductionmentioning

confidence: 99%

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M-Theory with Framed Corners and Tertiary Index Invariants

Sati¹

2014

SIGMA

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Abstract. The study of the partition function in M-theory involves the use of index theory on a twelve-dimensional bounding manifold. In eleven dimensions, viewed as a boundary, this is given by secondary index invariants such as the Atiyah-Patodi-Singer eta-invariant, the Chern-Simons invariant, or the Adams e-invariant. If the eleven-dimensional manifold itself has a boundary, the resulting ten-dimensional manifold can be viewed as a codimension two corner. The partition function in this context has been studied by the author in relation to index theory for manifolds with corners, essentially on the product of two intervals. In this paper, we focus on the case of framed manifolds (which are automatically Spin) and provide a formulation of the refined partition function using a tertiary index invariant, namely the f -invariant introduced by Laures within elliptic cohomology. We describe the context globally, connecting the various spaces and theories around M-theory, and providing a physical realization and interpretation of some ingredients appearing in the constructions due to Bunke-Naumann and Bodecker. The formulation leads to a natural interpretation of anomalies using corners and uncovers some resulting constraints in the heterotic corner. The analysis for type IIA leads to a physical identification of various components of eta-forms appearing in the formula for the phase of the partition function.

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Floer Homotopy Theory, Realizing Chain Complexes by Module Spectra, and Manifolds with Corners

Cohen

2009

Algebraic Topology

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In this paper we describe and continue the study begun in [5] of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the celluar chain complex of a CW -spectrum or pro-spectrum, where the attaching maps are determined by the compactified moduli spaces of connecting orbits. The basic obstructions to the existence of this realization are the smoothness of these moduli spaces, and the existence of compatible collections of framings of their stable tangent bundles. In this note we describe a generalization of this, to show that when these moduli spaces are smooth, and are oriented with respect to a generalized cohomology theory E * , then a Floer E * -homology theory can be defined. In doing this we describe a functorial viewpoint on how chain complexes can be realized by E-module spectra, generalizing the stable homotopy realization criteria given in [5]. Since these moduli spaces, if smooth, will be manifolds with corners, we give a discussion about the appropriate notion of orientations of manifolds with corners.

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On cobordism of manifolds with corners

Cited by 80 publications

References 32 publications

Framed cobordism and flow category moves

Framed cobordism and flow category moves

M-Theory with Framed Corners and Tertiary Index Invariants

Floer Homotopy Theory, Realizing Chain Complexes by Module Spectra, and Manifolds with Corners

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