On Bordism Theory of Manifolds with Singularities.

Baas, Nils A.

doi:10.7146/math.scand.a-11491

Cited by 125 publications

(106 citation statements)

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“…The category Cob d, k (X) is particularly important in the 2 dimensional case where this category of "surfaces in X" plays a role in both Gromov-Witten theory as well as string topology. Variations on this category have been studied in several contexts; see Baas [Baa73], Sullivan [Sul71], Cohen-Madsen [CM], and Hanbury.…”

Section: −→ Bo(d)mentioning

confidence: 99%

Cobordism categories of manifolds with corners

Genauer

2011

Trans. Amer. Math. Soc.

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Abstract. In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if Cob d, k is the category whose objects are a fixed dimension d, with corners of codimension ≤ k, then we identify the homotopy type of the classifying space BCob d, k as the zero space of a homotopy colimit of a certain diagram of the Thom spectra. We also identify the homotopy type of the corresponding cobordism category when an extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss (2009), and their proofs are an adaptation of the methods of their paper. As an application we describe the homotopy type of the category of open and closed strings with a background space X, as well as its higher dimensional analogues. This generalizes work of Baas-Cohen-Ramirez (2006) and Hanbury.

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Section: −→ Bo(d)mentioning

confidence: 99%

Cobordism categories of manifolds with corners

Genauer

2011

Trans. Amer. Math. Soc.

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“…The second main result of this paper is that at least a certain weaker invariant, which Höhn called the twisted χ y genus, can be defined for arbitrary singular varieties (compatibly with IH-small resolutions): see Theorems 8.1 and 8. 2.…”

Section: Statementsmentioning

confidence: 99%

Chern Numbers for Singular Varieties and Elliptic Homology

Totaro¹

2000

The Annals of Mathematics

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A fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones. Intersection homology, for example, directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety. Minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so.Here we use ideas from minimal model theory to define some characteristic numbers for singular varieties, generalizing the Chern numbers of a smooth variety. This was suggested by Goresky and MacPherson as a next natural problem after the definition of intersection homology [11]. We find that only a subspace of the Chern numbers can be defined for singular varieties. A convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism, whereas a singular variety can at most have a fundamental class in a weaker homology theory, elliptic homology. We use this idea to give an algebro-geometric definition of elliptic homology: "complex bordism modulo flops equals elliptic homology."This paper was inspired by some questions asked by Jack Morava. The descriptions of elliptic homology given by Gerald Höhn [13] were also an important influence. Thanks to Dave Bayer, Mike Stillman, John Stembridge, Sheldon Katz, and Stein Stromme for their computer algebra programs Macaulay, SF, and Schubert, which helped in guessing the right answer. Contents 1. Statements 2. Weakly complex manifolds 3. The complex elliptic genus 4. Complex bordism modulo flops 5. Complex bordism modulo flops, continued 6. SU-bordism modulo flops 7. Saito's homology classes χ n−k i 8. The Chern numbers c k 1 χ n−k i 9. The twisted χ y genus

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“…See Rudyak [7], Baas [2], Botvinnik [4], Sullivan [8], Vershinin [11]. Below we introduce differentiable stratified spaces which we consider as the general form of stratified manifolds If M is a differentiable manifold then the tangent bundle T M → M restricted to the boundary T M |∂M is isomorphic to the sum of the tangent bundle T ∂M of the boundary of M and the normal bundle ν∂M = ∂M × R as in the diagram…”

Section: Stratified Manifoldsmentioning

confidence: 99%

K-Theory of Stratified Vector Bundles

Baues

Ferrario

2003

K-Theory

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We show that the Atiyah-Hirzebruch K-theory of spaces admits a canonical generalization for stratified spaces. For this we study algebraic constructions on stratified vector bundles. In particular the tangent bundle of a stratified manifold is such a stratified vector bundle. Families of vector spacesLet R be the field of real numbers and R n = R ⊕ · · · ⊕ R be the standard n-dimensional R-vector space. Let Vect denote the category of finite dimensional R-vector spaces and linear maps and V a subcategory of Vect, termed the structure category. For example let V be the subcategory of the surjective maps in Vect. Or let G be a subgroup of the automorphism group GL n (R) of R n . Then G yields the subcategory G ⊂ Vect consisting of one object R n and morphisms given by elements in G. The category Vect is a topological category. We say that V is a closed subcategory if for objects V , W ∈ V the inclusion of morphisms setsis a closed subspace. Here hom Vect (V, W ) is homeomorphic to R N while hom V (V, W ) needs not to be a vector space.Following Atiyah [1] a family of vector spaces in V is a topological space X together with a map p X : X →X and for every b ∈X a homeomorphism, named chart,where X b is a suitable object of V depending upon b ∈X. We denote the family (p X : X →X, X b , Φ b , b ∈X) simply by X (it will be clear from the context whether X is the underlying space or the family). The map p X is termed the projection, the space X the total space of the family of vector spaces, the spaceX the base space, and for every b the vector space X b is termed fiber over b. A family X is also termed V-family, to make explicit the choice of V. The V-family X is discrete ifX 1

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On Bordism Theory of Manifolds with Singularities.

Cited by 125 publications

References 0 publications

Cobordism categories of manifolds with corners

Cobordism categories of manifolds with corners

Chern Numbers for Singular Varieties and Elliptic Homology

K-Theory of Stratified Vector Bundles

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