cobordism category (Rev #20) in nLab
Context
Manifolds and Cobordisms
manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
Functorial Quantum Field Theory
Contents
Idea
The notion of cobordism category is an abstract one intended to capture important features of (many variants of) the category of cobordisms and include in the same formalism cobordisms for closed manifolds with various kinds of structure.
The passage from a manifold MM to its boundary ∂M\partial M has some formal properties which are preserved in the presence of orientation, for manifolds with additional structure and so on. The category of compact smooth manifolds with boundary D=Diff cD = Diff_c has finite coproducts and the boundary operator ∂:D→D\partial:D\to D, M↦∂MM\mapsto \partial M is an endofunctor commuting with coproducts. (Often these coproducts are referred to as direct sums, and some say that ∂\partial is an additive functor, but DD is not actually an additive category). The inclusions i M:∂M→Mi_M:\partial M\to M form a natural transformation of functors i:∂→Idi:\partial\to Id. Finally, the isomorphism classes of objects in DD form a set, so DD is essentially small (svelte).
Definition
Axiomatization
Definition
A cobordism category is a triple (D,∂,i)(D,\partial,i) where
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DD is a svelte category (i.e. an essentially small category)
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with finite coproducts (called direct sums, often denoted by ++),
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including an initial object 00 (also often denoted by ∅\emptyset),
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∂:D→D\partial:D\to D is an additive (direct-sum-preserving) functor
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and i:∂→Id Di:\partial\to Id_D is a natural transformation such that ∂∂M=0\partial\partial M = 0 for all objects M∈DM\in D.
Note that ii is not required to be a subfunctor of the identity, i.e. the components i Mi_M are not required to be monic, which is however often the case in examples.
Definition
Two objects MM and NN in a cobordism category (D,∂,i)(D,\partial,i) are said to be cobordant, written M∼ cobNM\sim_{cob} N, if there are objects U,V∈DU,V\in D such that M+∂U≅N+∂VM+\partial U \cong N+\partial V where ≅\cong denotes the relation of being isomorphic in DD.
Definition
Objects of the form ∂M\partial M where MM is an object in DD are said to be boundaries and the objects VV such that ∂V=0\partial V = 0 are said to be closed.
Definition
By the above, the relation of being cobordant is compatible with the direct sum, in the sense that the direct sum induces an associative commutative operation on the set of equivalence classes, which hence becomes a commutative monoid called the cobordism semigroup
Ω(D,∂,i), \Omega(D,\partial,i) \,,
of the cobordism category (D,∂,i)(D,\partial,i).
Properties
The homotopy type of the cobordism category
Topological case
Theorem
There is a weak homotopy equivalence
Ω|Cob d|≃Ω ∞(MTSO(d)) \Omega |Cob_d| \simeq \Omega^\infty(MTSO(d))
between the loop space of the geometric realization of the dd-cobordism category and the Thom spectrum-kind spectrum
Ω ∞MTSO(d):=lim → n→∞Ω n+dTh(U d,n ⊥) \Omega^\infty MTSO(d) := {\lim_\to}_{n \to \infty} \Omega^{n+d} Th(U_{d,n}^\perp)
where
U d,n ⊥={...} U_{d,n}^\perp = \{ ... \}
This is (Galatius-Tillmann-Madsen-Weiss 06, main theorem).
Geometric case
- (Ayala)
The Thom group? 𝒩 *\mathcal{N}_* of cobordism classes of unoriented compact smooth manifolds is the cobordism semigroup for D=Diff cD=Diff_c.
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category of cobordisms
References
General
A classical reference is
- Robert Stong, Notes on cobordism theory, Princeton University Press 1968 (Russian transl., Mir 1973) (toc pdf, publisher page)
The GMTW theorem about the homotopy type of the cobordisms category with topological structures on the cobordisms appears in
- Søren Galatius, Ib Madsen, Ulrike Tillmann, and Michael Weiss, The homotopy type of the cobordism category Acta Math. 202 (2009), no. 2, 195–239 (arXiv:math/0605249)
A generalization to geometric structure on the cobordisms is discussed in
- David Ayala, Geometric cobordism categories thesis (2009) (arXiv:0811.2280)
Embedded cobordism category
- Oscar Randal-Williams, Embedded Cobordism Categories and Spaces of Manifolds, Int. Math. Res. Not. IMRN 2011, no. 3, 572-608 (arXiv:0912.2505)
On the homotopy groups of the embedded cobordism category:
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Marcel Bökstedt, Anne Marie Svane, A geometric interpretation of the homotopy groups of the cobordism category, Algebr. Geom. Topol. 14 (2014) 1649-1676 (arXiv:1208.3370)
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Marcel Bökstedt, Johan Dupont, Anne Marie Svane, Cobordism obstructions to independent vector fields, Q. J. Math. 66 (2015), no. 1, 13-61 (arXiv:1208.3542)
Revision on June 20, 2020 at 12:17:18 by Urs Schreiber See the history of this page for a list of all contributions to it.