Cobordism category

In mathematics , the cobordism category is a term from algebraic topology .

These are categories for whose objects are the closed -dimensional smooth submanifolds of a high-dimensional Euclidean space and whose morphisms are the -dimensional, embedded cobordisms with a collar edge . ${\displaystyle {\mathcal {C}}_{d}}$ ${\displaystyle d\in \mathbb {N} }$ ${\displaystyle (d-1)}$ ${\displaystyle d}$

Definition of the category

An object of is a pair with such that a closed, -dimensional -submanifold ${\displaystyle {\mathcal {C}}_{d}}$ ${\displaystyle (M,a)}$ ${\displaystyle a\in \mathbb {R} }$ ${\displaystyle M}$ ${\displaystyle (d-1)}$ ${\displaystyle C^{\infty }}$

${\displaystyle M\subset \mathbb {R} ^{d-1+\infty }:=\operatorname {colim} _{n\to \infty }\mathbb {R} ^{d-1+n}}$

is.

The identity morphism of is the triple . A morphism of to different from the identity is a triple of real numbers with and a -dimensional compact -submanifold ${\displaystyle (M,a)}$ ${\displaystyle (\left\{a\right\}\times M,a,a)}$ ${\displaystyle (M_{0},a_{0})}$ ${\displaystyle (M_{1},a_{1})}$ ${\displaystyle (W,a_{0},a_{1})}$ ${\displaystyle a_{0},a_{1}}$ ${\displaystyle a_{0}<a_{1}}$ ${\displaystyle d}$ ${\displaystyle C^{\infty }}$

${\displaystyle W\subset \left[a_{0},a_{1}\right]\times \mathbb {R} ^{d-1+\infty }}$,

so there is one with ${\displaystyle \epsilon >0}$

${\displaystyle W\cap (\left[a_{0},a_{0}+\epsilon \right)\times \mathbb {R} ^{d-1+\infty })=\left[a_{0},a_{0}+\epsilon \right)\times M_{0}}$,

${\displaystyle W\cap (\left(a_{1}-\epsilon ,a_{1}\right]\times \mathbb {R} ^{d-1+\infty })=\left(a_{1}-\epsilon ,a_{1}\right]\times M_{1}}$,

${\displaystyle \partial W=W\cap (\left\{a_{0},a_{1}\right\}\times \mathbb {R} ^{d-1+\infty })}$.

The composition of two morphisms is made through the union

${\displaystyle (W_{1},a_{0},a_{1})\circ (W_{2},a_{1},a_{2}):=(W_{1}\cup W_{2},a_{0},a_{2})}$

defined by subsets in . ${\displaystyle \mathbb {R} \times \mathbb {R} ^{d-1+\infty }}$

Topological enrichment of the category

Objects and morphisms are given a topology through the identifications

${\displaystyle \operatorname {ob} {\mathcal {C}}_{d}\cong \mathbb {R} \times \bigcup _{M}\operatorname {Emb} (M,\mathbb {R} ^{d-1+\infty })/\operatorname {Diff} (M)}$

and

${\displaystyle \operatorname {mor} {\mathcal {C}}_{d}\cong \operatorname {ob} {\mathcal {C}}_{d}\cup \bigcup _{W}\mathbb {R} _{+}^{2}\times \operatorname {Emb} (W,\left[0,1\right]\times \mathbb {R} ^{d-1+\infty })/\operatorname {Diff} (W)}$.

Here referred to the space of the embedding in the topology. The diffeomorphism group works by composing embeddings with diffeomorphisms. The factor space is provided with the quotient topology . ${\displaystyle \operatorname {Emb} (.,\mathbb {R} ^{d-1+\infty })}$ ${\displaystyle \mathbb {R} ^{d-1+\infty }}$ ${\displaystyle C^{\infty }}$ ${\displaystyle Diff(.)}$ ${\displaystyle \operatorname {Emb} (.,\mathbb {R} ^{d-1+\infty })/\operatorname {Diff} (.)}$

literature

• Galatius , Madsen , Tillmann , Weiss : The homotopy type of the cobordism category , Acta Math. 202 (2009), no. 2, pp. 195-239.
• Galatius, Randal-Williams : Stable moduli spaces of high-dimensional manifolds , Acta Math. 212 (2014), no. 2, pp. 257-377.