[UC Top] Proseminar practice: Cobordism

[Alexander F. Ritter]

Hi,

The proseminar practice talk on cobordism will take place at 4.30pm
today (Monday 15). An informal abstract follows - it is non-technical,
so have a glance at it.

All the best,
Alex


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INTRODUCTION TO COBORDISM

The notion of cobordism is simply that two manifolds are cobordant when
their disjoint union is the boundary of some manifold.

More precisely, let D be a collection of manifolds (smooth, in general
with boundary, and not necessarily connected). Say that two manifolds
M,N in D are cobordant iff their disjoint union M+N is the boundary of
some manifold in D. It turns out that for most choices of D this is an
equivalence relation and the collection of equivalence classes (called
cobordism classes) form an abelian group under disjoint union. Via
Cartesian products of manifolds, this can usually also be endowed with a
ring structure.

The obvious question to ask is: what is the zero class? In other words,
what manifolds arise as boundaries? Is there a feasible way to determine
whether M and N are cobordant or not? The reader may wish to ponder
whether RP^n is the boundary of some manifold or not (can you give an
easy answer when n is even, without using characteristic classes?).

Cobordism comes in many flavours depending on what collection D you
choose. This is why the definitions above were chosen to be rather vague
(for instance, you ought to require that the manifolds in D are compact
- why?). Indeed, D is a category and we have not specified the morphisms
yet. If you choose smooth compact oriented manifolds with smooth
orientation-preserving maps then you obtain the oriented cobordism ring,
whereas dropping the orientation requirements yields the unoriented
cobordism ring. These are the simplest two examples and we will try to
carry them through our talk. Indeed in the first part of the talk we
will compute the first few cobordism groups explicitly. This should
convince us that the explicit geometrical problem of determining wheter
a manifold is a boundary is actually fiendishly hard.

We will mention some of the highlights of the theory - the complete
description of the unoriented cobordism ring and a description of the
oriented cobordism ring modulo torsion. It turns out that there is a
beautiful connection between these rings and characteristic classes.

The second half of the talk is an introduction to category theory in the
language of category theory. We will define cobordism categories,
cobordism semirings, and (B,f) structures on manifolds. These are very
powerful notions which allow us to study several cobordism theories at
once (depending on the choice of B and f - for instance B might be BO,
BSO, BU, BSp, etc. and the corresponding cobordism groups are the
unoriented, oriented, complex, symplectic, etc. groups).

This will allow us to state the Pontrjagin-Thom isomorphism theorem in a
very general setting, and if time allows we will sketch the proof.
Essentially this isomorphism translates the geometrical problem of
determining cobordism groups into the algebraic problem of computing the
homotopy groups of a spectrum. There is no need to despair though (at
least not in the unoriented case) since it turns out that these
computations are often feasible.

In order to avoid disappointing anyone, let me stress that we will NOT
mention Adams spectral sequences - there will be a talk on this topic in
the future by a surely more competent speaker. Since time will be short,
I may choose to forfeit sketching the proof of the Pontrjagin-Thom
theorem in order to pave the way for the talk on Adams spectral
sequences - so I will mention the homology and cohomology of Thom
spaces, and I will vaguely sketch how the machinery of the Adams
spectral sequence can yield the oriented and complex cobordism groups.