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2 votes
0 answers
55 views

Tangential normal invariant isomorphism

Recently, I was reading the paper "Finite Group Actions on Kervaire Manifold" by Crowley, Hambolton. But I am having problem understanding a definition. Here it is, In page 15-16 they are ...
4 votes
1 answer
230 views

Gluing a manifold along its boundary, via chain complexes

Given closed oriented $n$-manifolds $M, M', M''$ and bordisms $W, W'$ with $\partial W = M \sqcup - M'$ and $\partial W' = M' \sqcup - M''$, we can collar-glue them to obtain a bordism from $M$ to $M''...
23 votes
2 answers
882 views

Vanishing of characteristic numbers vs vanishing of characteristic classes

A famous result by Thom states that Oriented Bordism classes are determined by characteristic numbers; specifically, two closed manifolds are orientedly bordant if and only if they have the same ...
5 votes
0 answers
350 views

Are there alternate descriptions of `elementary cobordisms'?

Let $M^d$, $N^d$ be cobordant $d$-manifolds. Then $M^d \sqcup \bar{N}^d = \partial W^{d+1}$ for some $(d+1)$-manifold $W$. This cobordism can be implemented via an elementary set of 'moves' called ...
6 votes
0 answers
184 views

Visualize how the 5d Dold manifold and Wu manifold are cobordant via a 6d manifolds with boundaries

Are there simple intuitions and arguments to visualize why the following two 5-manifolds are cobordant to each other with the oriented structures? (They can be two boundaries of 6-dimensional oriented ...
9 votes
1 answer
305 views

Topological Spin manifolds in dimension 4

In his ICM Adress at Nice (Proceedings of the International Congress of Mathematicians Nice, September, 1970, Gauthier-Villars, editeur, Paris 6 e ,1971, Volume 2, pp. 133-163.), Robion Kirby ...
9 votes
2 answers
641 views

Künneth formulas/theorem for bordism groups and cobordisms?

We are familiar with Künneth theorem: The Kunneth formula is given by $R$ as a ring, $M,M'$ as the R-modules, $X,X'$ are some chain complex. The Kunneth formula shows the cohomology of a chain ...