# Questions tagged [cobordism]

Literally "together boundary", a cobordism is a relation between two compact manifolds stating that their disjoint union forms the boundary of a higher dimensional manifold. This defines an equivalence relation between compact manifolds that is very coarse: two manifolds may be cobordant but not homeomorphic. This flexibility makes it the right notion for classifying manifolds in higher dimensions. See also (geometric-topology) and (algebraic-topology).

150 questions
Filter by
Sorted by
Tagged with
29 views

62 views

### Need help understanding the existence of a projection map $pr_2:M \wedge X_+ \rightarrow X$

I am reading these introductory notes to cobordism theory, however cobordism theory is not required to understand my question. I just need help understanding a certain map. On page number 8 the author ...
• 4,474
63 views

### Lifting a map to a homeomorphism of coverings

This is a part of the proof of Lemma 4 of "Cobordism of classical knots" by Casson and Gordon. Here, $\widetilde{X}$ denotes a prime-fold cyclic covering of $X$. Let $h\colon X\to X$ ...
• 541
107 views

### Computing $\pi_3(S^2)$ using framed cobordism.

Review of framed cobordism Let's say $(M^m,g)$ is a compact Riemannian manifold. Let $N$ and $N'$ be $n$-dimensional compact submanifolds of $M$. We say $N$ and $N'$ are cobordant within $M$ if the ...
94 views

### $\mathbf{RP}^m \times \mathbf{RP}^n$ bounds a $(m+n+1)$-d manifold?

How to show whether $\mathbf{RP}^m \times \mathbf{RP}^n$ bounds a $(m+n+1)$-d manifold or not? For example, could we prove or disprove $\mathbf{RP}^2 \times \mathbf{RP}^2$ bounds a 5d manifold?
• 3,379
1 vote
30 views

### Quadratic enhancements of the intersection form and spin structures

If we search spin structures and quadratic enhancements of the intersection form, there are many papers discussing the issue. What are the relations and basic intuitions between the quadratic ...
• 3,379
57 views

### all unoriented bordism groups are $\mathbb{Z}/2$s

Is it true that all unoriented bordism groups are $\mathbb{Z}/2$ vector spaces? This means that any unoriented bordism groups must be of the form:  \mathbb{Z}/2 \oplus \mathbb{Z}/2 \oplus \mathbb{Z}/...
• 3,379
38 views

### Smooth functions defined on connected sums

The following is Theorem 9.29 in Lee's Introducton to Smooth Manifolds. Theorem 9.29. Let $M$ and $N$ be smooth $n$-manifolds with nonempty boundaries, and suppose $h:\partial N\to \partial M$ is a ...
• 1,710
1 vote
78 views

### A non modern theory of generalized Thom spectra

I've already posted this question on mathoverflow a few days ago and I had no reaction, I hope it is not illegal to repost it on math.stackexchange. I'm new in the subject of stable homotopy theory, ...
• 2,089
1 vote
34 views

• 8,289
79 views

### Bordism invariants as integrals of Stiefel-Whitney classes

I am trying to understand this mathematical physics paper by A. Kapustin, which assumes knowledge of bordism invariants of smooth compact manifolds: https://arxiv.org/abs/1403.1467v3 For example, ...
86 views

### Existence of a transversal map prevents density?

Let $S$ be a $C^{\infty}$-submanifold of $N$ and suppose that $N-S$ is dense in $N$, where $M,N$ are $m$ and $n$ dimensional $C^{\infty}$-manifolds, respectively. In this post the answering poster ...
1 vote
55 views

• 964
1 vote
37 views

### About natural identifications in knot theory

Let's consider a knot $K$ in a general closed oriented 3-manifold $Y$. And for technical simplicity assume $K$ is rationally nullhomologous, ie, $[K]\in H_1(Y)$ is a torsion element. Now choose a ...
• 2,113