# 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 ...
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 $h\colon X\to X$ ...
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?
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 ...
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}/...
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 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, ...
1 vote
34 views

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

So I want to understand does the following addition give stricter condition for equivalence of two stably-complex manifolds? Assume the following relation: let two stably-complex $n$-manifolds $M^n,... 1 vote 1 answer 95 views ### Cobordism of points On the wiki page about cobordism, it is stated that the cobordism of oriented 0-dimensional manifolds is$\mathbb Z$. That seem surprising since One can always draw a line between two points. I ... 4 votes 2 answers 173 views ### Does the normal bundle of a manifold depend on embedding? In the proof of unoriented cobordism ring being isomorphic to homotopy group of Thom spectra, one considers a large enough dimensional Euclidean space where a given manifold has all the embeddings ... 1 vote 1 answer 215 views ### Mapping cylinder of homeomorphism is not homeomorphic to product Let$X$be a topological manifold, and$f:X\to X$be a homeomorphism. The mapping cylinder is defined as$M_f:=(X\times[0,1]\sqcup X)/(x,1)\sim f(x)$. I am told somewhere that there exists an example ... 0 votes 1 answer 52 views ### Question on concept of homology in calibrated geometry The fundamental lemma of calibrated geometry states that calibrated submanifolds are absolutely volume minimising in their homology class. In the proof, homology equivalent is used synonymously with ... 0 votes 1 answer 161 views ### J. Milnor: smooth manifold triad vs cobordism I started studying Milnor's "Lectures on the H-cobordism theorem". On page 2 he provides the following definitions Definition 1.3.$(W;V_0,V_1)$is a smooth manifold triad if$W$is a compact ... 4 votes 1 answer 151 views ### Unreadable lines of Milnor's book What are the (exact sentence of) Unreadable lines of the following images of Milnor's Lecture on h-cobordism theorem? (pages number: 28, 30, 15 respectively) 0 votes 0 answers 35 views ### What is the orientation of the normalized boundary$\partial(M\times N)$of product manifold? Assume$M$and$N$are two oriented smooth manifold with or without boundaries. Then$M\times N$is an oriented manifold with corners. Inspired by the theory of cobordism or differential forms, the ... 1 vote 1 answer 113 views ### Which spaces are homotopy equivalent in an h-cobordism? All the definitions I've found for h-cobordisms define it as a cobordism$(W,M,N)$such that the inclusions$i_M:M\rightarrow W$and$i_N:N\rightarrow W$are homotopy equivalences. But this doesn't ... 12 votes 1 answer 248 views ### What is Bordism good for? I am doing a course in algebraic topology, and recently we have defined bordism in the following way: Let$M_1,M_2$be connected oriented n-dimensional manifolds,$f_i:M_i\rightarrow X$be continuous ... 0 votes 0 answers 112 views ### Orientation of the unit interval Let the unit interval have postive orientation induced from the standard orientation of$\mathbb{R}$, and let$\{0\}$and$\{1\}$be equipped with orientation$+$. Why is$\{0\}$then an in-boundary ... 1 vote 1 answer 58 views ### Factorization of the orientation map$MU\to H\mathbb{Z}$through$ku$? Let$MU$denote the complex cobordism spectrum and$ku$the connective cover of the complex$K$-theory spectrum. Is it true that the orientation map$MU\to H\mathbb{Z}$factors through$ku$? 0 votes 1 answer 28 views ### Examples of codim-2 objects in extended TQFT I'm scratching my head trying to understand what an extended TQFT associates to$(n-2)$-hypersurfaces. Here's some intuition that I've developed. For an$(n-1)$-hypersurface chopped into$(n-2)$-... 6 votes 2 answers 303 views ### Stiefel-Whitney numbers of manifolds that are boundaries of non-smoothable manifolds Can a smooth compact manifold be the boundary of a non-smoothable manifold? If so can any of its Stiefel-Whitney numbers be non-zero? Thom's theorem says that a compact smooth manifold has zero ... 1 vote 0 answers 91 views ### Spin structure and characteristic classes I do not know if anyone can help me with these doubts of spin structures and characteristic classes. 1) Is there an orientable manifold that is not spin? 2) Is there a finite group$ G $such that ... 5 votes 1 answer 268 views ### Spin structure and bordism I have some questions about bordism and spin structures on manifolds. If you have any examples or references I would appreciate it. Is there a 3-manifold$ M $, orientable, which does not support 3 ... 2 votes 1 answer 66 views ### Cobordant Map from May's Book I have a question about an argument that occurred in the discussion about consequences of Bott periodicity in A Concise Course in Algebraic Topology by P. May on page 221. Here is the excerpt: ... 2 votes 1 answer 34 views ### Contruction of a New Manifold via One-Parameter Families of Embeddings Let$M$,$N$be two manifolds of dimension$n$. Fix$m>n$. Let$i:[0,1)\times M \to \mathbb{R}^m$,$j:(1,2]\times N \to \mathbb{R}^m$be embeddings such that for each$s\in[0,1)$,$t\in(1,2]$,$i_s:...
1 vote
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 ...