# 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).

163 questions
Filter by
Sorted by
Tagged with
1 vote
42 views

• 810
41 views

### Double of a manifold and cobordism

I read here the following: Given a manifold $M$, the double of $M$ is the boundary of $M \times [0,1]$. This gives doubles a special role in cobordism. What is this special role? Moreover, is it ...
44 views

### Disjoint closed sets separated by a submanifold

I came across the following statement in Connor and Floyd's book Differentiable Periodic Maps, page 10, proposition 3.1. (3.1) Suppose $P$ and $Q$ are closed disjoint subsets of the compact $n$-...
• 769
1 vote
35 views

• 321
31 views

### Spaces distinguished by cobordism homology

Recently I have been learning about cobordisms. While I have seen many applications of cobordism to stable homotopy theory and immersion problems, I couldn't find an example of spaces that can be ...
• 141
107 views

### Relation of Algebraic Cobordism to Cobordism of Smooth Manifolds

(I have almost no background in algebraic geometry, so excuse the naivete of my question.) Some ideas I've been thinking about recently led to the question of whether there is a notion of 'cobordism' ...
1 vote
107 views

### Equivalent definitions of cobordism

I am reading Stong's notes on Cobordism, and he defines a Cobordism category $\mathcal{C}$, with direct sums and a boundary functor $\partial:\mathcal{C}\longrightarrow \mathcal{C}$, and says that two ...
• 1,260
45 views

### A $\pi_1$-neglibility criterion in 4-dimensional manifolds

I'm reading about the h-cobordism theorem in boundary dimension 4. Most of the steps are the same as in the classical statement, but finding whitney disks to homotope the 2- and 3-handles into ...
65 views

### A question about the theorem of Kervaire and Milnor that the group $\Theta^n$ is finite for $n\neq 3$

In 1963, Kervaire and Milnor proved that the group $\Theta^n$ is finite for $n\neq 3$ (https://www.math.kit.edu/iag5/lehre/semgeo2014w/media/kervaire%20milnor.pdf). Here $\Theta^n$ is the group of h-...
• 1,562
66 views

I am now reading Thom's famous paper Quelques propriétés globales des variétés différentiables. In page 48, Thom used an auxiliary space $K$, which is a principal fiber bundle with base space $K(\... 4 votes 1 answer 135 views ### Are invertibly cobordant manifolds diffeomorphic Let$M$and$N$be oriented, closed,$n-1$manifolds and$F$a cobordism from$M$to$N$and$G$a cobordism from$N$to$M$such that the composite cobordism$G\circ F\cong M\times I$and$F\circ G\...
• 2,534
1 vote
116 views

Good time of day. I have the following question Milnor hypersurface $H_{ij}$ is a smooth hypersufrace in $\mathbb CP^i \times \mathbb CP^j$ for fix pair of integers $j \ge i\ge 0$. Its algebraic ...
• 411
1 vote
30 views

### Does dimensional reduction of TQFTs have an adjoint?

If $Cob_n$ is $n$-dimensional cobordisms, and $Z \colon Cob_n \to \mathcal{V}$ is an $n$-dimensional TQFT (i.e. a symmetric monoidal functor - in particular, $\mathcal{V}$ is symmetric), then one can ...
• 2,075
116 views

### Doubt in proof of Thom's Cobordism theorem

When proving the Thom's Cobordism theorem for unoriented manifolds, at some point we are able to create a map $$\phi: MO\rightarrow \bigvee_{i}\Sigma^{|d_i|}K(\mathbb{Z_2})$$ where $MO$ is the Thom ...
• 4,767
111 views

### cancellation theorem in h-cobordism

I'm reading milnor's book h-cobordism, in beginning of the section cancellation theorem, milnor give an example that composition of two elementary cobordism with index $0$ and $1$ may be a product ...
• 1,157
1 vote
59 views

### Problems on definition of genus with multiplicative sequence.

I’m reading Manifolds and Modular Forms. According to its introduction chapter, the definition of genus is a ring homomorphism $\varphi:\Omega\otimes\mathbb{Q}\rightarrow R$ with $R$ an integral ...
• 138
148 views

### Milnor's Lectures on h-cobordism theorem: Lemma 6.2

In the book, Lemma 6.2 (stated below) talks about a corollary of the Thom's isomorphism theorem and Tubular neighbourhood theorem. The proof of the lemma is not provided by the author. And the ...
162 views

### Embedded null-homologous circle in high dimension is bound by disc

Assume we have an oriented manifold $M$ of dimension say $n > 2$ (or if necessary $n > 3$) and an embedded circle $S$ whose fundamental class $[S]$ is zero in homology. Is that circle always the ...
• 591
105 views

### Elementary Morse Cobordism of Diffeomorphic Boundary Components

Let $(M,V,V')$ be a smooth manifold triads. I would like to find a Morse cobordism which is elementary, i.e. there exists Morse function $f:M\to[0,1]$ such that $f^{-1}(0)=V, f^{-1}(1)=V'$ and of only ...
3k views

• 2,271
233 views

### What is the zipper?

1. Defining an "open-closed TFT" Consider the following category of open-closed cobordisms $Cob_2^{o/c}$: Objects are compact oriented smooth one-dimensional manifolds possibly with ...
• 3,058
58 views

### Rigidity in the monoidal cobordism category

1. Defining $Cob_n$ Let $n$ be a positive integer. Define the cobordism category $C=Cob_n$ as follows: Objects are $(n-1)$-dimensional closed, smooth oriented manifolds. A morphism from $M \in Obj(C)$...
• 3,058
1 vote
62 views

### Embedding of a boundary component induces isomorphism on homologies

Let $(A,\partial)$ be a surface with boundary. Pick a relative bordism $W$ from $A$ to itself, i.e. $\partial W \cong (A\cup_{\partial}-A)$. Is it true that inclusion $i:A\hookrightarrow W$ induces ...
• 2,271
58 views

1 vote
180 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 ...
• 1,079
300 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 ...
• 1,063
1 vote
409 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 ...
86 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 ...
• 456
257 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 ...
• 2,438
171 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)
• 8,561
44 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 ...
• 2,802
1 vote
186 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 ...
• 2,110
466 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 ...
220 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 ...
• 115
1 vote
### 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 $ku$?