Questions tagged [cobordism]

Cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold.

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Why does the bordism category have duals?

The proof of the Cobordism Hypothesis outlined by Lurie seems to assume that the $(\infty, n)$-category $\mathbf{Bord}_n^{(X, \zeta)}$ has duals, i.e. duals for objects and adjoints for all $k$-...
1 vote
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Reshetikhin-Turaev invariants from extended 3d TQFTs

Attached to any object $V\in \mathcal{C}$ of a ribbon category $\mathcal{C}$, Reshetikhin and Turaev have defined knot invariants $$\tau_V(K)\ \in\ \text{End}_{\mathcal{C}}(1_{\mathcal{C}})$$ for ...
2 votes
2 answers
184 views

Inverse of a smooth concordance of smooth knots

We say that a smooth concordance of smooth knots C' is inverse to C if the concatenation C•C' is smoothly isotopic to the trivial cylinder. I wonder if there are any known ways of inverting smooth ...
0 votes
1 answer
340 views

Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]

We know that framing structure means the trivialization of tangent bundle of manifold $M$. string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
1 vote
1 answer
138 views

Lie group framing and framed bordism

What is the definition of Lie group framing, in simple terms? Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...
2 votes
0 answers
62 views

Framed bordism and string bordism in 3-dimensions vs topological modular form

In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable ...
2 votes
1 answer
142 views

string bordism group and framed bordism group for $d \leq 6$ and $d \geq 7$

Why do the string bordism group and the framed bordism group coincide the same in dimensions lower than 7 ($d = 0,1,2,3,4,5, 6$)? Why do the string bordism group and the framed bordism group differ ...
4 votes
1 answer
167 views

Version of pseudo-isotopy $\neq$ isotopy for $(n+1)$-framings

Let $M$ be a closed $n$-manifold and $\varphi$ be a self-diffeomorphisms of $M$. There is a bordism from $M$ to itself given by $M\times [0,1]$ with the identification $M \cong M \times \{0\}$ induced ...
6 votes
0 answers
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"Inclusion" between higher categories of framed bordisms?

Let $\mathrm{Bord}_n$ be the bordism $(\infty, n)$-category of unoriented manifolds. It can be viewed as an $(\infty, n+1)$-category whose $n+1$-morphisms are equivalences. If $n$ is large enough, ...
3 votes
1 answer
103 views

When are homologous embedded surfaces in 3-manifolds related by embedded cobordisms?

Let $M$ be an orientable closed 3-manifold and suppose $A$ and $B$ are embedded incompressible closed orientable surfaces in $M$ with $[A] = [B]$ in $H_2(M,\mathbb{Z})$. In general, there are a ...
3 votes
2 answers
271 views

Cut a homotopy in two via a "frontier"

Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$. (...
2 votes
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52 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 ...
2 votes
0 answers
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Turning cocycles in cobordism into an inclusion or a fibering

By the classical Pontryagin-Thom construction, we know that the cobordism group $\Omega^n_U(X)$ is represented by cocyles $$ M\hookrightarrow X\times \mathbb{R}^{2k}\rightarrow X,$$ where $M$ is a ...
5 votes
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132 views

Geometric interpretation of pairing between bordism and cobordism

In page 448 of these notes, a pairing between bordism and cobordism $$\langle \ ,\ \rangle: U^m(X)\otimes U_n(X)\rightarrow \Omega^U_{n-m}$$ is defined as follows. Assume $x\in U^m$ is represented by $...
9 votes
1 answer
305 views

Torsion and nilpotence of framed manifolds under the Pontryagin-Thom map

The framed Pontryagin Thom construction produces a graded ring isomorphism from the framed cobordism ring $\Omega^{fr}_*$ to the stable ring of homotopy groups of spheres $\pi_*(\mathbb{S})$. I have a ...
8 votes
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Generators for unstable cobordism

I am looking for explicit descriptions of generators of some low-dimensional unstable cobordism groups. For example, $\mathbb CP^2$ embeds into $\mathbb R^7$ by a result of Haefliger. Because it has ...
7 votes
1 answer
246 views

Bordism for oriented triangulable manifolds without smooth differentiable structures

We know the bordism group for oriented smooth differentiable structures such as $\Omega_d^{SO}$ that requires the special orthogonal group structure on the tangent bundle $TM$ of manifold $M$. $$\...
11 votes
1 answer
333 views

Lower bounds for Betti numbers of a manifold given its boundary?

Let $B$ be some compact, path connected $n$-manifold without boundary such that its cobordism class is trivial, so that there exists some other $n+1$ manifold $M$ with $\partial M= B$. While there is ...
1 vote
0 answers
176 views

Proving a Result About Pontryagin Numbers Without Forms

I've been reading the book Geometry of Differential Forms by Shigeyuki Morita, and I came across the following theorem on page 226 the other day: Proposition 5.53 (Pontryagin). Two cobordant closed (...
6 votes
2 answers
377 views

How to prove that Lie group framing on S^1 represents the Hopf map in framed cobordism

The Pontryagin-Thom construction gives an isomorphism from the stable homotopy groups of spheres and framed cobordism groups. It seems to be well-established that for dimension 1 (see this question), ...
7 votes
1 answer
252 views

Lens space bounding a topological, simply-connected 4-manifold with $b_2=1$

The following is written in section 1.6 (p.7) of this paper: https://arxiv.org/pdf/1010.6257.pdf. ($\cdots$) Which lens spaces bound a smooth, simply-connected 4-manifold $W$ with $b_2(W)=1$? ($\cdots$...
4 votes
1 answer
208 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''...
5 votes
1 answer
189 views

Is there a way to calculate the Froyshov $h$-invariant for Seifert homology spheres?

In 2002, by using Floer theory, Froyshov defined the $h$-invariant for intergal homology 3-spheres, which is a surjective group homomorphism $\Theta^3_{\Bbb Z}\to \Bbb Z$, where $\Theta^3_{\Bbb Z}$ is ...
3 votes
0 answers
118 views

Is there a framed nullbordism of $T^4$ with an action of $T^4$ that extends the self-action?

Under the identification of the stable homotopy groups with the (stably) framed bordism groups, it is well known that $\eta\in\pi_1\mathbb{S}$ is represented by $S^1$ with its Lie group framing. ...
6 votes
1 answer
545 views

Cobordism cohomology of Lie groups

Are there any results about cobordism cohomology of Lie groups?For example, $\mathrm{MU}^*(\mathrm{SU}(n))$.
2 votes
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Representative manifold of $\mathbb{Z}_4^T$

I want to know which manifold is the representative manifold (I do not know the correct terminology in math) $\mathcal{M}$ of $\mathbb{Z}_4^T$ in the following sense: $\mathcal{M}$ is unorientable, ...
14 votes
1 answer
876 views

What is known about exotic spheres up to stable diffeomorphism?

In even dimensions $n=2k$ we can define two smooth manifolds $M$ and $N$ to be stably diffeomorphic if they become diffeomorphic after the connect sum with $r$ many copies of $S^k \times S^k$ for some ...
6 votes
1 answer
363 views

Do $h$-cobordism groups arise from a 'Thom-like' spectrum?

Not thinking about $h$-cobordism, one usually defines a cobordism between manifolds, realizes it is an equivalence relation, chooses an appropriate class of structured manifolds (framed, unoriented, ...
2 votes
0 answers
109 views

About connected cobordism and surgery

I need to find various ways of performing two surgeries on a collection of circles so that the resulting 2-dimensional cobordism (the trace of the surgeries) is connected. How can I find these ? up ...
12 votes
1 answer
298 views

Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?

In my recent research, I need to know if the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ are always nontrivial in the unoriented and oriented bordism rings for $n>2$. (For the ...
4 votes
0 answers
240 views

Summary of different types of TQFT?

For the purposes of this question, a TQFT comprises the following data: An "upper dimenison" $n \in \mathbb N$. A "lower dimension" $0 \leq l \leq n$. A choice of structure ...
4 votes
1 answer
240 views

A cobordism theory from Hirsch's "Differential Topology" (reference request)

The following is exercise 5 on p. 176 in Hirsch's "Differential Topology" (corrected 6th printing): Let $\eta = (p,E,B)$ be a fixed vector bundle over a compact manifold $B$, $\partial B = \...
10 votes
1 answer
326 views

Simple examples of equivariant cobordism

Let $Y$ be an oriented 3-manifold with a free action by a finite group $G$. If I understand correctly, there exists a multiple of $NY$ of $Y$ and an oriented manifold $X$ such that $\partial X = NY$ ...
11 votes
0 answers
171 views

Natural knot homology

All knot homology theories I've seen share a flaw: their definitions explicitly use some combinatorial choices (such as a diagram presentation). The coin, however, has two sides and the other one is ...
1 vote
1 answer
165 views

Reference request: full classification of surfaces as being cobordant to $S^2$ or $\mathbb{R}P^2$

It is known that cobordism provides a complete classification of surfaces as: a surface is cobordant to either $S^2$ or $\mathbb{R}P^2$. I am looking for a reference with contains a proof of this fact....
8 votes
0 answers
175 views

Conner-Floyd Chern classes and $E$-(co)homology of $BU$

In his book, Stable homotopy and generalised homology, Adams computes the $E$-(co)homology of $BU$ for a complex oriented cohomology theory $E$. In II.4, he first describes the $E$-homology of $BU$ as ...
4 votes
1 answer
338 views

h-cobordisms between non-simply-connected 4-manifolds

Let $M_0^4$ and $M_1^4$ be two closed smooth 4-manifolds and let $M$ be an $h$-cobordism between them (i.e., a compact smooth 5-manifold with boundary the disjoint union of $M_0$ and $M_1$ and with ...
8 votes
1 answer
531 views

What motivated Thom to relate the cobordism groups with some homotopy groups?

I would like to know what motivated or led Thom to think that the (un)oriented cobordism groups would correspond with the homotopy groups of some structure (Thom spectum), or with the coefficient ...
49 votes
4 answers
6k views

Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?

I'm looking for an elegant proof that any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$.
9 votes
1 answer
417 views

Oriented bordism in higher dimensions (e.g. $12 \leq d \leq 28$)

The classification of oriented compact smooth manifolds up to oriented cobordism is one of the landmarks of 20th century topology. The techniques used there form the part of the foundations of ...
5 votes
0 answers
271 views

Factorization homology and topological conformal field theories

My question concerns some of the results of Costello's "Topological conformal field theories and Calabi-Yau categories" and how they are related/ can be rederived via the description of (...
6 votes
0 answers
217 views

Topological vs smooth (equivariant) bordism group

In Remark 1.31 of this work, it is claimed that "Standard arguments in Pontryagin-Thom theory imply that the relevant smooth and topological G bordism groups are isomorphic." The adjective &...
6 votes
0 answers
74 views

Rank of matrix coming from cobordism computations

In a computation of Pontryagin-numbers of certain manifolds (see the appendix of https://arxiv.org/pdf/2109.10306.pdf for more context) we came across the following elementary problem: Consider the ...
5 votes
1 answer
150 views

Is spin cobordism an invariant for surgery of codimension $q\ge3$?

Recall that a surgery of codimension $q$ on an $n$-manifold $X$ is a modification of $X$ of the following type. Let $\Sigma^{n-q}\subset X$ be a smoothly embedded $(n-q)$-sphere with a trivialized ...
4 votes
1 answer
203 views

Cobordism class of projectivization of a bundle

I was reading the book "Differentiable Periodic Maps" by P.E. Conner (1979). I am stuck at the following problem given at the end of section 21: Let $\xi\to V^n$ be a $k$-plane bundle over a ...
0 votes
1 answer
253 views

When do cobordism groups depend on differential structure? [closed]

I heard that cobordism group with structures sometimes depend on differential structure of space. Do you know any examples or references about this facts? I want to know when difference occur between ...
4 votes
0 answers
164 views

Infinite dimensional topological quantum field theories?

A topological quantum field theory is a functor $Z:Cob(n) \to Hilb$ mapping from the $n$-dimensional cobordisms to $Hilb$, the category of Hilbert spaces with morphisms being bounded linear operators. ...
8 votes
0 answers
216 views

Computation of the 3-dimensional $\mathbb{Z}/m$-equivariant spin cobordism group (with possibly non-empty fixed-point set)?

$\newcommand{\odd}{\mathrm{odd}}\newcommand{\ev}{\mathrm{ev}}$Consider tuples of the form $(Y,\mathfrak{s},\widehat{\sigma})$ where: $Y$ is a closed oriented 3-manifold, $\mathfrak{s}$ is a spin ...
8 votes
3 answers
2k views

Representing rational homology by manifolds

Let $X$ be a topological space (say a manifold). A result of R. Thom states that the pushforwards of fundamental classes of closed, smooth manifolds generate the rational homology of $X$. This work ...
1 vote
0 answers
168 views

Mapping class group and surgery theory

Given a smooth manifold $M$ of dimension $n$ and a diffeomorphism $\phi: M \to M$, we can construct a smooth cobordism of dimension $(n+1)$ from $M$ to $M$ by gluing $M \times [0,1]$ with itself by $\...

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