# 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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### 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 ...

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### 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 $...

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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 ...

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### 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 ...

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### 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 ...

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### 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 (...

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### 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 $b_2(W)=1$? ($\cdots$...

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### 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''...

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### 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), ...

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### 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 ...

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### 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. ...

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### Cobordism cohomology of Lie groups

Are there any results about cobordism cohomology of Lie groups？For example, $\mathrm{MU}^*(\mathrm{SU}(n))$.

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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, ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 = \...

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### 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 ...

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### 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....

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### 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 ...

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### 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 ...

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### 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 $M$.
$$\...

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### 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 ...

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### 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 (...

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### 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 &...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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. ...

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### 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 ...

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### 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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### 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 ...

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### Analog of Cerf theory in PL

Is there an analog of Cerf theory in PL?
More specifically, given two handle decompositions of a PL (relative) cobordism $W$, is it always possible to go from one handle decomposition to the other via ...

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### (Algebraic) cobordism and the rank function

I write the question for algebraic cobordism but I have the analogue question for classic cobordism.
The spectrum representing algebraic cobordism
$$
\mathbf{MGL}=(*, \mathrm{Th}(1) , \ldots , \mathrm{...

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### Extending an embedding with trivial normal bundle

I am recently studying the book Notes on Cobordism Theory by R. E. Stong and I have noticed that the proposition below is (implicitly) used (for example to extend a $(B,f)$ structure on a boundary of ...

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### How to show that, $ \mathrm{CH}_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \simeq \Omega_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} $?

Let $ X $ be a $ n $ - dimentional oriented closed real manifold ( i.e : compact and without boundary ).
Can you tell me how to show that,
$$ \mathrm{CH}_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \simeq ...

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### 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 ...

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### 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 ...

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### Can one show corbordism invariance of the Crane-Yetter state-sum using simplicial methods / are there 'Pachner-like' moves for cobordisms?

Let $\mathcal{C}$ denote some Unitary Braided Modular Fusion Category. It is well known that the Crane-Yetter state-sum, $Z_{CY}(\bullet|\mathcal{C})$ is an oriented-cobordism invariant. In other ...

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### 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 ...

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### Diffeomorphisms of manifolds with boundary

I repeat this, which I posted in Math Stack, where it got some attention but no answer.
If two compact manifolds have diffeomorphic interiors and diffeomorphic boundaries, are they then diffeomorphic? ...

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### The first Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure v.s. $H^1(M,\mathbb{Z}_2)$

$H^1(M,\mathbb{Z}_2)$ specifies the 1st cohomology class of manifold $M$ (can be regarded as spacetime) with $\mathbb{Z}_2$ coefficient,
it is often to see that we say the 1st Stiefel Whitney class
$$...

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### Prove or disprove that there exists no $G$ structure with its bordism group $\Omega_1^{G} =\mathbb{Z}/N$ for $N>2$

It can be found that there are the following bordism group $\Omega_0^{G}$ at $d=0$ and 1 dimensions by requiring $G$ structure for $d$-manifolds:
$$
\Omega_0^{SO} = \mathbb{Z} , \quad \Omega_1^{SO} = ...

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### Does the Gysin map in $K$-theory respect bordism?

Let $X_1$ and $X_2$ be two closed spin$^c$ manifolds that are bordant via a spin$^c$ manifold-with-boundary $W$.
Let $Z$ be a closed spin$^c$ manifold with $\dim Z=\dim X_1$ mod $2$. Let
$$f_1:X_1\to ...

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### 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 $\partial X = NY$ ...

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### Oriented cobordism classes represented by rational homology spheres

Any homology sphere is stably parallelizable, hence nullcobordant. However, rational homology spheres need not be nullcobordant, as the example of the Wu manifold shows, which generates $\text{torsion}...

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### Rational cobordism classes of manifolds fibered over spheres

Let us fix positive integers $k, m$. Let $A^k_{4m} \subset \Omega^{\text{SO}}_{4m} \otimes \mathbb Q$ be the subgroup generated by oriented cobordism classes of manifolds fibered over $S^k$.
The ...

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### Path integral derivation of extended TQFT

I know this isn't exactly a math question, but I am asking it here anyway. We define an extended TQFT to be a functor (preserving tensor products) from the $\left(\infty,n\right)$-category of ...