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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).

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Definition of genus: cobordism or Thom space

I know two definitions of a genus: a ring homomorphism $$\Omega_*^{U}\rightarrow R$$ or a ring homomorphism $$MU_*\rightarrow R$$ where $\Omega_*^{U}$ is the complex cobordism ring, and $MU_*=\...
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Let $f,g:M\rightarrow N$ be smooth maps transverse to $X\subset N$. Are $f^{-1}(X)$ and $g^{-1}(X)$ cobordant?

Let $F,G:M\rightarrow N$ be smoothly homotopic maps so that $F$ and $G$ are transverse to $X\subset N$, are $F^{-1}(X)$ and $G^{-1}(X)$ then cobordant to one another? I saw this statement in these ...
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Two manifolds are unoriented cobordant if and only if they have the same Stiefel–Whitney numbers

Could anyone explain this fact to me and/or refer me to a source with an adequate explanation? Assume that we know the dimension of each cobordism group $\mathfrak{N}_n$ as a vector space over $\...
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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 ...
Crash Bandicoot's user avatar
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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 $n$-...
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Torsion spin-c structures on $S^1\times S^2$

I have been reading the paper https://arxiv.org/pdf/1902.04050.pdf by Zemke and at some point we have the following: My question is, how can one make sense of torsion $\mathrm{Spin}^c$ structures on $...
horned-sphere's user avatar
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Help understanding Morse theory proof (Milnor)

I'll start with the statement of the theorem and its proof, and I'll end by explaining my difficulty understanding the proof. What follows are not Milnor's original words, but rather my best attempt ...
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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 $...
timaeus's user avatar
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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 ...
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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' ...
LeonardoOiler's user avatar
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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 ...
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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 ...
Timotheus Hauptinius's user avatar
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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-...
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How did Thom calculate $MSO(k)$ using Silber's polyhedron in his 1954's paper?

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(\...
Tongtong Liang's user avatar
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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\...
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Question about Milnor hypersurface

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 ...
Victory's user avatar
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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 ...
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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 ...
Someone's user avatar
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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 ...
taiat's user avatar
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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 ...
Sqr's user avatar
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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 ...
Prerak Deep's user avatar
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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 ...
YoungMath's user avatar
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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 ...
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Why care about homotopy equivalence up to $\vee S^2$ or $\# (S^2 \times S^2)$?

I have read that in the study of 4-manifolds (specifically, their classification up to homotopy) that a popular/useful notion of equivalence is that up to a wedge of 2-spheres or connected sum of $S^2\...
Descartes Before the Horse's user avatar
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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 ...
Noel Lundström's user avatar
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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 $h\colon X\to X$ ...
NothingInSense's user avatar
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$\mathbf{RP}^m \times \mathbf{RP}^n$ 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?
annie marie cœur's user avatar
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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 ...
annie marie cœur's user avatar
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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}/...
annie marie cœur's user avatar
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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 ...
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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, ...
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Inclusion of a punctured disk inside a cylinder with the strings cut off.

This is slightly related to my previous question Higher homotopy groups of a string complement in a cylinder., but now I'm only interested in a following one: We call the image of a smooth embedding $...
Mihail's user avatar
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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 ...
Max Demirdilek's user avatar
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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 $M \in Obj(C)$...
Max Demirdilek's user avatar
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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 ...
Mihail's user avatar
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Relationship between a certain scalar and any known notions of curvature in geometry

Disclaimer. I'm not an expert geometer, so feel free to fix my language if I use the wrong words... Let $S$ be a smooth $(n-1)$-dimensional surface in $\mathbb R^n$, with "inside" $A \subseteq \...
dohmatob's user avatar
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3 votes
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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, ...
Tomáš Bzdušek's user avatar
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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 ...
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1 vote
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Additional structure on complex cobordism?

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,...
Grisha Taroyan's user avatar
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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 ...
Anne O'Nyme's user avatar
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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 ...
Paladin's user avatar
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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 ...
Aliceyuenyuen's user avatar
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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 ...
deepfloe's user avatar
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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 ...
Zest's user avatar
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4 votes
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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)
C.F.G's user avatar
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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 ...
Hang's user avatar
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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 ...
Sam Jaques's user avatar
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12 votes
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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 ...
Angelo Brillante Romeo's user avatar
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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 ...
deeppurp's user avatar
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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 $ku$?
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