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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66 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 ...
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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\...
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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 ...
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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 $h\colon X\to X$ ...
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85 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?
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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 ...
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52 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}/...
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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 $...
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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 ...
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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 $M \in Obj(C)$...
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Why is this deformation retract well-defined?

I am trying to understand this proof from a paper, but I don't completely understand the deformation retract constructed in the last paragraph. For this deformation retract to be well-defined, we need ...
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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 ...
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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 \...
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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, ...
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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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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,...
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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 ...
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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 ...
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148 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 ...
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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 ...
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109 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 ...
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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)
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 $(n-2)$-...
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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 ...
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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 ...
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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 ...
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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: ...
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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:...
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About natural identifications in knot theory

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 ...
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Is the complex cobordism spectrum, $MU$, a finite spectrum?

Is the complex cobordism spectrum, $MU$, a finite spectrum? If yes, what other examples of finite spectrums there are? Is the Eilenberg-MacLane spectrum finite? What about the connective $K$-theory $...
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Computing $\Omega_1^{\text{Spin}}\cong \Bbb Z_2$

I'm trying to understand why $\Omega_1^{\text{Spin}}\cong \Bbb Z_2$. I know it's a pretty standard computations but I'd like to have an explicit description (and explanation) of what's going on. As ...
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When are homotopy-equivalent 4-manifolds s-cobordant?

Suppose $X$ and $Y$ are closed 4-manifolds, not necessarily simply connected. Such manifolds are said to be s-cobordant if there is a 5-manifold $W$ with $\partial W = X \sqcup Y$ such that the ...
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KO theory v.s. ko theory

It looks that there are different types of topological K-theories, with similar names but they are totally different outputs for the same input. The first theory is called the KO theory. There are ...
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Minimal definition of h-cobordism

Let $(W,M,N)$ be a cobordism between manifolds $M,N$. If the inclusion $M \to W$ is a homotopy equivalence, is the inclusion $N \to W$ also a homotopy equivalence?
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$H^1(M,\mathbb{Z}_2)$: 1st Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure

$H^1(M,\mathbb{Z}_2)$ 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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$\Omega_4^{SO}(K(\mathbb{Z}_2,2))$ v.s. $H^4(K(\mathbb{Z}_2,2),U(1))$: Cocycle form

The $SO$ bordism group of Eilenberg–MacLane space $K(\mathbb{Z}_2,2)$ is $\Omega_4^{SO}(K(\mathbb{Z}_2,2))=\mathbb{Z}_4$. The cohomology group of $K(\mathbb{Z}_2,2)$ with $\mathbb{R}/\mathbb{Z}=U(1)$...
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Cobordism theory for piecewise-linear (PL) and topological manifolds

The Cobordism theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise-linear and topological manifolds. I know the ...
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Cobordant of Dold manifold and Wu manifold via fibered classifying spaces

Background: I think, Dold manifold and Wu manifold are 5-dimensional manifolds which are cobordant to each other via 5-dimensional bordism group: $$ \Omega^{SO}_5. $$ Literally, cobordism theories ...
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Difference between bordism and cobordism

I have looked around for hours and although I have seen many definitions of bordism and cobordism (for some authors these two coincide and for some other not (without mentioning explicitly what's the ...
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How to pair the Arf with Stiefel-Whitney class?

The Arf invariant is a nonsingular quadratic form over a field of characteristic 2. The form that I looked at was: $$ S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[\pi \;...
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Thom space, homotopy group and cohomology group

In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...
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The generators of $\Omega_{10}^{Pin^-}(pt)=\mathbb{Z}_{128} \times \mathbb{Z}_{8} \times \mathbb{Z}_{2}$

From the literature I learned that the Pin$^-$ bordism group of a point in 10 dimensions is: $$\Omega_{10}^{Pin^-}(pt)=\mathbb{Z}_{128} \times \mathbb{Z}_{8} \times \mathbb{Z}_{2}$$ What are their ...
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Cobordant 1-manifolds are homologous?

Let $M$ be a smooth manifold and suppose that $\Sigma \subset M \times[0,1]$ is a surface such that $\partial \Sigma \subset \partial (M \times [0,1])= M \times \{0,1\}$. Denote by $\pi$ the ...