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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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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 ...
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Understanding Corollaries of the h-cobordism theorem

As I understand it the h-cobordism theorem says that if $M, N$ are closed $n$-manifolds and $W$ an $(n+1)$-manifold such that $\partial W = M \coprod \bar N $ then if W, N, M are mutually homotopy ...
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Why is the $J$-homomorphism an isomorphism for $n=1$?

I am trying to proove that $\pi_{n+1}(S^n) \cong \mathbb{Z}_2$ using the Pontryagin-Thom construction and the special case $n=1$ of the $J$-homomorphism $$ J_1:\pi_1(SO(n))\rightarrow \pi_{n+1}(S^n). $...
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Orientable double-cover of the Mobius strip

Let C0 be the cylinder, M1 the Mobius band (a cylinder with 1 twist), and C2 a cylinder with 2 twists (each embedded in R3). Although C0 and C2 are diffeomorphic, intuitively one should think that ...
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Pontryagin-Thom construction references for homotopy groups of spheres

I'm trying to find the details of the Pontriagin-Thom construction proof about the isomorphism between framed cobordism groups and homotopy groups of spheres and I can't find any good reference. I ...
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Formal group law of complex cobordism

Background: The underlying ring of the universal formal group law is, by a theorem of Quillen, the complex cobordism ring $\Omega_U^*$. Let $F_u(x,y) = x+y+\sum_{i,j\geq1}a_{ij}x^iy^j,\ a_{ij} \in \...
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Limitive result in constructing cobordisms for 3-manifolds

I'm just disovering cobordism theory and piecing together the subject from various resources, and the concept of explicitly constructing cobordisms between 3-manifolds is confusing me. Here's my ...
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Handle attachment and spin$^c$ structures

My apology for the uninformative title; I don't think my question can be compressed into one line. I'm trying to understand the relation between handle attaching and spin$^c$ structures. A particular ...
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Cobordism and Boundary Theorem (Guillemin-Pollack)

Prove that if $X$ and $Z$ are cobordant in $Y$, then for every closed manifold $C$ in $Y$ with dimension complementary to $X$ and $Z$, $I_2(X,C)=I_2(Z,C)$. [HINT: Let $f$ be the restriction to $W$ ...
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Cobordisms and compactness

Two compact $n$-manifolds $M_0, M_1$ are said to be cobordant if there is an $(n+1)$-dimensional compact manifold $M$ such that $\partial M = M_0 \sqcup M_1$. What is the necessity of compactness ...
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What are the possible boundaries of connected compact manifolds?

I'm trying to understand what types of manifolds can occur as the boundary of a compact manifold $M$. When $M$'s dimension $d$ is 1, the boundary is always either empty or two points (corresponding to ...
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Showing $\mathbb{R}P^2 \times \mathbb{R}P^2$ and $\mathbb{R}P^4$ do not have the same Stiefel-Whitney numbers.

I'm trying to show that $\mathbb{R}P^2 \times \mathbb{R}P^2$ and $\mathbb{R}P^4$ do not have the same Stiefel-Whitney numbers. I have a theorem that says "If $n+1$ is not a power of 2, then $\mathbb{...
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Disjoint union of the torus and the sphere is a boundary of a compact manifold.

What is the disjoint union of the torus $S^1\times S^1$ and the 2-sphere $S^2$? I ask, because I am trying to prove that the torus is cobordant to the sphere. By definition, this means that their ...
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canonical “simplest” cobordism for fixed boundary

For a given orientable $n$-manifold, is there a canonical choice of $n+1$-manifold that has this manifold as its boundary? For $n< 3$, every orientable manifold can be embedded into Euclidean $n+1$...
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Isomorphism of Principal G-Bundles and Hom($\pi_1(X)$,G)/G

I'm reading this paper and I am trying to wrap my head around the bijection found on the bottom of page 2, namely $$\operatorname{Hom}(\pi_1 (X),G)/G\xrightarrow{\sim} \operatorname{Prin}_G(X)$$ ...
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Does the smooth homotopy between diffeomorphisms have to be through diffeomorphisms in order to induce the same equivalence class of cobordisms?

I'm currently reading through Kock's notes on TQFTs, found here http://mat.uab.es/~kock/TQFT/FS.pdf, and I assume all of his definitions are standard. Suppose we have two diffeomorphic, closed, ...
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Spectral realization of the natural transformation from bordism to singular homology

There is a geometric version of the definition of singular homology $H_n$ in terms of continuous maps of "pseudomanifolds" ($n$-dimensional simplicial complexes such that every $(n-1)$-simplex is ...
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TQFT and Principle G-Bundles

I'm reading this paper. On page 27 near the middle the author writes Since $\pi_1(B)$ for the pair of pants $B$ is the free group on two generators $\mathbb{Z}*\mathbb{Z}$, a principal $G$-...
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Examples of higher dimensional TQFTs

1-dimensional TQFT's assign to every 1-manifold (disjoint union of circles) a vector space and to every surface a linear map between the vector spaces that correspond to the boundary manifolds. So ...
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Bimodules and Tesnors in TQFT

I'm reading this paper and am wondering if the quote below found on page 17 near the bottom makes sense. We need a way to produce a $B\otimes C-A\otimes C$ bimodule: to do this consider the ...
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Pontryagin-dual of the torsion subgroup of oriented bordism group

Consider the oriented cobordism group of $BG$ with $U(1)$ coefficient $\Omega_{SO}^d(BG,U(1))$ with a finite discrete group $G$. question: Explain or show that the Pontryagin-dual of the torsion ...
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Operation of knot cobordism group is well-defined

A knot is a $S^1$ embedded into $S^3$. Knots $K_0, K_1$ are concordant if there is a locally flat cylinder $C \cong S^1 \times [0,1]$ embedded in $S^3 \times [0,1]$ such that the ends $S^1 \times \{i\}...
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Equivalence class of cobordant manifolds is a set

How can it be seen that if we take the category of (oriented) $n$-dimensional compact smooth manifolds with boundary and identify them up to co-bordism that this is in fact a set? I am reading that ...
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On possible isomorphism between first bordism and homology

In geometric topology, one often encounters the following argument(or alike): If two (singular) 1-cycles $a_1,a_2:\Delta \to X$ are homologous, then there exists a bordism(singular surface) $F$ that ...
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Enriched Categories In TQFT

I'm reading the On the Classification of Topological Field Theories and have a question about the use of enriching categories in the definition of a strict 2-category found on page 9. These are ...
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Smith isomorphism between the Spin and the Pin- bordism group

How to show the isomorphism between the following Spin and the Pin- bordism group, known as the Smith isomorphism: $$ \Omega^{Spin}_{d}(B\mathbb{Z}_2)' \to \Omega^{Pin-}_{d-1}(pt) $$ in ...
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Reducing the spin bordism group to its Pontryagin dual of the torsion subgroup

The spin bordism group for the classifying space $BG$ of group $G$ is denoted as $\Omega^{Spin}_d(BG)$. $\Omega^{Spin}_d(pt)$ are computed by Anderson-Brown-Peterson (D. W. Anderson, E. H. Brown, Jr. ...
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Classify manifolds up to diff homeomorphism v.s. the word problem for groups

It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4 – because the word problem for groups cannot be solved. But it is possible to classify manifolds up to ...
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Why is the trace map the only evaluation map applicable to a TQFT

Below is a quote from this paper found on page 8. Dually, the evaluation map is ev(T) = tr(T) for some T ∈ hom(V, V ). Why does $ev(T)$ have to be the trace of $T$? It seems that there are a lot ...