# Tagged Questions

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 a equivalence relation between compact manifolds that is very coarse: two manifolds may be cobordant but not ...

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### rho invariant of manifold

If $G$ is a finite group, then the rational oriented cobordism group $\Omega_{2k-1}^{Stop}(BG)\otimes{\mathbb Q}=0$, so if $N^{2k-1}$ is an orientable odd-dimensional Top manifold with fundamental ...
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### On Steenrod's realization of cycles problem.

There is old problem of realization homology classes of (closed) manifold $M^n$ by fundamental classes of its submanifolds. Partially it was solved by René Thom in his "Quelques propriétés globales ...
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### Framed nullbordant and calculation of framing in coordinate chart

To prove the Hopf degree theorem (theorem 2.37) D. Freed in his notes proves the following lemma I have two questions about this lemma: (1) how to calculate the framing at time $s$? what is the ...
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### What motivated trying to express the signature of a manifold as a linear combination of pontrjagin numbers?

I have tried reading a proof of the signature theorem but it is way beyond me, is there a way to motivate, in english, why anyone even started searching for such a formula? Why would one assume that ...
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### surgery of type $(\lambda,n-\lambda)$ on manifold, h-cobordism theorem by Milnor

I am reading Milnor's Lectures on h-cobordism theorem, and I am stuck on Milnor's definition on surgery of type $(\lambda,n-\lambda)$ on manifold, where the definition following can be found on page ...
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### Proving that for a multiplicative (B,f)-structure $\mathfrak{B}$ (or X-strcture, or B-structure), Thom spectrum $M\mathfrak{B}$is a ring spectrum.

I'm interested in filling the detail of the claim I made above. I'm following Kochman's notation (page 14 for a def.). Actually he never claims it, (he never spoke about ring spectra), but I think ...
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### Building $MU$-spectrum via the definition of a $(B,f)$-structure

I want to construct the them spectrum $MU$ using the definition of spectrum associated to a $(B,f)$-structure. Here are the relevant definitions: A $(B,f)$-structure is a collection of strictly ...
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### Homotopy sets for a pushout of spaces (Seifert-van-Kampen?)

my problem is the following: I have two bordisms $M : \Sigma_0 \to \Sigma_1$ and $M' : \Sigma_1 \to \Sigma_2$, so I can glue them along $\Sigma_1$ to get $M'\circ M$. The manifold $M'\circ M$ is the ...
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### Todd genus as homomorphism from complex cobordism

It's well-known that the Todd genus/arithmetic genus $\chi(\mathcal{O}_X)$ (or probably preferably $\int_X \text{td}(T_X)$ so as to define it in purely terms of the complex structure) is a genus in ...
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### Another clarification about Thom-Pontrjagin construction

This is the second part of the following solved question. [I'm following Bredon's Book]. After explaining the idea behind the "desired" bijection we want to build, Bredon start dealing with the well-...
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### Examples of manifolds that are not boundaries

What are some examples of manifolds that do not have boundaries and are not boundaries of higher dimensional manifolds? Is any $n$-dimensional closed manifold a boundary of some $(n+1)$-dimensional ...
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### Why are homotopy spheres spin-cobordant in dimensions divisible by 4?

Manifolds are assumed to be smooth having dimensions $\geq5$. As usual, $\Omega^{Spin}_{*}$ denotes the spin bordism ring and $\Omega^{SO}_{*}$ the oriented bordism ring. Let $\Sigma^n$ be a closed ...
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### Notation and shorthand of cobordism G=O, SO, U

This is really a basic question: From Wikipedia: "The basic examples are G = O for unoriented cobordism, G = SO for oriented cobordism, and G = U for complex cobordism using stably complex manifolds."...
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### differential in AHSS for spin cobordism

According to these solutions, the differential $d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})$ is the dual of $Sq^2$. Why? This MO post asks a similar question (but about $d_3$ in ...
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### Bordism sanity check

Were framed cobordism and h-cobordism invented to use for different purposes? I have been slightly confused about all the different types of cobordism. Now I am wondering if h-cobordism was invented ...
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### Cobordant to sphere or homotopy sphere

In Ranicki's notes (here) remark 6.19 distinguishes between cobordant to a sphere and cobordant to a homotopy sphere. Earlier in these notes right after example 1.6 he writes that homotopy equivalent ...
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### On framed cobordism

If two manifolds are h-cobordant then their homotopy groups agree. The notion of framed cobordism is supposedly a weaker notion. How much weaker than h-cobordism is it? What can be said about two ...
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### Surgery and boundary

Let $L$ be a framed link in $S^3$ with $m$ components and let $U$ be a closed regular neighborhood of $L$ in $S^3$. Let $B^4$ be a closed 4-ball bounded by $S^3$ so that $U \subset S^3$. Gluing $m$ ...
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