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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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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Cobordism Groups an the Pontryagin-Thom Construction

I am confused by the statement that $\Omega^\text{framed}_1(S^3) \cong \mathbb{Z}$ which I came across as an application of the Pontryagin-Thom construction for showing that $\pi_3(S^2) \cong \mathbb{...
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Why is the homomorphism $\Omega_* \to H_*(BSO, Q)$ well defined.

Let $\Omega_*$ be the oriented bordism ring of a point. The correspondence of the map in the title is defined by $(K_{M^{N-1}})_*[M^{n-1}]$ where $K_{M^{N-1}}$ is the classifying map of the tangent ...
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Why are bordism groups of a point nontrivial

My definitions of Bordism are from Tom Dieck's Algebraic topology book. Briefly, a singular manifold, $M \xrightarrow{f} pt $, for a closed smooth oriented manifold $(M, \omega)$ without boundary is ...
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What is the map from $H_j( \Sigma MSO(k)) \to H_{j-k}(BSO(k))$ on Tom Dieck page 537

I am reading Tom Dieck's page 537 and I am not sure what the vertical map that I put in the title is in the diagram in the bottom of the page. This map is labeled Thom Isomorphism. Here $MSO(k)$ is ...
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Why is the fibration $BSO(n-1) \to BSO(n)$, $n-1$ connected?

In other words I want to show that the induced map $\pi_k BSO(n-1) \to \pi_k BSO(n)$ is 0 for $k\leq n-1$. The fiber of this fibration is $S^{n-1}$. I am having trouble with the case $k=n-1$. I am ...
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Cobordism and h-cobordism

Is there a way to simply explain cobordism and h-cobordism? I am not looking for a math based explanation, but rather, just the main ideas behind the concepts.
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Is $\mathsf{nCob}$ bicomplete?

Let $\mathsf{nCob}$ be the category of $n$-cobordisms, whose objects are $(n-1)$-dimensional closed manifolds and morphisms are bordisms. Is this category bicomplete, or even finitely bicomplete? ...
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Proof of general Poincaré Conjecture $\dim > 5$

Given $M$ is simply connected, $\dim(M) > 5$ and homotopy equivalent to a sphere. Let $W := M - D_1 \cup D_2$ for two smoothly embedded disjoint disks. How does one see that $W$ is simply ...
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Set consisting of all unoriented cobordism classes of smooth closed $n$-manifolds can be made into additive group? [closed]

How do I see that the set $\mathfrak{N}_n$ consisting of all unoriented cobordism classes of smooth closed $n$-manifolds can be made into an additive group?
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Equivalency of h-cobordisms

I'm reading lectures on the h-cobordism theorem by Milnor and I have a little problem understanding some basic points. I can't understand this theorem , not the theorem , not the proof. I appreciate ...
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Oriented Bordism Abelian Group structure proof

I've started studying cobordism and I found difficult formalising the abelian group structure of the oriented bordism group $\Omega_n$. The main problem I encountered is dealing with the orientation ...
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Bordism between bordisms

I am reading about $2$-categories of bordisms, denoted $Cob_2(n)$. This paper by Lurie (see page 10) states that the objects of such a category are closed $(n-2)$-manifolds and and for $M, N \in Cob_2(...
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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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Clarification about the Thom-Pontrjagin construction as explained in Bredon's book

In Bredon's book, at page 118-119, there is a little chapter about the Thom-Pontrjagin construction, and I'm trying to follow the reason depicted there. He starts with a map $f \colon R^{n+k}\to R^...
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Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure?

So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ...
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Cobordism: Reference Request.

Where does one learn Cobordism theory - is there some canonical text or reference at the beginning graduate level? More general sources on modern differential topology (surgery etc.) are fine; any ...
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Almost complex structure gluing

Consider smooth complex manifold $X$ of complex dimension $n$ and its smooth submanifold $Z$ of codimension $k$. Denote the normal bundle of the inclusion $Z\subset X$ by $\nu.$ One can blow up $X$ ...
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Grassmanians and boudaries of manifolds

Let $M$ be a smooth, compact manifold without boundary. I will say that $M$ is a boundary when there is a smooth, compact manifold with boundary $W$ such that $\partial W=M$. After some lectures I ...
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Hurewicz map factors through bordism homology

I've read in multiple sources that the hurewicz map $h \colon \pi_n(X) \to H_n(X)$ factors through oriented bordism homology. I'm particularly interested in the injectivity of the map $h \colon \pi_n(...
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The trivial cobordism and orientations

I have just come across the definition of a cobordism between two closed oriented $n$-dimensional manifolds $M,M'$ as an oriented $(n+1)$-dimensional manifold $W$ with boundary such that $\partial W =...
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Uncertainties on the details of the Connor-Floyd isomorphism and Formal Group Laws

Let $\Omega^{\bullet}(-)$ be the complex cobordism cohomology. $\Omega^n(X) = \{ (M, f) | f: M \to X \}$ where cobordant maps are identified, $M$ and $X$ are smooth manifolds, and dim($M $) = $n$. ...
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Is $H^*(\mathbf{C} P^\infty)=R[X]$ or $R[[X]]$?

The first ring seems to be what one learns first: the underlying group is the cohomology of the total singular cochain complex $C^*(\mathbf{C} P^\infty)$, which is defined as $\oplus C^n(\mathbf{C} P^\...
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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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About existence of Morse functions

Let's consider 4-manifold $M$, $\partial M = \partial M_1 + \partial M_2 = S^1 \times S^2 + \mathbb{RP}^3$. Is it true that there exist a Morse function $$f\colon M^4 \to [0,1],\quad f^{-1}(0) = \...
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Differential topology question involving cobordism

Prove that if $X$ and $Z$ are cobordant in $Y$, then for every compact 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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Identity in Thom spaces.

Let $T$ be the one-point compattification, $E$ a real vector bundle, $\epsilon$ the trivial line bundle and $\Sigma$ the suspension operation. How can I prove that $$ T(\epsilon \oplus E) \simeq \...
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Bordism classes of real projective plane

I have to prove that doesn't exist a compact $3$-manifold such that $\partial X= \mathbb{R}P^2$. My book suggests to use Euler characteristic and define $Y = X \cup_{\mathbb{R}P^2} X$. What is $X \...
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If $Z$ is an oriented manifold with boundary such that $\partial Z=M$ where $M$ is a compact and oriented manifold then $\chi(M)=0\mod(2)$..

I need some help for showing the following result: Let $M$ be a compact ($\partial M= \emptyset$) and oriented manifold of dimension $n$ and $Z$ an oriented manifold with boundary such that $\...
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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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Axiomatizing oriented cobordism

According to the nLab entry for abstract cobordism categories, the natural way of axiomatizing the relation of two oriented manifolds being cobordant is the following: Definition 1 Two objects $M,...
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Suspension Operation on the Pontryagin-Thom Construction

I have a feeling that this is well-known: View the Pontryagin-Thom construction as the bijective correspondence between $[M,S^r]$ and the set of (appropriate equivalence classes of) framed ...
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Why is Lemma 6.3 of Milnor's Lectures on the h-cobordism Theorem True?

Milnor's statement is: "Let $M^r$ and $N^s$ be sub-manifolds of $V^{r+s}$ which are all smooth, compact, oriented and without boundary. If $p$ is a point of $M^r$ contained in an $r$-cell $U$,...
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How to do this surgery?

Let $L$ be a $0$-framed trivial knot in $S^3 \subset B^4$. Take $B^3 \subset B^4$ such that $B^3$ splits $B^4$ into two and $\partial B^3$ intersects $L$ only two points. Take a neighborhood $U$ of $...
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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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Definition of a 4-cobordism with boundary

Definition of a 3-cobordism (in my context) is a pair $(M, \partial_{-} M, \partial_{+}M)$, where $M$ is a closed orientable topological 3-manifold and $\partial M$ is a disjoint union of $\partial_{-}...
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Cylindrical structure and homology

Let us consider a cobordism $(M, \partial_{-} M, \partial_{+}M)$, where $M$ is homeomorphic to $T \times I$, here $T$ is a torus $S^1 \times S^1$ and $I=[0, 1]$. I encountered the statement "...
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Neutral elements in the unoriented cobordism.

Let $M$ be an $n$-dimensional manifold and let $[M]$ denote the unoriented bordism class of $M$. Forming the usual commutative graded ring $\text{MO}_n$ we know that $$\text{MO}_* \simeq \mathbb{Z}_2[...
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Opposite Orientation of Boundary in Bordisms

In Lurie's "On the Classification of Topological Field Theories" (and certainly other places) he defines the category $\mathbf{Cob}(n)$ who objects are oriented $(n-1)$ manifolds. Given $M,N\in\mathrm{...
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Embedding a manifold in the disk

I don't understand a sentence made by Hirsch in his Differential Topology at page 175: If $k > n+1$ and $M^n = \partial W^{n+1}$, then an embedding $M^n \hookrightarrow S^{n+k}$ extends to a neat ...
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Definition of Cobordism

I am currently trying to read Quillen's Cobordism paper - On the formal group laws of unoriented and complex cobordism theory Bull. Amer. Math. Soc, 75 (1969) pp. 1293–1298. Still on the first page ...