Cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold.
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Super-cobordisms
One can construct the $d$-dimensional bordism category by declaring the objects to be the $(d-1)$-dimensional compact manifolds without boundary and the morphisms the $d$-dimensional bordisms between ...
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String Orientation and Level Structures
Atiyah, Bott and Shapiro defined orientations of real and complex K-theory that were later refined to maps of ($E_\infty$-ring) spectra
$$MSpin \to KO$$
and
$$MSpin^c \to KU.$$
Likewise, but more ...
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K-theory of the h-cobordism category
I was reading through Kervaire and Milnor's "Groups of Homotopy Spheres", in which the authors begin to compute the groups $\Theta_n$ of h-cobordism classes of homotopy $n$-spheres (with group ...
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Chromatic Spectra and Cobordism
I apologize in advance, if some of the things I've written are incorrect.
The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with ...
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$n$-Fold Framed Functions
Suppose that $M$ is a manifold. One can consider a suitably constructed space of generalized framed Morse functions on $M$, let's call it $\mathrm{Fun}^\mathrm{fr}(M)$. This space is known to be ...
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Is a Morse function always the height function of some embedding? [closed]
Pictures in introductory texts to Morse theory are often drawn as to interpret a Morse function as a height function. Typically, an embedding of a torus into $\mathbb{R}^3$ is drawn, and the Morse ...
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Homology theory represented by Madsen-Tillman spectra
The generalized homology theory of the Thom spectrum $MO=\varinjlim\Sigma^nMTO_n$ is bordism theory:\begin{equation*}\pi_k(MO\wedge X)=\Omega^O_k(X)\end{equation*}These groups form the ring of ...
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75 views
Connected representant of a framed cobordism class (reference needed)
Let $N^n\subseteq M^m$ be a submanifold with a framing of the normal bundle, $2n<m$. Then $N^n$ is framed cobordant (in $M^m$) to something connected.
I believe it could be proved by directly ...
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An orthogonal factorization system on 1-Cob?
Let $1-Cob$ denote the category of oriented 0-manifolds and oriented cobordisms between them. If $W:A\to B$ is a cobordism, i.e. $\partial W\cong A+B$, we write $i^W_{dom}:A\to W$ and $i^W_{cod}:B\to ...
5
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217 views
Connection between framed cobordisms and zero sets
Let $W\subseteq M\times[0,1]$ be a framed submanifold (a framed cobordism in $M$) and $2w<m-1$ where $w,m=\dim W,M$. Assume that $M$ is compact and that $W\cap M\times \{0\}$ is the zero set of a ...
4
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1answer
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How do we handle the symmetry condition in nCob and TQFTs?
A $(n+1)$-topological quantum field theory $\mathcal{T}$ is a rigid symmetric monoidal functor from the category $(n+1)$-Cob of $n$-manifolds and $(n+1)$-cobordisms to FdVect.
My question is about the ...
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extension problem for the Atiyah-Hirzebruch spectral sequence
For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...
4
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2answers
279 views
stability results for the Atiyah-Hirzebruch spectral sequence
For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...
4
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1answer
207 views
Lower dimensional Pin cobordisms
I'm studying Pin cobordism groups of a point for some low dimensions. I found a general result by Anderson, Brown, Peterson in Theorem 5.1 of their paper "Pin cobordism and related topics" ...
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When does a cobordism factorize the sphere?
Let $W: \emptyset \Rightarrow M$ be a smooth cobordism from $\emptyset$ to a smooth closed $n$-manifold $M$. Are there reasonably simple conditions on $W$ which guarantee the existence of another ...
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Question about lower homology class of cobordism
Assume there are three differential oriented manifold $M_0$, $M_1$, $W$ with $\partial W= M_0 \coprod -M_1$. Denote dim $M_0$=dim $M_1$=n, and dim W=n+1.
We know that for the highest homology class, ...
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spectral sequence differential for cobordism
From page 6 of these solutions:
the differential\begin{equation}d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})\end{equation}connecting the 1-st and the 2-nd row is the ...
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“Naïve”cobordism?
The naive view of (say, complex) cobordism of a space $X$ is that it should be the group generated by continuous families of closed manifolds parametrized by $X$. Acting even more naively, one may try ...
4
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124 views
Oriented cobordism group generated by mapping torus
The 4-dimensional oriented cobordism group of closed manifolds is $\Omega^{SO}_4=Z$, and we know that it cannot be generated by a mapping torus, since the Pontryagin for $p_1$ is zero for any mapping ...
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1answer
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Cobordism modulated by a cohomology operation
I've recently encountered the following cobordism theory modulated by a class $\sigma \in H^{d+1}(B^2\mathbb{Z}/2,U(1))$.
My objects are $d$-dimensional spin manifolds with chosen spin structure. ...
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Computing the spin cobordism groups of a CW complex from its cohomology groups
It is shown in Conner and Floyd's book "Differentiable periodic maps", Theorem 14.2, that the oriented bordism group of a CW-complex $X$ can be computed by
$\Omega_p(X) = \sum_{q = 0}^p ...
3
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1answer
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$(BU,f)$ structures on manifolds via stable normal bundles and stable tangent bundles
Background
Consider $BU=colim \, BU_k$ where we take $BU_k$ to be the specific model of classifying space for the group $U(k)\subseteq O(2k)$ given by the quotient space of the infinite real Stiefel ...
5
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1answer
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Construction of Thom-Spectrum for G_2-Structures
The motivation to this question is the paper of Crowley and Nordstrøm "A New Invariant of $G_2$-Structures". I am trying to find a homotopy theoretic interpretation of the following geometric ...
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(∞,n)-category of triangulated cobordisms
What is an accepted definition of a (∞,n)-category of triangulated cobordisms?
Is there one that has a forgetful functor to (Rezk - Hopkins -) Lurie's smooth cobordisms? Does it shed light on how ...
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1answer
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Does a smooth homeomorphism of closed manifolds preserve cobordism fundamental class?
Let $f:M\to N$ be a smooth map of closed oriented smooth manifolds which is also a homeomorphism. Let $[M]\in H_\bullet(M;\mathbb Z)$ denote the fundamental class (and similarly for $N$). It is ...
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2answers
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Computing a cobordism group of manifolds endowed with a real vector bundle with constraints on the Stiefel-Whitney classes
I am interested in computing the cobordism group of oriented manifolds $M$ of dimension 7 endowed with real vector bundles $N$ of rank 5 with the following conditions on the Siefel-Whitney classes:
$ ...
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1answer
322 views
Does the signature admit a homotopy coherent refinement?
Cobordism genera can often be refined to $E_\infty$-orientations in the sense of Ando-Blumberg-Gepner-Hopkins-Rezk:
1) the mod 2 Euler characteristic $MO\to H\mathbb{F}_2$;
2) the $\widehat A$-genus ...
10
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1answer
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Does the paper “On the cobordism ring $\Omega_*$ and a complex analogue II” exist?
I've been investigating the Milnor hypersurfaces, and every reference seems to point to the paper by Milnor, "On the cobordism ring $\Omega_*$ and a complex analogue II". Despite my best efforts, I ...
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1answer
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Units of MO and MU
Real (or complex) cobordism is described by a symmetric ring spectrum MO (or MU respectively) as explained in examples 2.8 and 2.9 here. Associated to such a ring spectrum $R$, we have a unit spectrum ...
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Are Thom spectra MU, MSO and K-theory spectra KU, KO modules over some truncations of the sphere spectrum?
The Thom spectrum MO is a module over the ring spectrum π≤0S=HZ, where S is the sphere spectrum.
In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ*(MO).
On the other hand, MU and ...
3
votes
1answer
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Connection between complex orientations and R-orientations for a ring spectrum R?
We have a well defined notion of complex orientation for a spectrum (coh. theory) $E$, that is, we have a class $x_E\in \tilde{E}^2(\mathbb{C}P^\infty)$ which restricts to identity along the inclusion ...
4
votes
1answer
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Seifert surfaces via Alexander duality
If we take a knot $K$ in $S^3$, there are several ways to construct the associated Seifert surface. One way, which I am not familiar with, I just came across in a paper I am reading. It goes like ...
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1answer
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Diffeomorphism coming from the s-cobordism theorem
Let $f:M_0\rightarrow M_1$ be a diffeomorphism between two compact $n$-dimensional manifolds. Let $W^{n+1}$ be an $h$-cobordism between $M_0$ and $M_1$. Assume that the cobordism has no torsion and ...
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0answers
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Is the pushout of smooth varieties along a smooth closed subvariety again a variety?
The following question is motivated by a desire to find a rough analog in algebraic geometry of the usual notion of gluing of smooth bordisms.
Suppose k is an algebraically closed field of ...
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Cobordism of orbifolds?
Is it possible to setup classical cobordism theory in the context of orbifolds? For example, let's consider the free abelian group generated by oriented smooth orbifolds and quotient by those which ...
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For a finite flat (etale?) morphism $f:Y\to X$, is $f_*1_Y-\deg f . 1_X$ nilpotent in $A^0(X)$, where $A^*$ is the algebraic cobordism?
Let $A^\ast$ be an algebraic oriented cohomology theory (i.e. it is equipped with certain push-forwards for projective morphisms of smooth varieties over the base field $k$; see section 2 of ...
4
votes
2answers
346 views
(Infinite) Suspension Functor on the Pontryagin-Thom Construction
This is a slightly revamped version of a question I asked on the stackexchange forum. That question was asking if the Pontryagin-Thom constructon respects the suspension operation, alluding to stable ...
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votes
1answer
347 views
Geometry Realization of Homology Class
Hello!
My question is about the realization of homology class.
The definition of the realizaion of homology class is: for manifold M and a homology class $z\in H_k(M)$, k is an integer. If we find a ...
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votes
2answers
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Coherent MU_*-Modules
It is proven by Thom that for a finite cw-complex $X$, its $MU$-homology, which, in honor of the authors I'm currently reading, I'll denote by $\Omega_\ast^U(X)$, is a coherent module over ...
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votes
2answers
381 views
Orientation of complex bordism spectrum
I have the following question: If $E$ is a ring spectrum, then a complex orientation of $E$ is an element of $E^2(\mathbb{C}P^{\infty})$ that is mapped to $1$ in $E^2(\mathbb{C}P^{1})$.
I have read ...
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0answers
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Extending psc metrics
Let $S^1$ denote the circle with the non-trivial spin structure, i.e. $0\neq[S^1]\in\Omega^{Spin}_1$. Considering characteristic numbers it is easy so see that $S^1\times\mathbb{H}P^3$ is spin null ...
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1answer
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Atiyah-Bott-Shapiro Orientation
Dear community,
there are so-called orientation maps $a:MSpin\to ko$ and $b:MSpin^c \to k$, "defined" in ABS's paper "Clifford modules". Unfortunately I am not familiar with representation theory.
...
4
votes
1answer
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Computation of KO characteristic classes/numbers
How to compute KO characteristic classes/numbers?
They were introduced by Anderson/Brown/Peterson to study the structure of the spin cobordism ring. I looked through the literature but I did not find ...
4
votes
4answers
530 views
Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles
I am looking for characteristic classes of vector bundles (either real or complex) with values in generalized multiplicative cohomology theories such that:
i) they vanish if the bundle of unit ...
13
votes
1answer
676 views
Does the bordism homology theory satisfy the weak equivalence axiom?
There is an interesting and important homology theory called bordism. Briefly speaking, a singular manifold in a space $X$ is a pair $(M, f)$ where $M$ is a closed smooth manifold and $f : M \to X$ is ...
10
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1answer
859 views
Reference request for TQFT, functoriality
I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds.
It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says;
The function $(M, \partial_{-}M, ...
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votes
0answers
175 views
Extension of homeomorphism of boundaries to a homeomorphism of a cobordism
Suppos we have a cobordism $(M, \partial_{-}M, \partial_{+}M)$, where $M$ is a oriented compact (topological) 3-manifold.
Assume we have orientation preserving homeomorphism $f_{\pm}: \Sigma \to ...
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1answer
238 views
Mapping class group and cylindrical structure
Let us fix a torus $\Sigma=S^1 \times S^1$. We consider a cylinder $\Sigma \times I$ and a data $(\Sigma\times I, \Sigma\times 0, \Sigma\times 1, f_{0},f_{1})$. Here $f_{i}$, called parametrization, ...
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Isomorphism of cobordisms
Let $(M, \partial_{-}M, \partial_{+}M)$ be a decorated 3-cobordism, where $M$ be a (topological) 3-manifold and $\partial M=-\partial_{-}M \cup \partial_{+}M$.
(decorated in a sense of Turaev, Quantum ...
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1answer
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Embedded (framed) cobordisms
[The title initially was "Actions of gauge groups on framed cobordisms. This has been changed.]
This question is a follow-up to my answer to When is a submanifold of $\mathbf R^n$ given by global ...
