**17**

votes

**5**answers

901 views

### Book recommendation for cobordism theory

I am planning to organize a seminar on cobordism theory and I'm looking for a reference. Such a reference is preferably a book, but I'm open to other ideas.
The audience is familiar with ...

**3**

votes

**0**answers

97 views

### Constructing a “nice” cobordism

Denote by $\Sigma_g$ the closed, orientable surface of genus $g$. I want to construct a cobordism $M_g$ between $\Sigma_g$ and $\Sigma_{g+1}$ with the following two nice properties:
1) $M_g$ is an ...

**7**

votes

**2**answers

192 views

### Does $\mathfrak{N}_4$ contain at least four distinct elements?

How do I see that the set $\mathfrak{N}_4$ consisting of all unoriented cobordism classes of smooth closed $4$-manifolds contains at least four distinct elements?

**2**

votes

**1**answer

87 views

### open book decompositions and being a boundary

Are there examples of smooth closed manifolds (not necessarily oriented) that admit an open book decomposition but that are not the boundary of any compact smooth manifold?

**3**

votes

**1**answer

85 views

### If two knots in $S^3$ are invertible cobordant (from both ends), are they equivalent?

Let $K_1,K_2$ be two knots in $S^3$ and assume that there exists a cobordism $(W;K_1,K_2)$ which is invertible from both ends. Does this imply that $K_1, K_2$ are equivalent? In the paper by D.W. ...

**2**

votes

**1**answer

113 views

### Which bordism classes fiber over the circle?

Let $\mathcal{G}$ denote a (stable) tangential structure such as $O$, $SO$, $Spin$, or $Pin^\pm$. Which bordism classes $[M,f]\in\Omega_*^\mathcal{G}(X)$ are represented by an $f:M\rightarrow X$ where ...

**2**

votes

**0**answers

77 views

### Choice of framing in Gravitational Chern Simons

I was trying to understand formula(2.21) in Witten's paper "Quantum Field Theory and Jones Polynomial"(link: https://projecteuclid.org/euclid.cmp/1104178138) (Page 360).
There, it was mentioned, the ...

**6**

votes

**1**answer

248 views

### Naive G-spectrum representing geometric equivariant cobordism

Let $G$ be a finite group. By the transversality results of Wasserman $G$-equivariant bordism (say real) should be a naive homology theory, and as such it should be represented by a naive G-spectrum.
...

**1**

vote

**0**answers

97 views

### 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 $X$ ...

**3**

votes

**1**answer

191 views

### Why is an extended T(Q)FT called fully local?

Hopefully this question does not double another. If so, don't bother to close this.
An extended topological quantum field theory is sometimes called, 'fully local".
Why is that? I can imagine that ...

**0**

votes

**0**answers

60 views

### Product structure on manifolds via lifting classifying maps

Let's say you want to study $d$-dimensional manifolds $M$ which decompose functorially into $M\cong N\times P$ for a fixed $P$. Can this structure be expressed by a lift of the stable normal bundle?
...

**3**

votes

**1**answer

206 views

### How to write the Thom spectrum representing cobordism as an $\Omega$-spectrum?

It is often said [e.g. Atiyah, "Bordism and Cobordism" (1961)] that the Thom spectrum $MSO(i)$ represents oriented cobordism, in the following sense:
\begin{eqnarray}
MSO^n(X,Y) &:=& \lim_{i ...

**26**

votes

**1**answer

529 views

### Nilpotence of the stable Hopf map via framed cobordism

The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...

**18**

votes

**1**answer

472 views

### 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 ...

**10**

votes

**1**answer

249 views

### 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 ...

**7**

votes

**1**answer

428 views

### 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 ...

**17**

votes

**0**answers

315 views

### 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 ...

**4**

votes

**0**answers

186 views

### $n$-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 ...

**3**

votes

**2**answers

379 views

### 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 ...

**13**

votes

**1**answer

572 views

### 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 ...

**1**

vote

**1**answer

81 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 ...

**4**

votes

**3**answers

239 views

### 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**

votes

**1**answer

223 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**

votes

**1**answer

236 views

### 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 ...

**2**

votes

**0**answers

194 views

### 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**

votes

**2**answers

317 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**

votes

**1**answer

232 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" ...

**9**

votes

**0**answers

122 views

### 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 ...

**4**

votes

**2**answers

211 views

### 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, ...

**4**

votes

**0**answers

130 views

### 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 ...

**4**

votes

**0**answers

214 views

### “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 ...

**5**

votes

**0**answers

151 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 ...

**5**

votes

**1**answer

575 views

### 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. ...

**5**

votes

**0**answers

175 views

### 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**

votes

**1**answer

291 views

### $(BU,f)$ 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**

votes

**1**answer

277 views

### 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 ...

**4**

votes

**0**answers

195 views

### (∞,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 ...

**9**

votes

**1**answer

553 views

### 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 ...

**8**

votes

**2**answers

351 views

### 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:
$ ...

**12**

votes

**1**answer

343 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 ...

**12**

votes

**1**answer

426 views

### 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 ...

**13**

votes

**1**answer

540 views

### 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 ...

**9**

votes

**1**answer

820 views

### 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

**1**answer

316 views

### 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 ...

**5**

votes

**1**answer

252 views

### 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 ...

**0**

votes

**1**answer

306 views

### 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 ...

**4**

votes

**0**answers

246 views

### 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 ...

**23**

votes

**2**answers

965 views

### 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 ...

**3**

votes

**0**answers

162 views

### 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

**2**answers

359 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 ...