# Tagged Questions

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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### Does the theorem that genera vanishing on even-dim complex projective bundles are elliptic also apply for integral-valued genera?

Ochanine proved in this paper that for genera taking values in $\mathbb{Q}$-algebras, vanishing on even-dimensional projective bundles is equivalent to being an elliptic genus (i.e. a specialization ...

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votes

**1**answer

211 views

+50

### Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely,
$$
\mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x.
$$
...

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votes

**0**answers

86 views

### Sphere spectrum, Thom spectrum, and Madsen-Tillmann bordism spectrum

This is a following up question of Sphere spectrum, Character dual and Anderson dual.
What are the differences and the significances of the following:
(1). Homotopy classes of maps from a Thom ...

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votes

**2**answers

390 views

### Sphere spectrum, Character dual and Anderson dual

The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.
However, could you help me to appreciate the mathematical meanings of the following:
What is the significance of ...

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**0**answers

74 views

### Twisted spin cobordism v.s. KO theory in low dimensions

Based on the background info and this this webpage, here is a more advanced problem:
Question: If we consider a different more subtle twisted structure, like
$${\Omega_d^{(\mathrm{spin} \times G)/N}},...

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votes

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114 views

### Spin cobordism v.s. KO theory in low or in any dimensions

It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.
If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$
...

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88 views

### Relating bordism groups of $\Omega_{d}^{Spin_c}$ and $\Omega_{d}^{(Spin \times SU(N))/\mathbb{Z}_2}$ to that of $U(N)$

I felt that the earlier question may be too challenging, so let me provide a different angle and more infos to tackle an easier and separate problem.
Let us consider a more explicit a short exact ...

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votes

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93 views

### Bordism groups and a short exact sequence

Let us consider a short exact sequence:
$$
1\to N\to G\to Q \to 1,
$$
where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups).
Suppose I have the data and the computations ...

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votes

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112 views

### Pairing 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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votes

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83 views

### Cobordant of 5d manifolds, and the generalization of bordisms

Some of the 5-dimensional manifolds are (co)bordant via oriented cobordism.
For example, if I understand correctly, 5-dimensional Dold manifold and Wu manifold are manifolds which are cobordant to ...

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votes

**1**answer

128 views

### Stable normal framings of parallelizable manifolds

Suppose $M$ is a compact, connected, orientable manifold ($\dim M=m$) with trivial tangent bundle and let $j \colon M \to \mathbb R^n$ be an embedding. Suppose we choose a trivialization of $TM$. Then ...

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votes

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70 views

### Reference Request: Equivariant Symplectic bordism

Non-equivariantly, symplectic bordism has been developed extensively by Ray, Gorbunov, and specially S. Kochman in this memoir: http://dx.doi.org/10.1090/memo/0496 Yet the coefficients ...

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votes

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277 views

### 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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votes

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371 views

### Is the boundary of a manifold topologically unique? [duplicate]

Let $X$ be a manifold without boundary and let $Y$ and $Z$ be two manifolds with boundary such that $X$ is homeomorphic to their interiors: $X \cong Y^\circ \cong Z^\circ$. Does it follow that $Y \...

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votes

**1**answer

107 views

### 2-morphisms for Bord(n)

I am currently reading in Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors, and have a (I guess) pretty basic question for my understanding of the (...

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**1**answer

364 views

### Is there a PL, or topological, bordism hypothesis?

The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with ...

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vote

**0**answers

52 views

### Does this condition imply symplectic birational cobordism?

From the definition of symplectic birationality given here (https://arxiv.org/pdf/0906.3265.pdf, Definition 2.1), two compact symplectic $2n$-manifolds $(M_{1},\omega_{1}),(M_{2},\omega_{2})$ are ...

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votes

**1**answer

316 views

### Twisting bordism classes

Let $X$ be a reasonable topological space (I'd be happy to assume that $X$ is a smooth closed manifold) and let $f\colon M^n \rightarrow X$ be a continuous map from a smooth oriented $n$-manifold $M^n$...

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votes

**1**answer

280 views

### Relative Steenrod's problem

Thom's theorem states that for every homology class $\alpha \in H_{*}(X)$ there exists an integer $k = k(\alpha)$ such that the class $k\, \alpha$ comes from the fundamental class of an orientable ...

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130 views

### Simply connected homology cobordisms

I'm looking for interesting examples of a homology 3-sphere $Y$ for which there exists a smooth, simply connected homology cobordism from $Y$ to itself (or simply to another homology 3-sphere $Y'$, ...

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345 views

### Functor of points definition of the Thom space

Let $X$ be a space (CW complex) and let $E \to X$ be a vector bundle.
Using the language of $\infty$-categories we can can define the Thom space $T(E)$ as the pointed space representing the ...

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127 views

### Can the cobordism hypothesis be formulated as a statement about adjoint functors?

I would like to formulate the cobordism hypothesis for general tangential structure as a statement about adjoint $(\infty,1)$-functors.
For a space $Y$ with an action of $O(n)$ let $X=Y\times_{O(n)} ...

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votes

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89 views

### Landweber Exact Functor Theorem for Cohomology

I have seen the Landweber exact functor theorem beeing used to retrieve cohomology theories, in particular singular cohomology and K-Theory.
However the statement of the theorem itself is always ...

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votes

**1**answer

326 views

### Is there a closed 5-manifold $M$ with $w_1(M)w_2(M)\ne 0$?

I'm trying to find generating manifolds for the cobordism group $\mathit{MO}_5(K(\mathbb Z/2, 2))\cong (\mathbb Z/2)^4$, which can be represented as the cobordism group of closed 5-manifolds $M$ ...

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votes

**1**answer

207 views

### Oriented Bordism Group and Un-Oriented Bordism Group of points $pt$

Do we know, or are there any References that list down complete oriented and unoriented Bordism Group $Ω_{n,O}(pt)$ and $Ω_{n,SO}(pt)$ of points $pt$ for dimensions $n=1,2,...,10$?
Here are some ...

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votes

**1**answer

209 views

### What is the growth rate of the number of unoriented cobordism classes?

Let $\Omega_n^O$ denote the abelian group of cobordism classes of closed, unoriented manifolds of dimension $n$, and let $d(n) := \lvert\Omega_n^O\rvert$. What are the asymptotics of $d(n)$?
It's ...

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votes

**1**answer

136 views

### Manifold bounded by Sp(2) realisable inside End(H^2)?

The Lie group $Sp(2) = \{A\in GL(2,\mathbb{H})\mid A^\dagger A = I \}$ has a variety of nice geometric aspects. One of which is that it is the boundary of the disk bundle $D(V)$ of the rank-1 ...

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224 views

### More on categories of modules over the algebraic cobordism spectrum

I have the following questions on monoidal model structure(s) for the motivic stable homotopy category $SH(k)$ (where $k$ is a field); certainly, I am also interested in general statements concerning ...

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124 views

### Explicit cobordism between Wu manifold and Dold manifold P(1,2)?

The Wu manifold $SU(3)/SO(3)$ and the Dold manifold $P(1,2)$, the latter being defined as $(S^1\times \mathbb{C}P^2) / (p, x) \sim (-p, \overline{x})$, are cobordant because they are both generators ...

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votes

**1**answer

110 views

### Submanifold generator of Pontryagin-dual of the torsion subgroup of the $Pin^+$ bordism group (dimension 4th)

I am interested in figuring out the following submanifold generator of a $Pin^+$ Bordism or Cobordism group in the dimension 4, say for a $Pin^+$ cobordism group of the classifying space $BG=BSU(2)$:
...

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votes

**1**answer

380 views

### Cobordisms and Euler characteristics

I am trying to understand exactly which role the Euler characteristic plays in (smooth) cobordism theory, and especially why the answer seems to depend on the dimensions of the manifolds in question. ...

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votes

**1**answer

138 views

### Cobordism/bordism group based on orbifolds with corners

We define a geometric homology group of a topological space $X$ as follows: the chain complex $C_{\bullet}$is freely generated by the maps $f$ from a compact oriented orbifold with corners $P$ to $X$, ...

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votes

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364 views

### Is the bordism from disjoint union to connected sum universal for connected manifolds?

Let $M_1$ and $M_2$ be two oriented, connected, closed $n$-manifolds. It is known that the disjoint union $M_1 \sqcup M_2$ and the connected sum $M_1 \# M_2$ are cobordant, via a bordism $\Sigma_{M_1, ...

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215 views

### Building examples of elements of $\Omega_4(\xi)$ via surgery theory: how to do it?

When computing special bordism groups, I often need to determine existence of (singular) smooth $4$-manifolds with fixed fundamental group and certain properties like the spin behaviour (i.e. being ...

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votes

**1**answer

248 views

### Is equivariant oriented cobordism finite?

It is known that for $n \not\equiv 0 \mod 4$, the oriented cobordism ring $MSO_n$ is finite. That is, for oriented n-dimensional manifold $Y$, there exists $m\in \mathbb{N}$, such that $mY$ bounds.
...

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votes

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153 views

### Variants and Generalizations of Arf (-Brown-Kervaire) invariants

(1) I encounter the Arf invariants in Kirby-Taylor, Pin structures on low-dimensional manifolds. 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)} ...

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votes

**1**answer

269 views

### Computation of $\Omega^{Pin-}_{d}(B\mathbb{Z}_2)$ and Smith isomorphism

question: I am looking for the literature with the result or the computation of Pin- bordism group: $\Omega^{Pin-}_{d}(B\mathbb{Z}_2)$. Can someone point out some useful ways to do this or any helpful ...

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votes

**1**answer

258 views

### Spin bordism group of classifying space $BG$ with a finite Abelian $G$

The spin bordism group for the classifying space $BG$ of group $G$ can be denoted as $\Omega^{Spin}_d(BG)$.
For example, $\Omega^{Spin}_d(pt)$ are computed by Anderson-Brown-Peterson (D. W. Anderson, ...

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votes

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### Low dimensional homotopy fibration TOP(M) -> TOP(int(M))

In the thesis of Nancy Cardim she proves that for $M$ a topological manifold of dim $\geq 5$ with connected boundary, there exists a homotopy fiber sequence
$C(\partial M)\rightarrow TOP(M) \...

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votes

**1**answer

336 views

### Vanishing of characteristic numbers vs vanishing of characteristic classes

A famous result by Thom states that Oriented Bordism classes are determined by characteristic numbers; specifically, two closed manifolds are orientedly bordant if and only if they have the same ...

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votes

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381 views

### Spin 4-manifold bounded by a mapping torus of tori

Consider a smooth torus endowed with the non-bounding spin structure. Pick a basis of its first homology and a diffeomorphism inducing the S-transformation
$\left(\begin{array}{cc} 0 & 1 \\-1 &...

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votes

**1**answer

329 views

### A proof of Theorem 9.2.12. in the Gompf-Stipsicz

I'm seeking for a proof of Theorem 9.2.12. in the Gompf-Stipsicz "4-Manifolds and Kirby Calculus" (for the statement, see the following image). But the textbook omits any proofs and only gives a ...

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votes

**1**answer

222 views

### Cohomology operations on unoriented cobordism

In unoriented cobordism there exist stable cohomology operations looking similar to Steenrod squares (they were used by Quillen to compute the unoriented cobordism ring with its formal group law ...

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votes

**1**answer

391 views

### Under what condition is a fiber bundle cobordant to the trivial bundle?

Let $E$ be the total space of a fiber bundle with base $B$ and fiber $F$, where $B$ and $F$ are smooth manifolds.
Under what condition is $E$ unoriented cobordant to $B\times F$?
And what happens ...

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votes

**1**answer

337 views

### Is $MGL$ an $H\mathbb{Z}$-algebra?

Let $\mathrm{MGL}$ be the $\mathbb{P}^1$-ring spectrum over a field $k$ representing algebraic cobordism. Suppose, for simplicity, that $k$ is of characteristic 0. Let $H\mathbb{Z}$ be the motivic ...

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votes

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158 views

### For which cobordism theories framed manifolds not bound?

If $X$ is a complex and $\xi:X\to BO$ is a map, when is the natural map from the stable stem $\pi_*^S\to \pi_{*}( M\xi)$ injective, where $M\xi$ is the associated Thom spectrum?
For $MO$ or $MU$, ...

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votes

**1**answer

184 views

### Algebraic cobordism (of a point) outside the geometric diagonal

This question is about the state of current knowledge regarding Voevodsky's algebraic cobordism of a point $\mathrm{MGL}^{*,*}(\mathrm{Spec}\,k)$. That the geometric diagonal $\mathrm{MGL}^{2*,*}(\...

**27**

votes

**1**answer

592 views

### Is there an explicit description of a cobordism between $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$?

With a little bit of work, one can show that $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$ have the same Stiefel-Whitney numbers, so by a theorem of Thom, they are (unorientedly) cobordant.
...

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votes

**6**answers

2k 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 ...

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votes

**1**answer

151 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 ...