# Questions tagged [cobordism]

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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### Extending an embedding with trivial normal bundle

I am recently studying the book Notes on Cobordism Theory by R. E. Stong and I have noticed that the proposition below is (implicitly) used (for example to extend a $(B,f)$ structure on a boundary of ...

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### How to show that, $ \mathrm{CH}_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \simeq \Omega_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} $?

Let $ X $ be a $ n $ - dimentional oriented closed real manifold ( i.e : compact and without boundary ).
Can you tell me how to show that,
$$ \mathrm{CH}_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \simeq ...

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### What is known about exotic spheres up to stable diffeomorphism?

In even dimensions $n=2k$ we can define two smooth manifolds $M$ and $N$ to be stably diffeomorphic if they become diffeomorphic after the connect sum with $r$ many copies of $S^k \times S^k$ for some ...

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### Are there alternate descriptions of `elementary cobordisms'?

Let $M^d$, $N^d$ be cobordant $d$-manifolds. Then $M^d \sqcup \bar{N}^d = \partial W^{d+1}$ for some $(d+1)$-manifold $W$. This cobordism can be implemented via an elementary set of 'moves' called ...

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### Can one show corbordism invariance of the Crane-Yetter state-sum using simplicial methods / are there 'Pachner-like' moves for cobordisms?

Let $\mathcal{C}$ denote some Unitary Braided Modular Fusion Category. It is well known that the Crane-Yetter state-sum, $Z_{CY}(\bullet|\mathcal{C})$ is an oriented-cobordism invariant. In other ...

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### Visualize how the 5d Dold manifold and Wu manifold are cobordant via a 6d manifolds with boundaries

Are there simple intuitions and arguments to visualize why the following two 5-manifolds are cobordant to each other with the oriented structures? (They can be two boundaries of 6-dimensional oriented ...

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### Diffeomorphisms of manifolds with boundary

I repeat this, which I posted in Math Stack, where it got some attention but no answer.
If two compact manifolds have diffeomorphic interiors and diffeomorphic boundaries, are they then diffeomorphic? ...

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### The first Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure v.s. $H^1(M,\mathbb{Z}_2)$

$H^1(M,\mathbb{Z}_2)$ specifies the 1st cohomology class of manifold $M$ (can be regarded as spacetime) with $\mathbb{Z}_2$ coefficient,
it is often to see that we say the 1st Stiefel Whitney class
$$...

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### Prove or disprove that there exists no $G$ structure with its bordism group $\Omega_1^{G} =\mathbb{Z}/N$ for $N>2$

It can be found that there are the following bordism group $\Omega_0^{G}$ at $d=0$ and 1 dimensions by requiring $G$ structure for $d$-manifolds:
$$
\Omega_0^{SO} = \mathbb{Z} , \quad \Omega_1^{SO} = ...

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### Does the Gysin map in $K$-theory respect bordism?

Let $X_1$ and $X_2$ be two closed spin$^c$ manifolds that are bordant via a spin$^c$ manifold-with-boundary $W$.
Let $Z$ be a closed spin$^c$ manifold with $\dim Z=\dim X_1$ mod $2$. Let
$$f_1:X_1\to ...

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### Simple examples of equivariant cobordism

Let $Y$ be an oriented 3-manifold with a free action by a finite group $G$. If I understand correctly, there exists a multiple of $NY$ of $Y$ and an oriented manifold $X$ such that $\partial X = NY$ ...

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### Oriented cobordism classes represented by rational homology spheres

Any homology sphere is stably parallelizable, hence nullcobordant. However, rational homology spheres need not be nullcobordant, as the example of the Wu manifold shows, which generates $\text{torsion}...

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### Path integral derivation of extended TQFT

I know this isn't exactly a math question, but I am asking it here anyway. We define an extended TQFT to be a functor (preserving tensor products) from the $\left(\infty,n\right)$-category of ...

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### Generators for unitary bordism ring $\pi_*(MU)$

I’m reading Pengelley’s paper “The mod 2 homology of $MSO$ and $MSU$ as $\mathfrak A^*$ comodule algebras, and the cobordism ring”.
He has chosen very special generators $z_n \in H_n(MO; \mathbb F_2)$...

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### Generators and relations for the 2-dimensional unoriented cobordism category

It is very well known in the field of TQFT that the 2-dimensional oriented cobordism category is generated by the disk and the pair of pants (each going in both directions), subject to a finite set of ...

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### About the cohomology of $BG^\delta$. Making a Lie group discrete

Let $G$ be a connected Lie group. Recall that the topological group $G^\delta$ is $G$ endowed with the discrete topology. The inclusion $G^\delta \to G$ induces a map between the classifying spaces $\...

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### Any cobordism invariant made of “characteristic classes”, on unorientable manifolds, must be a mod 2 class?

For any cobordism invariant (or simply bordism invariant) quantity $\omega$ that satisfy the conditions:
$\omega$ can be fully decomposed from the cup product of characteristic classes (such as ...

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### Cobordism invariants: topological v.s. geometric

Some cobordism invariants are not cohomology classes. Such as the $\mathbb{Z}_{16}$-valued eta invariant $\eta$ of $\Omega_4^{Pin^+}$, the $\mathbb{Z}_8$-valued Arf-Brown-Kervaire invariant ABK of $\...

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### mod $p$ homology of Thom spectra MSU

Using pairing in Atiyah-Hirzebruch spectral sequence one can show that homology of $BU(n)$ is a free abelian group with basis $\alpha_{k_1}\cdots\alpha_{k_t}$, $k\leqslant n$, where $\alpha_{i} = \big(...

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### Bordism groups of $X$, Thom isomorphism and characteristic numbers

Recap: bordism group
An oriented singular $n$-manifold in $X$is a map $f:M^n\to X$ where $M$ is a finite disjoint union of $n$-dimensional smooth manifolds.
The empty set is an admissible oriented ...

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### Does every $SL_2\mathbb{C}$ representation of a closed oriented surface extend over a compact oriented three-manifold?

Let $F$ be a compact oriented surface and $\rho:\pi_1(F)\rightarrow SL_2\mathbb{C}$ be a representation. Does there exist a compact oriented three-manifold $M$ with $\partial M=F$ and a homomorphism $...

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### Boundary map for Mayer-Vietoris sequence for Bordism

I am trying to reproduce some of the details in how the Mayer-Vietoris sequence for bordism should go, especially in showing exactness using this definition of the boundary operator. I have tried to ...

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### Cobordism and Kirby calculus

It may be a simple question but I wonder to ask:
Is it possible to draw a homology cobordism between $3$-manifolds by using the techniques of Kirby calculus?
At least, for instance, Brieskorn ...

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### Computation of mod p homology of $MSU$

I am trying to proof Novikov theorem
\begin{equation}
MSU_*\otimes \mathbb Z[\frac 1 2] \cong \mathbb Z[\frac 1 2][y_2, y_4, \ldots],\quad \deg y_i = 2i.
\end{equation}
This can be proved by using ...

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### Topological Spin manifolds in dimension 4

In his ICM Adress at Nice (Proceedings of the International Congress of Mathematicians Nice, September, 1970, Gauthier-Villars, editeur, Paris 6 e ,1971, Volume 2, pp. 133-163.),
Robion Kirby ...

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### Realizing an amalgamated product of groups by splitting a closed manifold along a codimension 1 submanifold

In the paper "A splitting theorem for manifolds" by S.E. Cappell,
https://www.maths.ed.ac.uk/~v1ranick/papers/capsplit.pdf
the following "inverse" of the Seifert-van Kampen theorem for closed ...

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### About the proof of Milnor-Novikov theorem about multiplicative generators of (complex) bordism ring

I am trying to understand part of Milnor-Novikov theorem about multiplicative generators of $MU_* \cong \mathbb Z[x_1, x_2, \dots]$ using S.Kochman’s “Bordsim, Stable Homotopy and Adams Spectral ...

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### Flatness, “Continuously varying fibers”, and bordism

It is commonly said that a flat map of schemes $f : X \rightarrow Y$ is like a map with "continuously varying fibers". We see a hint at this in the result that $\text{dim} X_y$ is constant when $f$ is ...

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### Correction to Milnor's h-cobordism book

This is a cross-post from MSE.
These four screenshots from milnor's book baffled me a bit (pages 24, 50, 51 and i-iii resp.):
In first one, there is no Theorem 3.1 in the book, but there is def 3.1.
...

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### Positive scalar curvature and $\mathbf{H}P^2$-bundles

Let $M$ be a simply-connected spin-manifold of dimension $n\geq 5$. The Atiyah-Bott-Shapiro orientation $\mathrm{MSpin} \to KO$ produces an element $\alpha(M)$ of $\pi_n KO$. Results of Gromov-Lawson ...

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### Cobordism modelling fibration over $S^1$

Let $X$ be a closed oriented manifold which is a fibration over $S^1$ whose fiber $F$ is connected, i.e. $X\cong F\times[0,1]/\sim h$, for an $h\in \mathrm{Diff}(F)$.
Suppose that $b_1(X)=1$. ...

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### Diffeomorphism type of the added sphere in simply connected surgery

A classical result of simply connected surgery theory, is that if two normal maps $f:M_i\rightarrow X$, $i=0,1$ are normally cobordant and if the dimension of the manifolds is odd, there exists a ...

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### Cobordism Theory of Topological Manifolds

Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds.
Cobordism Theory for DIFF/Differentiable/smooth manifolds
However, there are ...

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### Orientable with respect to complex cobordism?

I have learned that an orientation of a manifold $M$ with respect to ordinary cohomology is an ordinary orientation, that an orientation with respect to complex K-theory is a Spin$^c$ structure, and ...

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### Third differential in the homology AHSS

I need some guidance in identifying the third differential in the homology AHSS for $\Omega_{\ast}^{\text{Spin}^c}(X)$ in degrees $\leq 4$.
Remember that $\pi_0(M\text{Spin}^c)=\Bbb Z$, $\pi_2(M\...

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### Unoriented bordism with twisted orientation

The computation of the unoriented bordism group of the point $N_*=\Omega_*^O$ is a classic result.
I would like to know a related bordism group, where we specify the twisted fundamental class $[M]\in ...

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### Decompose $MT(E(d)\times_{\mathbb Z_2} SU(2))$ as the wedge sum or smash product of spectra

Consider the extension
$$1\to SU(2)\to X\to O\to1,$$
there are 4 possibilities for $X$:
$X=O\times SU(2)$ or $E\times_{\mathbb{Z}_2}SU(2)$ or $Pin^+\times_{\mathbb{Z}_2}SU(2)$ or $Pin^-\times_{\...

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### Basic question on the cobordism spectrum

I am reading a little about cobordism and I have a basic question, which makes sense both in the topological and motivic setting. Let $\mathrm{Gr}_{n,\infty}$ denote the infinite $n$-Grassmanian and ...

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### What mathematical background to i need in order to understand proofs of the h-cobordism theorem?

I am about to finish my undergraduate studies and I really enjoyed the topology and differential-geometry classes. I'd love to continue studying differentialtopology and i considered doing some ...

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### Reference on complex cobordism

I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism.
...

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### Conformal group and cobordism

In this post, I am exploring my thoughts on the implementation of conformal symmetry group structure and cobordism relations.
Namely, I like to know what has been done and explored in the past?
on ...

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### Relate two different mod 2 indices: $\eta$ invariant and the number of zero modes of Dirac operator, associated to SU(2)

My major question in this post here is that:
How can we relate the following two mod 2 indices:
$\eta$ invariant,
the number of the zero modes of the Dirac operator $N_0'$ mod 2,
associated to ...

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### Madsen-Tillmann spectrum $MTE$ of the group $E$ which is defined in Freed-Hopkins's paper

In Freed-Hopkins's paper, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...

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### Twisted spin-bordism invariant and a possible Postnikov square from $d=2$ to $d=5$

This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post.
I had discussed my computation of
$$
\Omega_5^{...

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### Twisted spin bordism invariants in 5 dimensions

[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance!
The spin $G$-bordism invariant can be twisted in the way that ...

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### How does quotienting by a finite subgroup act on the framed-cobordism class of a group manifold?

Let $G$ be a connected simple connected compact Lie group, and $\Gamma \subset G$ a finite subgroup. Then (the underlying manifold of) $G$ can be framed by right-invariant vector fields, and this ...

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### Critical dimensions D for “smooth manifolds iff triangulable manifolds”

I am aware that at least for lower dimensions,
"smooth manifolds iff triangulable manifolds"
at least for dimensions below a certain critical dimensions D.
My question is that for
For ...

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### Are there non trivial maps from $H\mathbb{Z}$ to $MGL$?

Let $k$ be a field of characteristic $0$. Let us denote by $\mathbf{1}_{k}$ the sphere spectrum. Let $MGL$ be the algebraic cobordism spectrum.
We have the following diagram
$$H\mathbb{Z}\...

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### Decomposition of bordism groups for $BG$ where $G$ is a product of two groups

Let a group $G=G_1 \times G_2$, where
$G_1$ is a discrete group (can be finite or infinite),
$G_2$ be any compact Lie group or finite group.
Question: Is there some simple result that we can ...

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### Categorification-like statement in the cobordism group?

Suppose we consider a $d$-cobordism group classifying manifolds with $H$-structure and with a classifying space $BG$ of a group $G$, written as
$$
\Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots,
$$
...