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.
185
questions
7
votes
2answers
199 views
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 ...
5
votes
0answers
168 views
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 $z_n \in H_n(MO; \mathbb F_2)$...
8
votes
1answer
156 views
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 ...
9
votes
1answer
418 views
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 $\...
4
votes
0answers
87 views
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 ...
8
votes
1answer
314 views
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 $\...
3
votes
0answers
250 views
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(...
4
votes
1answer
290 views
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 ...
13
votes
3answers
403 views
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 $...
1
vote
0answers
94 views
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 ...
12
votes
2answers
802 views
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 ...
2
votes
0answers
82 views
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 ...
9
votes
1answer
211 views
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 ...
2
votes
0answers
70 views
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 ...
3
votes
0answers
99 views
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 ...
6
votes
0answers
194 views
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 ...
3
votes
0answers
238 views
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 ...
2
votes
0answers
77 views
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 ...
4
votes
1answer
186 views
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 $b_1(X)=1$. ...
6
votes
1answer
135 views
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 ...
8
votes
1answer
777 views
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 ...
9
votes
1answer
297 views
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 ...
8
votes
1answer
302 views
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\...
11
votes
1answer
234 views
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 ...
5
votes
1answer
130 views
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_{\...
3
votes
1answer
119 views
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 ...
3
votes
0answers
218 views
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 ...
11
votes
1answer
360 views
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.
...
4
votes
0answers
100 views
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 ...
7
votes
0answers
87 views
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 ...
11
votes
0answers
214 views
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 $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
5
votes
0answers
127 views
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
$$
\Omega_5^{...
7
votes
1answer
276 views
Twisted spin bordism invariants in 5 dimensions
[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance!
The spin $G$-bordism invariant can be twisted in the way that ...
14
votes
0answers
280 views
How does quotienting by a finite subgroup act on the framed-cobordism class of a group manifold?
Let $G$ be a connected simple connected compact Lie group, and $\Gamma \subset G$ a finite subgroup. Then (the underlying manifold of) $G$ can be framed by right-invariant vector fields, and this ...
8
votes
1answer
473 views
Critical dimensions D for “smooth manifolds iff triangulable manifolds”
I am aware that at least for lower dimensions,
"smooth manifolds iff triangulable manifolds"
at least for dimensions below a certain critical dimensions D.
My question is that for
...
8
votes
1answer
301 views
Are there non trivial maps from $H\mathbb{Z}$ to $MGL$?
Let $k$ be a field of characteristic $0$. Let us denote by $\mathbf{1}_{k}$ the sphere spectrum. Let $MGL$ be the algebraic cobordism spectrum.
We have the following diagram
$$H\mathbb{Z}\...
4
votes
0answers
80 views
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 ...
5
votes
0answers
137 views
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
$$
\Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots,
$$
...
6
votes
0answers
115 views
Pin cobordism v.s. “KO” theory in low or in any dimensions
Fact: The spin cobordism is equivalent to "KO" theory in low dimension if we only consider the 2-torsion.
This is related to a question and an answer supports the claim.
Here we denote the $p$-...
6
votes
0answers
171 views
Arf-Brown-Kervaire invariant and a surjective map $G \to Pin^-$
We know that the Arf-Brown-Kervaire (abk) invariant is a bordism invariant of
$$
\Omega_2^{Pin^-}(pt)=\mathbb{Z}/(8\mathbb{Z}),
$$
where the $\mathbb{Z}/(8\mathbb{Z})$ is generated by a 2-manifold $M^...
5
votes
0answers
61 views
Triple data for Pontrjagin dual of the Spin bordism group
It is said that the Pontrjagin dual of the 3-dimensional Spin bordism group of $BG$ for $G$ a finite group,
$$
\text{Hom}(Ω^{spin}_3(BG),\mathbb{R/Z}),
$$
can be expressed by triples of cochains $$(w, ...
14
votes
1answer
664 views
The structure of complex cobordism cohomology of the Eilenberg-Maclane spectrum
Let $MU$ be the complex bordism spectrum and let $H\mathbb{Z}$ be the Eilenberg-Maclane spectrum.
Is it know what the structure of the complex cobordism cohomology $MU^{*}(H\mathbb{Z})$ is?
EDIT: ...
4
votes
0answers
90 views
Linear Independence & Integral Homology Cobordism Group
The set of integral homology spheres up to integral homology cobordism forms
an abelian group with the operation induced by the connected sum. This group is called integral homology cobordism group ...
8
votes
0answers
110 views
Relating bordism generators in d and d+2 dimensions — an explicit example
This is an attempt to make my relation between bordism invariants in $d$
and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...
2
votes
1answer
134 views
Extending the maps between the bordism groups: NONE exsistence of a certain kind of extended group
Let $M^d$ be a nontrivial bordism generator for the bordism group
$$
\Omega_d^G= \mathbb{Z}_n,
$$
suppose $G$ (like O, SO, Spin, etc) specify the group structure of the boridsm group. The $\mathbb{Z}...
8
votes
0answers
164 views
“Gerbes” in the cobordism theory
In a lecture I attended today, I heard the use of gerbes in the cobordism theory.
Previously, I use cobordism theory, but I never encounter the term "gerbes" when I work on bordism or cobordism group ...
4
votes
0answers
67 views
Relating bordism invairants in $d$ and $d+2$ dimensions
Are there some relationship between mapping the bordism invairants of eq.1 and eq.2?
$$\Omega_{O}^{d}(B(PSU(2^n)\rtimes\mathbb{Z}_2)) \tag{eq.1}$$
and
$$\Omega_{O}^{d+2}(K(\mathbb{Z}/{2^n},2)) \...
5
votes
1answer
217 views
Manifold generators of O-bordism invariants
If I understand correctly, I can obtain the $O$-cobordism group of
$$
\Omega^{O}_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4,
$$
The 3d cobordism invariants have 4 generators of mod 2 classes, are generated ...
9
votes
2answers
353 views
Künneth formulas/theorem for bordism groups and cobordisms?
We are familiar with Künneth theorem:
The Kunneth formula is given by $R$ as a ring, $M,M'$ as the R-modules, $X,X'$ are some chain complex. The Kunneth formula shows the cohomology of a chain ...
5
votes
1answer
137 views
Relating bordism groups of different dimensions
Let
$M_d$
be a $d$-manifold generator of a subgroup of bordism group
$$
\Omega_d^{G},
$$
or further generalization
$$
\Omega_d^{G}(K(\mathcal{G},n+1)),
$$
which $G$ is the given structure ...
