All Questions
21 questions with no upvoted or accepted answers
12
votes
0
answers
266
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 $Y'$, ...
- 779
9
votes
0
answers
287
views
Rational cobordism classes of manifolds fibered over spheres
Let us fix positive integers $k, m$. Let $A^k_{4m} \subset \Omega^{\text{SO}}_{4m} \otimes \mathbb Q$ be the subgroup generated by oriented cobordism classes of manifolds fibered over $S^k$.
The ...
- 11.9k
8
votes
0
answers
219
views
Computation of the 3-dimensional $\mathbb{Z}/m$-equivariant spin cobordism group (with possibly non-empty fixed-point set)?
$\newcommand{\odd}{\mathrm{odd}}\newcommand{\ev}{\mathrm{ev}}$Consider tuples of the form $(Y,\mathfrak{s},\widehat{\sigma})$ where: $Y$ is a closed oriented 3-manifold, $\mathfrak{s}$ is a spin ...
- 299
8
votes
0
answers
125
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 ...
- 2,147
8
votes
0
answers
198
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 ...
- 10.5k
6
votes
0
answers
209
views
"Inclusion" between higher categories of framed bordisms?
Let $\mathrm{Bord}_n$ be the bordism $(\infty, n)$-category of unoriented manifolds.
It can be viewed as an $(\infty, n+1)$-category whose $n+1$-morphisms are equivalences.
If $n$ is large enough, ...
- 890
6
votes
0
answers
184
views
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 ...
- 10.5k
6
votes
0
answers
142
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 $p$-...
- 10.5k
6
votes
0
answers
224
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^...
- 2,147
6
votes
0
answers
84
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 $$(w, ...
- 10.5k
5
votes
0
answers
348
views
Are there alternate descriptions of `elementary cobordisms'?
Let $M^d$, $N^d$ be cobordant $d$-manifolds. Then $M^d \sqcup \bar{N}^d = \partial W^{d+1}$ for some $(d+1)$-manifold $W$. This cobordism can be implemented via an elementary set of 'moves' called ...
- 545
5
votes
0
answers
161
views
Twisted spin-bordism invariant and a possible Postnikov square from $d=2$ to $d=5$
This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post.
I had discussed my computation of
$$
\Omega_5^{...
- 10.5k
5
votes
0
answers
150
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
$$
\Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots,
$$
...
- 10.5k
5
votes
0
answers
67
views
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 ...
- 2,044
5
votes
0
answers
155
views
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) \...
- 93
4
votes
0
answers
243
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 ...
- 10.5k
3
votes
0
answers
122
views
Is there a framed nullbordism of $T^4$ with an action of $T^4$ that extends the self-action?
Under the identification of the stable homotopy groups with the (stably) framed bordism groups, it is well known that $\eta\in\pi_1\mathbb{S}$ is represented by $S^1$ with its Lie group framing. ...
- 2,052
3
votes
0
answers
194
views
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
$$...
- 2,147
3
votes
0
answers
170
views
Pairing the Arf with Stiefel-Whitney class
The Arf invariant is a nonsingular quadratic form over a field of characteristic 2.
The form that I looked at was:
$$
S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[\pi \;...
- 10.5k
2
votes
0
answers
55
views
Tangential normal invariant isomorphism
Recently, I was reading the paper "Finite Group Actions on Kervaire Manifold" by Crowley, Hambolton. But I am having problem understanding a definition. Here it is,
In page 15-16 they are ...
2
votes
0
answers
430
views
The signature of a mapping torus
Consider a manifold $M$ of dimension $4k + 2$, $k$ an integer. Pick a diffeomorphism $\phi$ of $M$ and construct the mapping torus $T$ of $\phi$. Suppose that there is a $4k+4$ dimensional manifold $B$...
- 1,582