All Questions
42 questions
6
votes
1
answer
373
views
Fivebrane bordism $\Omega_d^{\mathrm{Fivebrane}}$
$\newcommand{\Fr}{\mathrm{Fr}}\newcommand{\Fivebrane}{\mathrm{Fivebrane}}\newcommand{\String}{\mathrm{String}}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$What ...
- 10.5k
2
votes
0
answers
175
views
Minimal first Pontryagin class $p_1=1$?
From Hirzbuch theorem,
the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$.
I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$.
Is ...
- 447
4
votes
1
answer
341
views
Induced fiber sequence and Eilenberg–MacLane space in Whitehead tower of $BO$
In Whitehead tower of $BO$, there is a induced fiber sequence:
1.
$$
Z_2 \to B SO \to BO \overset{w_1}{\rightarrow} B Z_2
$$
How does this map $\overset{w_1}{\rightarrow}$ from $BO$ to $B Z_2$?
...
- 447
1
vote
0
answers
145
views
Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
Namely, how do we know
$$
K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)?
$$
Naively -- in each step ...
- 447
11
votes
3
answers
1k
views
Computation on characteristic classes
I am organizing a reading seminar on characteristic classes. The audience in the seminar is interested in symplectic and contact manifolds. I work in categorification and would like to compute some ...
- 341
5
votes
1
answer
244
views
Coefficient of the top Pontryagin class in $L$-genus
The $L$ genus can be expressed as combinations of the Pontryagin classes with the first few terms as follows:
$$L_1=\frac{1}{3}p_1,$$
$$L_2=\frac{1}{45}(7p_2-p_1^2),$$
$$L_3=\frac{1}{945}(62p_3-...
- 707
0
votes
0
answers
266
views
Define a characteristic class on a simplicial complex (non-manifold)
Given a simplicial complex with only triangulation and only branching structure, is it enough to define Stiefel–Whitney class?
(Please provide Yes or No answers, and reasonings.)
Given a fixed ...
- 10.5k
6
votes
0
answers
184
views
Visualize how the 5d Dold manifold and Wu manifold are cobordant via a 6d manifolds with boundaries
Are there simple intuitions and arguments to visualize why the following two 5-manifolds are cobordant to each other with the oriented structures? (They can be two boundaries of 6-dimensional oriented ...
- 10.5k
3
votes
0
answers
194
views
The first Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure v.s. $H^1(M,\mathbb{Z}_2)$
$H^1(M,\mathbb{Z}_2)$ specifies the 1st cohomology class of manifold $M$ (can be regarded as spacetime) with $\mathbb{Z}_2$ coefficient,
it is often to see that we say the 1st Stiefel Whitney class
$$...
- 2,147
4
votes
0
answers
273
views
Instanton numbers for diverse gauge bundles on diverse manifolds --- their relations to characteristic classes
It is standard (?) that the $SU(N)$ gauge theory has the instanton number $n$ quantized as $n \in \mathbb{Z}$
$$
n = { 1 \over 8\pi^2} \int_{\mathcal{M}_{4}} \text{tr} \left(F \wedge F\right) = {1 \...
- 10.5k
18
votes
1
answer
1k
views
Wu formula for manifolds with boundary
The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=...
- 1,329
5
votes
0
answers
225
views
Generalizing the formula between Wu class and the Steenrod square
I know that on the tangent bundle of $M^d$, the corresponding Wu class and the Steenrod square satisfy
$$
Sq^{d-j}(x_j)=u_{d-j} x_j, \text{ for any } x_j \in H^j(M^d;\mathbb Z_2) .
\tag{eq.1}$$
...
- 10.5k
7
votes
0
answers
319
views
Different definitions of Stiefel-Whitney classes
It is quite easy to show that different definitions of the Stiefel-Whitney classes agree by showing that they satisfy the well-known axioms. Nevertheless I have been asking myself wether one can prove ...
- 1,174
8
votes
2
answers
478
views
Can one disjoin any submanifold in $\mathbb R^n$ from itself by a $C^{\infty}$-small isotopy?
Let $M$ be a manifold and $V$ be an oriented vector bundle. It's well known that if the Euler class of $V$ is non zero, then $V$ can't have a non-vanishing section. The converse is not true, see ...
- 14.3k
10
votes
1
answer
378
views
Discrete Pin structures
It is clear that an oriented manifold $M^n$ (with dimension $n$) admits spin structures if and only if its second Stiefel-Whitney class $[w^2]\in H^2(M,\mathbb Z_2)$ vanishes. In the construction of ...
- 10.5k
4
votes
0
answers
290
views
Generalized Postnikov square
Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, ...
- 1,329
4
votes
1
answer
354
views
(Co)bordism invariant of Eilenberg–MacLane space becomes vanished
Consider a (co)bordism invariant
$$
u_2 Sq^1 u_2+Sq^2 Sq^1 u_2
$$
obtained from
$$
\Omega^5_{O}(K(\mathbb{Z}/2,2)).
$$
Here $u \in H^2(K(\mathbb{Z}/2,2),\mathbb{Z}_2)$. The $K(\mathbb{Z}/2,2)$ is ...
- 2,147
5
votes
1
answer
263
views
Pontryagin square, Postnikov square and their consistency formulas
$\mathcal{P}_2$ is Pontryagin square
$$H^{2i}(M,\mathbb Z_{2^k})\to H^{4i}(M,\mathbb{Z}_{2^{k+1}}).$$
$\mathfrak{P}$ is the Postnikov square $$H^2(M,\mathbb Z_3)\to H^5(M,\mathbb Z_9).$$
question (i)...
- 10.5k
3
votes
0
answers
170
views
Pairing the Arf with Stiefel-Whitney class
The Arf invariant is a nonsingular quadratic form over a field of characteristic 2.
The form that I looked at was:
$$
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
11
votes
1
answer
454
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 $M$ ...
- 6,881
12
votes
2
answers
1k
views
A Compact Manifold with odd Euler characteristic whose tangent bundle admits a field of lines
I understand that the top Stiefel Whitney class is an obstruction for the tangent bundle of a manifold to have a trivial line sub-bundle. I am looking for a counterexample when removing the word "...
- 953
3
votes
1
answer
135
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 $BG=BSU(2)$:
...
- 2,147
8
votes
1
answer
573
views
Majorana modes and the first Stiefel–Whitney class
The first Stiefel–Whitney class of a vector bundle is an element in the first cohomology group of the base space. Namely, the first Stiefel–Whitney class for a vector bundle $E$ over a $d$-dimensional ...
- 10.5k
16
votes
1
answer
797
views
Do vanishing characteristic classes of the tangent bundle imply a manifold is stably frameable?
Suppose we know that all stable characteristic classes of the tangent bundle of a manifold $M$ vanish, i.e. the map $f:M\rightarrow BO$ stably classifying the tangent bundle is trivial on cohomology, ...
- 323
5
votes
1
answer
403
views
covering map from spheres to projective spaces and the associated vector bundle
Let $S^n$ be the $n$-sphere and consider a $2$-sheeted covering
$$
S^n\longrightarrow\mathbb{R}P^n.
$$
We have an associated vector bundle
$$
\xi: \mathbb{R}^2\longrightarrow S^n\times_{\mathbb{Z}/2}\...
- 519
2
votes
0
answers
175
views
Chern character of finite $CW$-complexes and rational Pontrjagin class of vector bundles
Let $K$ be a finite $CW$-complex. Could you give any references or explanations for the following two items? I do not understand. Thanks!
(1). The Chern character from $\tilde{KO}^0(K)$ to the ...
- 519
8
votes
0
answers
277
views
Which map realizes the isomorphism $KO_n(X)\otimes \mathbb{Q}\to \bigoplus_{i\in\mathbb{Z}}H_{n-4i}(X;\mathbb{Q})$?
The description of the real $KO$-homology groups $KO_n(X)$ can be given abstractly as maps to the real K-theory spectrum $KO$ smash $X$, or via triples $(M,x,\phi)$ where $M$ is a closed manifold, $\...
- 362
13
votes
2
answers
900
views
References for Stiefel-Whitney class of Stiefel manifolds and Grassmannians
Let $M$ be a manifold. The total Stiefel-Whitney class of $M$ is defined to be the Stiefel-Whitney class of the tangent bundle $TM$
$$
w(M)=1+w_1(TM)+w_2(TM)+\cdots
$$
I want to find references for
$$
...
- 2,223
7
votes
1
answer
541
views
vector bundles associated to a covering space
Let $M$ be a $m$-dimensional manifold whose cohomology ring and cell structure are well-understood, such that there is a free action of the symmetric group $S_n$ on $M$. Then we have a $n!$-sheeted ...
- 2,223
5
votes
0
answers
242
views
characteristic classes of a covering space with symmetric group action
Let $S_n$ be the $n$-th symmetric group. Suppose we have a $n!$-sheeted covering space
$$
S_n\to M\to M/S_n
$$
where $M$ is a manifold. Let $\mathbb{K}$ be the real numbers $\mathbb{R}$, complex ...
- 2,223
12
votes
1
answer
825
views
Stiefel-Whitney class of fibre bundles
Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are ...
- 2,223
3
votes
1
answer
1k
views
characteristic classes of homotopy equivalent manifolds
Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal
$$
w(TM)=w(...
- 2,223
3
votes
1
answer
493
views
Four-dimensional vector bundles over $S^4$, intuition?
I know that $\pi_3(SO(4)) = \mathbb{Z} \oplus \mathbb{Z}$. We can choose an explicit identification as follows: given $(i, j) \in \mathbb{Z}$, we have a map $\phi: S^3 \to SO(4)$ which sends a unit ...
- 31
1
vote
1
answer
214
views
Chern classes of three (two) dimensional complex vector bundles
Let $M$ be a manifold.
Let $F(M,3)=\{(m_1,m_2,m_3)\mid m_1, m_2, m_3\in M, m_i\neq m_j, \text{ for any } i\neq j\}$.
Let $S_3$ be the symmetric group of order $3$.
Let $S_3$ act on $F(M,3)$ by $\...
- 1,990
8
votes
2
answers
1k
views
rational cohomology of finite real grassmannian
Let $p_j$ to be the $j$-th Pontryagin class of the universal $n$-plane bundle $E_n(\mathbb{R}^\infty)\to G_n(\mathbb{R}^\infty)$. Then according to Theorem 1.6, The Cohomology of BSO n and BO n with ...
- 2,223
5
votes
4
answers
3k
views
integral or rational cohomology of real grassmannians
I have obtained that the cohomology rings
$$
H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k].
$$
Also
$$
H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
- 1,990
2
votes
3
answers
490
views
Is true that $\left[\frac{\hat{A}(\mathbb HP^m)} { \hat{M}(\mathbb HP^m) }\right]_{4m} = 0$?
There are two questions:
How to prove that in general
$[\hat{A}(\mathbb HP^m)]_{4m} = 0$
It is possible to verify it for low values of $m$.
How to prove that in general
$\left[\frac{\hat{A}(\...
- 165
0
votes
1
answer
213
views
An integrality theorem for immersions of complex projective spaces in the euclidean space
There are three questions:
Please let me know your proof of the following theorem:
If $CP^3$ can be immersed in $R^8$ with an Euler class $W_{2}(\nu)$ for the normal bundle of $CP^3$ respect to $R^8$...
- 165
12
votes
2
answers
660
views
Vector bundle for prescribed Stiefel-Whitney classes
I hope this is not trivial.
Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice)
For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology ...
- 2,554
6
votes
1
answer
753
views
Rigidity of secondary characteristic classes
For a representation $\rho:\pi_1M\rightarrow GL(n,C)$ and the associated flat $GL(n,C)$-bundle $E_\rho\rightarrow M$ one has the Cheeger-Chern-Simons classes
$$\hat{c}_k(E_\rho)\in H^{2k-1}(M,R/Z)$$
...
- 10.4k
20
votes
2
answers
1k
views
Characteristic classes for block bundles
Block bundles are PL analogs of vector bundles, see e.g. Rourke-Sanderson's
article
in Bulletin of AMS, 1966. There is a classifying space $B\widetilde{PL}_q$ for rank $q\ $ block bundles, which is ...
- 29.1k
8
votes
1
answer
679
views
topological type of smooth manifolds with prescribed homotopy type and pontryagin class
Can someone help explain the following result:
If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
Thank ...
- 179