All Questions
Tagged with cohomology characteristic-classes
47
questions
6
votes
1
answer
355
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 ...
- 10.4k
6
votes
0
answers
350
views
Reference request: cohomology of BTOP with mod $2^m$ coefficients
I am searching for a reference with information pertaining to the $\mathbb{Z}/{2^m}$ cohomology of ${\rm{BTOP}}(n)$, for $n \geq 8$ and $m=1,2$, where
$${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \...
3
votes
0
answers
78
views
Tautological ring for moduli of flat connections
Let $X$ be a smooth complex manifold and $G$ a connected complex algebraic group. Let $M$ denote the moduli stack of flat $G$-connections on $X$. Over $M\times X$, we have the tautological $G$-bundle, ...
- 2,711
19
votes
3
answers
1k
views
Are Chern classes well defined up to contractible choice?
The Chern classes are, by definition, cohomology classes. And
cocycle representatives of the Chern classes are not unique.
But it might be the case that cocycle representatives of the Chern classes ...
- 42.7k
1
vote
0
answers
34
views
Are there local maps of simplicial (co-)cycles on $d$-manifolds beyond cohomology operations?
I'm interested in locally defined maps of cocycles/chains on manifolds of a fixed dimension $d$ which are compatible with cohomology. To be concrete about what "local" means, let me consider ...
- 2,921
4
votes
0
answers
186
views
Multi-variable cohomology operations
Intuitively, cohomology operations are ways to locally compute a cocycle $\alpha\in H^i(X, G)$ from any cocycle $\beta\in H^j(X, H)$. Formally, they are in one-to-one correspondence with homotopy ...
- 2,921
3
votes
0
answers
207
views
What characteristic classes are there?
Can someone concisely list all characteristic classes (i.e., the cohomology classes $H^*(BX,A)$ of the corresponding classifying spaces) for the most relevant structure groups $X$ such as $O(n)$, $SO(...
- 2,921
4
votes
2
answers
522
views
How much do characteristic classes fail to characterize bundles?
Given a group $G$, let $E \to B$ be a principal $G$-bundle. It is
well-known that when $B$ is a nice enough topological space (e.g.
CW-complex), such a thing corresponds to a connected component of
$...
- 5,038
8
votes
1
answer
612
views
Motivation for the definition of complex orientable cohomology theory
PRELIMINARY DEFINITIONS:
Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have:
$$
\tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt)
$$
So there is a special ...
- 563
3
votes
0
answers
276
views
Evaluating the Euler class of a circle bundle on fibers
I am trying to understand what kind on information the Euler class provides about certain submanifolds of a given circle bundle.
This might be completely obvious, but I don't see how to answer the ...
- 1,227
5
votes
0
answers
288
views
Chern-Weil theory in the cohomological Atiyah-Singer theorem
I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer.
Let $D:\...
- 2,237
5
votes
0
answers
254
views
Integration on an non-orientable manifold [closed]
Suppose $M_n$ is a $n$ dimensional non-orientable manifold.
I am interesting in knowing whether the following statements are true:
A characteristic class $w_{n}^{(p)} \in H^{n}(M_n, \mathbb{Z}_p)$...
- 487
3
votes
1
answer
865
views
Chern classes of complex vector bundle
I'm reading characteristic classes form the book Differential forms in Algebraic Topology by Bott and Tu. The Chern classes are defined as follows:
$E\xrightarrow{\rho} M$ is a vector bundle and $E_p$...
18
votes
1
answer
985
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
8
votes
1
answer
373
views
On the classification of $\mathrm{SU}(mn)/\mathbb{Z}_n$ principal bundles over 4-complexes
In
The Classification of Principal PU(n)-bundles Over a 4-complex, J. London Math. Soc. 2nd ser. 25 (1982) 513–524, doi:10.1112/jlms/s2-25.3.513
Woodward proposed a classification of $\mathrm{PU}...
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