The characteristic-classes tag has no usage guidance.

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### Existence of flat connections via characteristic classes, for nice groups

I have two questions about what I write below (which honestly seems pretty elementary).
Is it true (more or less)?
Is there a clean reference that I can cite.
Let $G$ be a compact Lie group, $M$ a ...

**12**

votes

**1**answer

288 views

### Measuring the failure of pushforward to commute with Steenrod squares

Let $f \colon X \rightarrow Y$ be a map of topological spaces. Let's say that they are (closed) manifolds (not necessarily orientable), though to be honest I'm really interested in the more general ...

**14**

votes

**1**answer

207 views

### Vector bundles with exactly one nonzero SW-class

I am interested in seeing examples of a space $X$ (preferably a closed smooth manifold, but any finite-dimensional CW-complex would also be of interest) with a vector bundle $\xi\colon E \to X$ on it, ...

**10**

votes

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445 views

### Local proof of Grothendieck-Riemann-Roch theorem

There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem.
Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ ...

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votes

**3**answers

538 views

### Nice things that can be proved easily with characteristic classes

A bit of context for this question: as a project for my master's degree my supervisor asked me to understand the construction of Milnor's exotic spheres. After learning the heavy material (I knew very ...

**3**

votes

**1**answer

116 views

### Is there a matrix representation of the permutation group whose character is the Markov trace?

Let $g$ be an element in the permutation group (symmetric group) $S_N$. Define the Markov trace of $g$ (denoted $\text{tr}_k g$) as
$$\text{tr}_kg = k^\text{number of cycles in $g$} ,$$
which depends ...

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votes

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124 views

### Equivariant and orbifold Chern classes

Edit. After thinking about this problem a bit longer, I am not so sure anymore that the Bredon cohomology proposed by Adem and Ruan gives me the invariants I am looking for. I have therefore moved ...

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votes

**1**answer

391 views

### Differential geometric interpretation of cohomology

I'm not sure whether this is an appropriate question for this forum. I'm afraid that this is not a research level question however:
1. It's about reference request therefore the answer does not ...

**7**

votes

**1**answer

194 views

### Source request for $H^*(B\mathrm{TOP},\mathbb{Q}) \cong H^*(BO,\mathbb{Q})$

Let $B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $B\mathrm{TOP}$ via the inclusion. Let $f$ denote the ...

**12**

votes

**1**answer

309 views

### Characteristic classes for odd $K$-theory

There are different models of odd $K$-theory. In one case,
one takes the group $U=\lim\limits_{\longrightarrow}U(n)$ as classifying space. Similarly, if $\mathcal U$ denotes the unitary group of a ...

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votes

**2**answers

212 views

### non-triviality of the underlying real vector bundle of the complexification of a real vector bundle

Let $M$ be a given manifold and $\xi$ be a given $k$-dimensional vector bundle over $M$. How to determine whether the underlying real vector bundle of $\xi\otimes\mathbb{C}$, i.e. the Whitney sum $\xi\...

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votes

**1**answer

137 views

### Smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$ without a point admits symplectic structure?

See my previous question here.
Let $M$ be a smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$. Can $TM$ be trivialized in the complement of a point?
This was answered in the ...

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votes

**2**answers

146 views

### Tangent bundle of smooth closed simply-connected $4$-manifold $w_1 = w_2 = 0$ can be trivialized in complement of point?

Let $M$ be a smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$. Can $TM$ be trivialized in the complement of a point?

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votes

**1**answer

269 views

### Primary obstruction to the existence of a cross-section of $V_{n - q}(\omega)$ is a cohomology class in $H^{2q+2}(B, \pi_{2q+1} V_{n - q}(F))$?

Let $V_{n - q}(\mathbb{C}^n)$ denote the complex Stiefel manifold consisting of all complex $(n - q)$-frames in $\mathbb{C}^n$, where $0 \le q < n$. This manifold is $2q$-connected, and$$\pi_{2q + ...

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votes

**1**answer

254 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, ...

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votes

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159 views

### functor from complex algebraic variety to constructible function

I asked the same question on mathstackexchange but didn't get any response so I guess I should ask here. If you think it is a duplicate and it is not appropriate to post it here, I will delete this ...

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votes

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369 views

### What is the “real” meaning of the $\hat A$ class (or the Todd class)?

In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the ...

**3**

votes

**1**answer

279 views

### Differential characters, Chern-Simons forms, and differential cohomology

I've read through the classic Chern-Simons paper where they introduce the Chern-Simons forms. These are differential forms whose exterior derivative gives you the characteristic forms for any given ...

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309 views

### Milnor-Stasheff Characteristic Classes Problem 7B, Borel 1953

There is the following Proposition 11.1 from Borel's 1953 paper La cohomologie mod 2 de certains espaces homogènes (see here).
Proposition 11.1 The classes $w^i$ and $\overline{w}^j$ are related ...

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votes

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572 views

### Intuition/idea behind a proof of the splitting principle?

The splitting principle is as follows.
Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map $p^*...

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votes

**0**answers

41 views

### Does anyone know if there is a generalization of symplectic Kodaira dimension beyond 4-manifolds?

I'm aware that in algebraic geometry, one has the Kodaira-Iitaka dimension, which generalizes the Kodaira dimension, but does anyone know if a correspondent generalization in the symplectic category ...

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157 views

### Cohomology algebra generated by $n$ Steifel whitney classes and and $k$ dual classes subject only to $n+k$ defining relations? [closed]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, \...

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votes

**1**answer

138 views

### Stiefel-Whitney classes of closed topological manifolds with no smooth structure

If $M$ is a smooth manifold, the total Stiefel-Whitney class of $M$ is defined to by the total Stiefel-Whitney class of the tangent bundle, i.e. $w(M) := w(TM)$.
If $M$ is a closed smooth manifold, ...

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votes

**1**answer

841 views

### A dictionary of Characteristic classes and obstructions

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians.
In an effort to ...

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votes

**1**answer

163 views

### Triviality of certain vector bundles

Let $M$ be a smooth manifold and let $SM$ be the bundle of symmetric bi-linear forms on $TM.$ Riemannian metrics are a particular kind of sections in this bundle. Since any manifold admits a global ...

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votes

**1**answer

249 views

### Stiefel-Whitney classes of tensor product $\xi^m \otimes \eta^n$, computation

Let $\xi^m$ and $\eta^n$ be vector bundles over a paracompact base space. Where can I find a reference to the Stiefel-Whitney classes of the tensor product $\xi^m \otimes \eta^n$ being computed as ...

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votes

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507 views

### Vector bundles, finitely generated projective module?

Let $B$ be a Tychonoff space and let $\mathbb{R}(B)$ denote the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $S(\xi)$ denote the $\mathbb{R}(B)$-module ...

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173 views

### Vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type? [closed]

Using Stiefel-Whitney classes, how do I see that the vector bundle $\gamma^1$ over $\textbf{P}^\infty$ does not have finite type?

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votes

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249 views

### Relative Characteristic classes

A pair of vector bundles over a base space $X$ is a pair $(E,F)$ where $E$ is a vector bundle over $X$ and $F$ is a sub-bundle of $E$. Two pairs $(E_{1},F_{1})$ and $(E_{2}, F_{2})$ are ...

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votes

**1**answer

145 views

### $n + 1 = 2^rm$ with $m$ odd $\implies$ do not exist $2^r$ vector fields on $\mathbb{P}^n$ that are everywhere linearly independent?

What is the easiest/quickest way to see the following?
If $n + 1 = 2^rm$ with $m$ odd, then there do not exist $2^r$ vector fields on the projective space $\mathbb{P}^n$ which are everywhere ...

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votes

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153 views

### Immersing spaces in $\mathbb{R}^{n+1}$, Stiefel-Whitney classes

Where can I find references to proofs/can anyone supply me a quick proof of the following facts?
If the $n$-dimensional manifold $M$ can be immersed in $\mathbb{R}^{n+1}$, then each $w_i(M)$ is ...

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votes

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50 views

### Isometry theorem, exists homeomorphism that carries each fiber isomorphically onto itself

Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see there exists a homeomorphism $f: E(\xi) \to E(\xi)$ which carries each fiber isomorphically onto ...

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votes

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142 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}\...

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votes

**2**answers

518 views

### Cohomological obstructions to lift $\pi_0$ of a topological group

Let $G$ be a topological group. Denote the same group with the discrete topology by $G^\delta$ and denote the group of connected components of $G$ by $\pi_0G$. I am interested in the question when we ...

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votes

**0**answers

96 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 ...

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171 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, $\...

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votes

**1**answer

188 views

### Whitney sum formula for Pontryagin classes II

I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand the 2-...

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votes

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271 views

### Whitney sum formula for Pontryagin classes I

I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand this ...

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votes

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218 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
$$
...

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votes

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263 views

### triviality of Whitney sums of a vector bundle

Let a $3$-dimensional subspace $V$ of $\mathbb{R}^4$ be $$V=\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\mid\sum_{i=1}^4x_i=0\}.$$ The alternating group $A_4$ acts on $V$ by
$$\sigma(x_1,x_2,x_3,x_4)=(x_{\...

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votes

**1**answer

254 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 ...

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votes

**0**answers

94 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 ...

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votes

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381 views

### first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles

Let $\xi$ be a (real) vector bundle of dimension $n$. Then the first Stiefel-Whitney class
$$
w_1(\xi)=0
$$
if and only if $\xi$ is orientable, i.e. the structure group of $\xi$ can be reduced to $SO(...

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votes

**1**answer

271 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 ...

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votes

**1**answer

233 views

### canonical action of symmetric groups on orthogonal groups

There is a canonical faithful orthogonal representation of the symmetric group $S_{n+1}$, for $n\geq 1$:
$$
S_{n+1}\to O(n)
$$
given as follows.
(1). I regard $O(n)$ as the isometry group of the unit ...

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votes

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412 views

### Examples of Stiefel-Whitney classes of manifolds

Let $M$ by an compact, connected $n$-dimensional manifold without boundary.
Are there any other computable examples of the Stiefel-Whitney class $w(M)$ except for $M=S^m, \mathbb{R}P^m,\mathbb{C}P^m, ...

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votes

**1**answer

201 views

### Stiefel-Whitney class of unordered configuration space

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. How to compute the total Stiefel-...

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votes

**1**answer

216 views

### characteristic classes of tangent bundle of 2-nd unordered configuration space

Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space
$$
B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2
$$
where
$$
\Delta=\{(m,m)\mid m\...

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votes

**1**answer

504 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(...

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votes

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226 views

### Schubert calculus and Pieri's formula

In the lecture notes Grassmannians: the first example of a moduli space. MIT Open Course Ware. page 7:
Are there any formal publications (books/papers) where I can find the formula?