Characteristic classes are invariants of bundles living in the cohomology of the base. The most common examples of characteristic classes are the Chern, Stiefel–Whitney, and Pontryagin classes.

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Explicit isomorphism $H (E, E^0) \cong H_{cv} (E)$

Yo! I've been trying to understand better why $$H (D, D-\{x\}) \cong H_c (D)$$ for a disk $D$ and, more generally, why $$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as ...
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How can I equip the universal vector bundle over $\mathbf{G}(k,n)$ with a connection?

When constructing the grassmannians $\mathbf{G}_\mathbb{F}(k,n)$ for $\mathbb{F} = \{\mathbb{R},\mathbb{C} \}$ there is a natural vector bundle $$ \mathbf{G}_{k,n} \to \mathbf{G}_\mathbb{F}(k,n) $$ ...
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Naive question on characteristic classes of $Gr(k,n)$

Let $Gr(k,n)$ be the Grassmann manifold of $k$-planes in $\mathbb{R}^n$ and $\gamma_k$ its tautological $k$-plane bundle. Is it obvious to see (or even true) that the Stiefel-Whitney classes $w_i(\...
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Is there a way to find the euler class of the tangent bundle of the sphere from the cohomology ring of real projective space?

So (I am pretty sure) that the tangent bundle over the sphere is the pull back of the tangent bundle of real projective space. I know that the sphere gives a double cover of real projective space and ...
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Under what type of transformations are characteristic classes and characteristic numbers of a manfold invariant?

When I say "characteristic class of a manifold" I mean the characteristic class of the tangent bundle. I assume that all Chern/Pontrjagin classes/numbers are invariant under diffeomorphism, if they ...
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What motivated trying to express the signature of a manifold as a linear combination of pontrjagin numbers?

I have tried reading a proof of the signature theorem but it is way beyond me, is there a way to motivate, in english, why anyone even started searching for such a formula? Why would one assume that ...
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Cohomology class of zero set of a section

Say we have a rank $r$ smooth vector bundle $E\to X$ and a smooth section $s:X\to E$ of it. Shortly after the 30 min mark in this video Joe Harris defines the $r$th Chern class of this bundle to be ...
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Bott's topological obstruction to integrability

Let $M$ be a smooth manifold, $E\subseteq TM$ be an integrable distribution and $\pi:TM\to Q=\frac{TM}{E}$ the quotient bundle with $\dim Q=q$. Bott showed that \begin{equation} Pont^k(Q)=0 \qquad \...
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Chern classes of a double cover

Let $X$ be a compact complex surface and let $D$ be a double cover of $X$. Let $\pi:D\to X$ be the double cover map (a 2:1) map. If $E$ is a vector bundle (rank at least 2) on $X$ with $c_1(E) = A$ ...
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Are there characteristic classes for symplectic vector bundles?

Given a real vector bundle, say the tangent bundle of a manifold, some obstructions to this being the underlying real bundle of a complex bundle come from characteristic classes. These can prove the ...
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Characteristic class invariant under bundle isomorphism

Let $c$ be a characteristic class for principal $G$-bundles and $p_1: E_1 \to X, p_2: E_2 \to X$ be isomorphic principal $G$-bundles, then $c(E_1) = c(E_2)$ Is this part of the defining naturality ...
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On Atiyah-Singer Index theorem and Atiyah-Bott fixed point theorem.

I would like to understand the statement of Atiyah-Singer and Atiyah-Bott theorems enough to see lots and lots of applications they have. My background is pretty thin with some basic algebraic ...
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Motivating Characteristic Classes Using $S^2$

Trying to understand characteristic classes, hoping someone can explain/fit my example below into the wider scheme of things: Chern's book says Characteristic classes are the simplest global ...
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33 views

Chern character of canonical line bundle over $\mathbb{CP}$

Let $H \to \mathbb{CP}$ be the canonical line bundle over $\mathbb{CP}=S^2$. Then from the text Vector Bundles and K-theory by Hatcher, given the chern character, $ch$, and first chern class $c_1(H)$,...
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Euler classes of oriented $2$-dimensional vector bundle, oriented $S^1$-bundle same?

As the question title suggests, are the Euler classes of an oriented $2$-dimensional vector bundle and of an oriented $S^1$-bundle the same?
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Chern Weil theory-independence from the choice of connection

In Milnor-Stasheff's book about characteristic classes there is an appendix about Chern-Weil theory. Suppose that $E \to M$ is a (complex) vector bundle with connection $\nabla$: this connection can ...
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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 ...
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Euler class in Morita's book

I was trying to follow the construction of the Euler class in Morita's 'Geometry of differential forms' for an $S^1$ bundle over a manifold M; $p: E \rightarrow M$. There the author builds the class ...
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32 views

Coefficient space of formal power series corresponding to characteristic classes

It is known that there is a bijection between the set of characteristic classes of rank $k$ complex vector bundles and the ring of symmetric formal power series in $k$ variables, given by identifying ...
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Intuitive interpration of the Thom isomorphism and relative cohomology

Let $p: E \rightarrow B $ be an an oriented real vector bundle or rank $n$. Then there exists a unique class $u \in H^n(E, E - B; \mathbb{Z})$, where $B$ is embedded into $E$ as the zero section, such ...
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Non-zero tangent vectors in Hermitian manifolds

I'm a physicist trying to learn about Chern's class and forms by myself. To that respect I feel the Chern's writings far more pleasant than esoteric literature written by and for physicists, which are ...
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Chern character in odd K-theory

I'm familiar with the definition of Chern character for a vector bundle. This leads to the definition of Chern character for $K$ theory (even theory) with values in even cohomology (the definition ...
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Chern class of Hopf fibration: $S^1 \hookrightarrow S^3 \xrightarrow{\ p \, } S^2$?

As definition, the first Chern class is an element in $H^2_{dR}(S^2)$, how can we represent it as an integer? If we change the orientation of $S^2$, then does the integer change sign?
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Restrictions of $E \to X×I$ over $X \times \{0\}$ and $X \times \{1\}$ are isomorphic if X is paracompact

The following assertion appears in Milnor's Characteristic Classes. The restrictions of a vector bundle $E \to X×I$ over $X \times \{0\}$ and $X \times \{1\}$ are isomorphic if $X$ is paracompact. ...
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Calculating the stiefel whitney class of Tangent Bundle of projective space

I was reading the proof of the fact that whitney sum of the tangent bundle of $RP^n$ with the trivial bundle is isomorphic to whitney sum of $n+1$ tautological line bundles on $RP^n$ from Hatcher's ...
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Prop 12.8 in Bott & Tu

Yo! This proposition in Bott & Tu have been haunting me for a year or so since I always have to come back to this book for references. More precisely, the second equality in Proposition 12.8 in ...
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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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discontinuous group homomorphism from $(\mathbb{R},+)$ to $S^1$,unit circle

It is well known that if $\gamma:(\mathbb{R},+)\rightarrow S^1$ is continuous homomorphism,then $\exists y\in\mathbb{R}$,such that $\gamma(x)=e^{ixy}$. Show that there is a discontinuous homomorphism ...
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How do we construct an associated bundle $V_{n, q}(\omega)$ over $B$ with typical fiber $V_{n - q}(F)$?

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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Finding the chern classes of line bundles over projective space using homotopy classes of clutching functions

I have just started to learn about characteristic classes and before learning more about the ways to compute them it would be nice to compute some examples using tools I already know. I only started ...
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Property of second Steifel-Whitney class?

Let $M$ be manifold, $n = 4$. Is $w_2$ special in in the regard it's the only thing of $H^2(M, \mathbb{Z}_2)$ where $w_2 \cup \tau = \tau \cup \tau$, $\tau \in H^2(M, \mathbb{Z}_2)$ or not? I wondered ...
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functor from complex algebraic variety to constructible function

I am reading MacPherson's paper "Chern Classes for singular varieties". Proposition1 : There is a unique covariant functor from compact complex algebraic variety to abelian groups whose value on a ...
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Is $[N]^\#([N])$ congruent to $w_n(\nu_N)([N])$ mod $2$, where $\nu_N$ is the normal bundle of the embedding of $N$ in $M$?

Let $M$ be a closed, smooth, orientable $2n$-manifold, and let $N$ be a closed, smooth, orientable $n$-submanifold. Let $[N]^\#$ denote the cohomology class (Poincaré) dual to the homology class $[N]$....
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Bott and Tu construction of chern classes

To quote from Differential Forms in Algebraic Topology, Set $x=c_1(S^*)$. Then $x$ is a cohomology class in $H^2(P(E))$. Since the restriction of the universal subbundle $S$ on $P(E)$ to a fiber $...
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Cohomology of a classifying space

I would like some advice on the following problem: I have a topological group $G=\langle H,g\rangle$, where $H$ is a subgroup and $h$ is an element. Using a result of classifying spaces $Bi:BH \to BG$...
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If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$?

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$? For starters, I know that if the $n$-dimensional $M$ can be immersed in $\...
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How is this “Stiefel manifold”-like fiber bundle called? How to characterize it?

Let $M$ be a smooth manifold of dimension $n$. Consider the space $V_k(M)$ of pairs $(x,\alpha)$ where $x \in M$ and $\alpha$ is a linear embedding $\mathbb{R}^k \hookrightarrow T_x M$ (or ...
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Is Hatcher's proof of thom isomorphism theorem flawed?: I don't believe that $H^n(E,E_0)\cong H^n(R^n,R^n-0)$

Let $E$ be an oriented vector bundle over $B$, a CW complex, with fiber of dimension $n$. Let $E_0$ be $E - B\times 0$. The main theorem Hatcher uses to prove the thom isomorphism theorem is that the ...
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Exists homeomorphism which carries each fiber isomorphically to itself, composition… make rigorous.

See here for a question I asked. Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see that there exists a homeomorphism $f: E(\xi) \to E(\xi)$ ...
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Why doesn't this construction of Chern classes generalize to real bundles?

Can we mimic the construction of Chern classes using real or quaternionic bundles? If so, do we get anything interesting? My question concerns the construction of Chern classes in Bott and Tu's ...
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Set consisting of all unoriented cobordism classes of smooth closed $n$-manifolds can be made into additive group? [closed]

How do I see that the set $\mathfrak{N}_n$ consisting of all unoriented cobordism classes of smooth closed $n$-manifolds can be made into an additive group?
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Recurrence equation for equivalent (charasteristic) classes in graphs

Is there any Recurrence equation to get the number of equivalent classes in graphs? For example if you have: 2 vertex in a graph there are 2 equivalent classes 3 vertex in a graph there are 4 ...
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Exists homeomorphism which carries each fiber isomorphically to itself, composition?

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

Let $X$ be some smooth manifold and let $p:E\to X$ be a complex vector bundle of finite rank. Our goal is to classify $E$ up to bundle-isomorphisms. According to Atiyah's K-Theory book, there is a a ...
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Computing Stiefel Whitney classes

I am computing the cohomology of $BO(2) \times B0(3)$ and I would like to identify the Stiefiel Whitney classes of this space. For instance, I know $H^*(BO(2);\mathbb{Z}/2)\cong \mathbb{Z}/2[w_1,w_2]$...
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Additive cohomology of BO(k)

we know that $H^*(BO(n);\mathbb{Z}/2) \cong \mathbb{Z}/2[w_1,\cdots,w_n]$, where $w_i \in H^i(BO(n);\mathbb{Z}/2)$ is the $i$-th Stiefel Whitney class. Is it correct to say $\langle w_i \rangle = H^i(...
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Is multiplication by a Stiefel-Whitney class an injective map?

I have a doubt: In cohomology, when you multiply by a Stiefel-Whitney class is it always an injective map? For example: is $$H^{j-1}(X)\xrightarrow{\smile\ w_1}H^{j}(X)$$ always injective? Thanks!
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Construction of Grassmann manifolds

Is there a way to construct the Grassmann manifold via block matrices? For example the upper triangular matrices stabilize the (coordinate) basis of $\mathbb R^n$.
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triviality of vector bundles with the reduced homology of base space entirely torsion

Let $\xi$ be an $n$-dimensional vector bundle over a manifold $M$ such that the reduced cohomology $\tilde H^*(M;\mathbb{Z})$ is entirely torsion (every element has finite order under addition). ...
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Why the combinatorial second Stiefel-Whitney class is a cocycle?

From the book "Spin geometry" by Lawson&Michaelson Appendix A or this literature we know that there is a nice combinatorial way to interpret the second SW class by the transition functions of a ...