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.

learn more… | top users | synonyms

3
votes
1answer
47 views

the top chern class of the holomorphic tangent bundle is the euler class

Is the following true? Let X be a complex manifold of complex dimension d and let V denote its holomorphic tangent bundle (ie it's $T^{1,0} \subset T \otimes_R C$, where T is the tangent bundle of ...
1
vote
1answer
69 views

Reference request: Zero set of global section

Things are in the complex algebraic setting. Assume that a vector bundle $V$ of rank $n$ over a $\mathbb{P}^n$ has a global section $\sigma$. Is it true that the zero set of $\sigma$ is a ...
4
votes
2answers
109 views

Elementary proof of the fact that any orientable 3-manifold is parallelizable

A parallelizable manifold $M$ is a smooth manifold such that there exist smooth vector fields $V_1,...,V_n$ where $n$ is the dimension of $M$, such that at any point $p\in M$, the tangent vectors ...
3
votes
2answers
75 views

cohomology of classifying space of cyclic group

(1). Let $p$ be a prime number. Let $B\mathbb{Z}_p$ be the classifying space of the discrete group $\mathbb{Z}_p$. How to obtain $$ H^*(B\mathbb{Z}_p;\mathbb{Z}_p)=\mathbb{Z}_p[t]\otimes \Lambda[e]? ...
2
votes
2answers
68 views

K-theory formulation of the index theorem

The Atiyah-Singer index theorem is one of the most important results in twentieth century's mathematics. It states that for an elliptic differential operator $D$ on a smooth, oriented compact manifold ...
7
votes
1answer
102 views

What can we say about a vector bundle with trivial unit sphere bundle?

If you want to avoid the back story to this question, feel free to skip the paragraphs between the horizontal lines. Yesterday Georges Elencwajg asked me the following question (I'm paraphrasing): ...
3
votes
0answers
32 views

cohomology of complex grassmannians and quaternion grassmannians (the finite dimensional case)

Let $G_k(\mathbb{C}^N)$ be the complex grassmannian manifold. Let $G_k(\mathbb{H}^N)$ be the quaternion grassmannian manifold. Let $p$ be a prime integer (we may also impose extra conditions, such as ...
5
votes
1answer
70 views

Manifolds with vanishing Stiefel-Whitney classes but are not stably parallelizable

It is known that if a manifold is stably parallelizable, then it's Stiefel-Whitney classes must vanish. Is the converse true? Note that we know that the converse cannot hold if stably parallelizable ...
0
votes
0answers
36 views

cohomology homomorphism between grassmannians induced by inclusion

Let $i:G_k(\mathbb{R}^n)\to G_k(\mathbb{R}^\infty)$ be inclusion of grassmannians. Then $H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]$. $ ...
1
vote
1answer
37 views

cohomology homomorphism induced by classifying map

Let $\xi=(E,p,B)$ be an $n$-dimensional vector bundle. Let $f: B\to G_n(\mathbb{R}^\infty)$ be the classifying map. Let $f^*: ...
1
vote
1answer
45 views

Reference request: second Chern class of P^2

I have heard that $c_2(T_{\mathbb{P}^2})=e(\mathbb{P}^2)$. What's the general result and where can I read about it? Thanks.
2
votes
0answers
37 views

cohomology of finite dimensional grassmannians

What is the cohomology algebra of finite dimensional grassmannian $$ H^*(G_k(\mathbb{R}^N);\mathbb{Z}_2)? $$ $$ H^*(G_k(\mathbb{C}^N);\mathbb{Z}_p)? $$ $$ H^*(G_k(\mathbb{H}^N);\mathbb{Z}_p)? $$ I ...
3
votes
2answers
32 views

The first Chern class of projective line $\mathbb{CP}^1$

I am studying the Chern class using by some textbooks and lecture notes. One day, I found an example of the first Chern class of $\mathbb{CP}^1$. Let $\xi$ be a tautological line bundle of ...
1
vote
0answers
42 views

The smallest $n> 0$ with the nonzero $n$th Stiefel-Whitney class is a power of 2 when total Stiefel-Whitney class is not trivial.

This is the Problem 8-B form the characteristic classes by John W. Milnor and James D. Stasheff. [Problem 8-B]. If the total Stiefel-Whitney class $w(\xi) \neq 1$, show that the smallest $n>0$ ...
7
votes
3answers
165 views

Why $\mathbb{RP}^2$ can not be embedded to $\mathbb{R}^3$?

Is there any answer of this question around basic theory of differentiable manifolds?
1
vote
0answers
78 views

Characteristic class integral: on what manifold does $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$ hold?

Characteristic class integral: when does the equality hold $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$, on what manifolds? Here $c_1$ is the first Chern class. Here $w_2$ is the 2nd ...
2
votes
1answer
21 views

Weyl Groups/Borel

Could someone tell me where to find a proof of the following statement that I found in some notes about characteristic classes I was reading? If $G$ is a compact connected Lie group with maximal ...
2
votes
2answers
40 views

Stably equivalent bundles and stable homotopy classes of maps

We know that there is a one-to-one correspondence between isomorphism classes of principal $G$-bundles over a base space $M$ and homotopy classes of maps $M \to BG$, where $BG$ is the classifying ...
4
votes
1answer
61 views

Integrality of the $L$-genus for a smooth manifold

For a compact, oriented, smooth manifold $M^{4k}$, the Hirzebruch signature theorem gives the signature $\sigma(M)$ in terms of a polynomial $P_k$ in the Pontryagin numbers of $M$ whose coefficients ...
2
votes
1answer
36 views

A question about characteristic classes

I have a map $\phi:BO(1)^n\rightarrow BO(n)$ which is given by sending any $n$-tuple in $BO(1)^n$ to an $n$-plane through the origin. Thus, this induces a group action on the symmetric group $S_n$ ...
4
votes
2answers
81 views

Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ ...
1
vote
0answers
10 views

Vector bundle base space map

Is it true, and if it is, is there some easy way to see the following? Suppose that $\xi = (\pi, E, B)$ is an $n$-vector bundle with $B$ paracompact but not necessarily compact. Is there a base ...
0
votes
0answers
22 views

whitney class of fiber bundles over classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $\rho: \Sigma_k\to GL(k)$ be regular representation by permuting basis. Let $\rho': B\Sigma_k\to BGL(k)=G_k(\mathbb{R}^\infty)$ be the induced ...
0
votes
0answers
28 views

classifying map of associated bundles

Let $G\leq GL(\mathbb{R}^n)$ be a group and $\xi$ be a principal $G$-bundle over a space $X$. Let $\eta=\xi[\mathbb{R}^n]$ be the associated vector bundle of $\xi$. Let $f_\xi: X\to BG$ be the ...
0
votes
1answer
14 views

If we construct an euler class, why is it that it behaves functoriality (Naturality)? That is:

Here is a very simple question that I can't seem to figure out for some reason. If we construct an euler class, why is it that it behaves functoriality (Naturality)? That is: $e(F) = f*e(E)$
0
votes
0answers
3 views

Let ${\Phi \in H^n(E, E - B; \mathbb{Z}/2)},$ restrict to $E$, and pullback to $B$, where $B \subset E$.

I am reading about characteristic classes, and I came upon this statement. Let ${\Phi \in H^n(E, E - B; \mathbb{Z}/2)},$ restrict to $E$, and pullback to $B$, where $B \subset E$. Can someone ...
0
votes
1answer
29 views

cup product of stiefel-whitney class

Let $\xi$ be a vector bundle. Let $w(\xi)$ be the total Stiefel-whitney class. Let $\bar w$ be the dual Stiefel-whitney class. In John Milnor's Characteristic class book, page 40-41 Chap.4, ...
3
votes
1answer
78 views

Analogue in algebra for characteristic classes?

By Swan's Theorem, we know that projective modules over a ring are an algebraic analogue of vector bundles over a base space. Is there some sort of cohomology theory of rings (or modules? or schemes, ...
5
votes
0answers
68 views

Determining an explicit line bundle over surface

The following is a explicitly defined complex line bundle $E\to\Sigma$ over a closed surface: View $\Sigma$ as a subset of $\mathbb{R}^3$ and consider its Gauss map $n:\Sigma\to S^2$ given by ...
0
votes
1answer
82 views

A question on Kronecker Index

I am reading A book by Milnor (Lectures on characteristic classes) and I can across this section on Stiefel-Whitney numbers (page 16) and he uses the Kronecker index but never defines it (He says to ...
2
votes
0answers
52 views

Connecting homomorphism in the Gysin sequence

Let $j : SO(2n) \to SO(2n+1)$ be the standard subgroup embedding and let $Bj : BSO(2n) \to BSO(2n+1)$ be the induced fibration obtained by factoring the universal $SO(2n+1)$-bundle by the subgroup ...
2
votes
1answer
69 views

Euler Classes, Chern Classes, $S^2$ Bundles, and $CP^1$ Bundles

I am just starting out learning about characteristic classes (Euler, Chern, etc.) from Bott and Tu's book, and I had the following question. Let $E$ be an oriented $S^2$ bundle over $M$ with ...
3
votes
1answer
117 views

Characteristic Class with arbitrary coefficient

I'm looking for some "natural definition" for characteristic class with arbitrary coefficient. For example, the chern class satisfies four conditions and Stiefel-Whitney class satisfies three ...
1
vote
0answers
47 views

Relative cohomology of a vector space module non-zero vectors

I am trying to explicitly compute the relative cohomology groups $H^m(\mathbb R^n,\mathbb R^n_0;\mathbb Z)$, where $\mathbb R^n_0$ is all the non-zero vectors in $\mathbb R^n$. I think that the answer ...
6
votes
1answer
181 views

Tensor product of real line bundles is trivial as a map $\mathbb{R}P^\infty\to\mathbb{R}P^\infty\times\mathbb{R}P^\infty\to\mathbb{R}P^\infty$

The tensor product of a real line bundle with itself is trivial, as is easily seen by looking at the transition functions or checking the Stiefel-Whitney class. Real line bundles are classified by the ...
6
votes
2answers
161 views

Why is the constant relating the chern class and curvature form always $2\pi i$?

I'm reading Milnor's book on Characteristic Classes. In Appendix C, Milnor shows the invariant polynomial of the curvature form and the Chern class differ by powers of $2\pi i$. He first shows that ...
2
votes
1answer
79 views

Characteristic classes not defined on vector bundles

If you read the definition on Wikipedia, you'll see that they allow characteristic classes to be defined on general principal $G$-bundles (vector bundles being subsumed in this general case by looking ...
2
votes
2answers
88 views

Is the following situation about first Chern numbers possible?

Let's consider complex vector bundles on a torus $T^2$ constructed in the following way: Suppose we have a map $f:T^2\to U(n)$, where $U(n)$ is the space of $n\times n$ unitary matrices. This ...
5
votes
1answer
254 views

Relations of Characteristic classes: Chern, Stiefel-Whitney, Pontryagin, Euler, Wu class. [closed]

We have a quite some characteristic classes: Chern class, Stiefel-Whitney class, Pontryagin class, Euler class, Wu class, etc. I wonder whether some math experts can use simple words and basic ...
5
votes
2answers
525 views

Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the ...
0
votes
1answer
31 views

Euler class is odd under orientation, thus its integral over a manifold will be even.

I learned a statement from others: "Euler class is odd under orientation, thus its integral over a manifold $M$ is even." I cannot fully appreciate it, can someone show this explicitly? The ...
4
votes
1answer
64 views

Calculate the Wu class from the Stiefel-Whitney class

The total Stiefel-Whitney class $w=1+w_1+w_2+\cdots$ is related to the total Wu class $u=1+u_1+u_2+\cdots$: The total Stiefel-Whitney class $w$ is the Steenrod square of the Wu class $u$: ...
0
votes
0answers
77 views

Two books defined two Chern(Euler) classes yet differed by a negative sign, what's wrong?

In the book 'Principles of Algebraic Geometry' P141 and the book 'Differential Forms in Algebraic Topology' P72-73, they defined the Chern(Euler) classes of line bundles using the patching data ...
5
votes
1answer
108 views

Chern classes via connections

Let $M$ be a smooth real manifold and $B$ an Hermitian vector bundle over it. Then one can define Chern classes as $$c(B)=\sum c_i(B)t^i=\det \left( I+\frac{it\Omega}{2\pi} \right) \in ...
1
vote
0answers
19 views

the Pontryagin number for 4-dim surface bundle

Corollary 1.8 in arxiv.org/pdf/1103.0218 implies that the Pontryagin number for a 4-dim surface bundle is non-zero only when the surface has a genus $g>2$. I would like to ask what is the minimal ...
1
vote
1answer
70 views

Geometric picture of Stiefel-Whitney class of a tangent bundle?

The first Stiefel-Whitney class $w_1$ of a tangent bundle of a manifold $M$ has a simple geometric picture: if there is a loop that the orientation of the tangent space reverses, then $w_1\neq 0$ and ...
7
votes
1answer
112 views

Relation between Stiefel-Whitney class and Chern class

A complex vector bundle of rank n can be viewed as a real vector bundle of rank 2n. From nLab, we have that the second Stiefel-Whitney class of the real vector bundle is given by the first Chern class ...
7
votes
2answers
123 views

Chern-Weil: why do we divide by $2\pi$?

So here's a somewhat incoherent question. To define characteristic classes in the Chern–Weil way, one takes a curvature form $\Omega$ on a vector bundle $E \to M$ and an invariant polynomial ...
3
votes
1answer
74 views

Naive question: what good are characteristic classes of principal bundles?

I recently read a development of characteristic classes on principal bundles through curvature forms and the Chern–Weil homomorphism. Unfortunately, this exposition concluded without listing any ...
5
votes
1answer
116 views

Topological invariance of chern classes

Are Chern classes topological invariants? To be more precise: Given two complex manifolds $M$ and $N$. Does a homeomophism $f:M\to N$ map Chern classes to Chern classes?