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14
votes
2answers
436 views
How to understand the Todd class?
I am reading the article "K-Theory and Elliptic Operators"(http://arxiv.org/abs/math/0504555), which is about Atiyah-Singer index theorem. In page 14 the article discussed the Thom isomorphism: ...
11
votes
2answers
276 views
Where do Chern classes live? $c_1(L)\in \textrm{?}$
If $X$ is a complex manifold, one can define the first Chern class of $L\in \textrm{Pic}\,X$ to be its image in $H^2(X,\textbf Z)$, by using the exponential sequence. So one can write something like ...
11
votes
2answers
141 views
Different ways of representing a second cohomology class
There are probably many ways of talking about a second (integral) cohomology class of a smooth, closed, orientable manifold $M$ of dimension $n$. Here are a few, with $\alpha\in H^2(M,\mathbb{Z})$:
...
9
votes
3answers
537 views
How to interpret the Euler class?
Although I (hardly) understand the formal definition of the Euler class, I have very little intuition of it. I understand that the Euler class of $E\to X$ is zero if and only if there is a section, ...
6
votes
1answer
150 views
obstruction cocycle of stiefel manifold
I was reading about the interpretation of Stiefel Whitney classes as obstructions from Milnor-Stasheff's book and I got stuck at a step. The context is the following. Let $E \to B$ be a vector bundle ...
6
votes
0answers
36 views
Chern classes of free quotient manoflds
Let $X$ be a compact complex manifold. Assume that a finite group acts on $X$ freely. Then the quotient $X/G$ is again a compact complex manifold. I wonder if there is a good way to compute Chern ...
6
votes
0answers
185 views
Cohomology ring of Grassmannians
I'm reading a paper called An Additive Basis for the Cohomology of Real Grassmannians, which begins by making the following claim (paraphrasing):
Let $w=1+w_1+ \ldots + w_m$ be the total ...
5
votes
2answers
211 views
Vanishing of the second Stiefel–Whitney classes of orientable surfaces
How does one see that the second Stiefel-Whitney class is zero for all orientable surfaces. For $S^2$ this can be seen by $TS^2$ being stably trivial, and for $S^1 \times S^1$ one can use $T (S^1 ...
5
votes
1answer
171 views
Why are characteristic classes well-defined?
In the definition of characteristic classes for a complex vector bundle $E$ ober a topological space $X$, we consider some space $X_S$ and a continuous map $p: X \rightarrow X'$ such that $E$ is ...
5
votes
1answer
128 views
Second Stiefel-Whitney Class of a 3 Manifold
This is exercise 12.4 in Characteristic Classes by Milnor and Stasheff.
The essential content of the exercise is to show that $w_2(TM)=0$, where $M$ is a closed, oriented 3-manifold, $TM$ its tangent ...
5
votes
1answer
142 views
How can we detect the existence of almost-complex structures?
Any smooth $2n$-manifold $M$ comes with a well-defined map $f:M\rightarrow BGL_{2n}(\mathbb{R})$ (up to homotopy) classifying its tangent bundle. Since $GL_{2n}(\mathbb{R})$ deformation-retracts onto ...
5
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0answers
54 views
explicit cocycle representing Stiefel-Whitney class in Milnor and Stasheff
I am trying to do Problem 7A in Characteristic Classes by Milnor and Stasheff which asks the reader to do the following:
Identify explicitly the cocycle $C^r(G_n) \cong H^r(G_n)$ which ...
5
votes
0answers
91 views
Exercise 4-A “Characteristic Classes” by Milnor and Stasheff
Exercise 4-A of Milnor and Stasheff's book "Characteristic Classes" reads:
Show that the Stiefel-Whitney classes of a Cartesian product are given by
$w_k(\xi\times\eta) = \sum^k_{i=0} ...
4
votes
1answer
255 views
Euler class and Vandermonde polynomial
I found the following in the wikipedia page for Euler class.
«If the rank $r$ is even, then this cohomology class $e(E) \cup e(E)$ equals the top Pontryagin class $p_{r/2}(E)$.
Under the splitting ...
4
votes
2answers
274 views
The “Wu formula” and Steenrod algebras
The Wikipedia page on Stiefel-Whitney classes includes the following paragraph:
Over the Steenrod algebra, the Stiefel–Whitney classes of a smooth
manifold (defined as the Stiefel–Whitney ...
4
votes
1answer
227 views
Chern Classes and Stiefel-Whitney Classes
I'm trying to understand the relationship between Chern classes and Stiefel-Whitney classes, and I came upon this problem (14-E) in Milnor-Stasheff's Characteristic Classes.
We are asked to define ...
4
votes
1answer
87 views
Stiefel-Whitney numbers for product bundle
I'm reading Milnor's "characteristic classes" and I want to compute Stiefel-Whitney numbers of $ P^2 \times P^2 $ (product of projective spaces) for one of the problems,
I know how Stiefel-Whitney ...
4
votes
0answers
147 views
Grothendieck on (topological) Chern Classes
I have been reading through the wikipedia article about Chern classes and it currently has a section devoted to the Alexander Grothendieck axiomatic approach. The language used throughout the section ...
4
votes
0answers
140 views
Characteristic Class exercises
I tagged this as "homework" because my supervisor told me I need to be better at computing characteristic classes.
The classic examples I can think of are tangent bundles and tautological line ...
4
votes
0answers
102 views
The action of the Steenrod algebra on $H^*(BU; \mathbb{Z}_p)$
By considering the classifying map $f \colon (\mathbb{C} P^{\infty})^n \rightarrow BU(n)$, its induced map on cohomology, and using the Cartan formula, we can derive the Wu formula for the action of ...
3
votes
3answers
727 views
Chern Classes of a Trivial Bundle
Could someone explain to me why the chern classes of a trivial bundle are zero?
(I'm studying it from Bott & Tu book) To be more specific I can't understand why, given the vector bundle $E$ on ...
3
votes
1answer
161 views
Does naturality imply isomorphism invariance for characteristic classes?
A property of characteristic classes $c$ is that $c(f^* E) = f^* c(E)$ where $E\to M$ is a bundle and $f^* E$ is the pullback of $E$ by some map $f: N\to M$. In Bott and Tu it is stated (for Chern ...
3
votes
1answer
192 views
What's the map $BU \times \mathbb{Z} \to \prod K(\mathbb{Z},n)$ representing the total Chern class?
Recall that complex topological $K$-theory is representable on reasonable spaces by the space $BU \times \mathbb{Z}$ (where $BU$ is a colimit of various infinite Grassmannians), and that the total ...
3
votes
2answers
104 views
An alternative description of the first Stiefel-Whitney class
I've heard this description of the first Stiefel-Whitney class of a vector bundle but I don't know why this is true. Can anyone help me with this, please?
The first Stiefel-Whitney class of a vector ...
3
votes
1answer
274 views
Determining the “positivity” or “negativity” of Chern class (number?) of zero-sets of homogeneous polynomials
If $\Omega$ is the curvature 2-form on a $n-$manifold, then I would think that the Chern classes (forms), $c_k$ are defined as,
$det(I + \frac{it\Omega}{2\pi}) = \sum c_k t^k$
I would like to ...
3
votes
1answer
103 views
Differing axioms for Stiefel-Whitney class
In Milnor and Stasheff, it is taken as part of the first axiom that all Stiefel-Whitney classes of a bundle vanish in dimensions greater than the rank of the bundle. However, in other sources this is ...
3
votes
1answer
78 views
Does the splitting principle define chern classes for vector bundles if they are known for line bundles?
Suppose we have definied chern classes $c_i$ for line bundles for all $i\geq 0$.
Let $E\to B$ be a rank $r$ vector bundle. By the splitting principle there exists a map $f:Y\to B$ such that $f^*E$ is ...
3
votes
1answer
61 views
On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.
This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set $F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$ where ...
3
votes
1answer
88 views
Pontryagin classes of a product manifold
I'm imagining there's a way to relate the pontryagin classes of $T(M\times N)$ to the pontryagin classes of $M$ and those of $N$, but I haven't been able to find a helpful reference. Could someone ...
3
votes
0answers
44 views
What is the geometric meaning of powers of the first Stiefel-Whitney class?
If $M$ is an $n$-dimensional manifold, I know $\\w_1^n(M)$ is a Stiefel-Whitney number of the manifold, but what does its vanishing geometrically mean?
More generally, does the Stiefel-Whitney height ...
3
votes
0answers
70 views
How to calculate characteristic classes of tensor products?
I was given the following as an execrise:
Prove that the tensor product of complex line bundles over $X$ satisfies the following relationship:
$$c_{1}(\otimes E_{i})=\sum c_{1}(E_{i})$$
It is ...
3
votes
0answers
109 views
Two Grassman manifold problems from Milnor and Stasheff's book
I'm stuck on the following two problems in Milnor and Stasheff's book Characteristic Classes. Really can't get my head around this material and I'm hoping that more worked examples would help. Even ...
3
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0answers
206 views
Intuition for multiplicative sequences
I have recently been reading about multiplicative sequences and genera from a couple of different sources, most notably "Spin Geometry" by Lawson and Michelsohn and "Characteristic Classes" by Milnor ...
2
votes
1answer
96 views
Codimension 1 immersions and Stiefel-Whitney classes
From Milnor and Stasheff:
If the $n$-dimensional manifold $M$ can be immersed in $\mathbb{R}^{n+1}$ show that each Stiefel-Whitney class $w_i(M)$ is equal to the $i$-fold cup product $w_1(M)^i$.
...
2
votes
1answer
74 views
Number of Zeros of a Section vs Integral First Chern Class
I've often read that the first chern class can be seen as "the number of zeroes a section must have".
How precise can this statement be made?
I'm only interested in Line bundles. I actually know ...
2
votes
1answer
66 views
Thom class: Why are the two definitions equivalent?
We know that the Thom class $\tau_W$ is defined on a disk bundle $W\rightarrow L$, where $L$ is a $p$-dimensional manifold and the rank of $W$ is $k$.
Let $[W]_0$ denote the fundamental class of the ...
2
votes
1answer
43 views
Curvature form, tangent bundle and structural group.
Let $T\mathbb{C}P^n$ the tangent bundle over $\mathbb{C}P^n$. We have that the Chern classes are the coefficients of the characteristic polynomial of curvature form $\Omega$ of $T\mathbb{C}P^n$:
$$ ...
2
votes
0answers
48 views
Chern class of tautological line bundle
I'm studying characteristic classes from the Chern-Weil construction (via connection and curvature). I'm trying to compute some simple examples. Let $E$ be the tautological line bundle over projective ...
2
votes
0answers
56 views
Chern classes tangent bundle.
I studied Chern classes but I don't find explicit examples of calculation. So I'd like to consider the tangent bundle over $\mathbb{C}P^n$ ($T\mathbb{C}P^n$) and develop the theory beginning from this ...
2
votes
0answers
54 views
A question on Chern character computation
Let $C$ be a smooth curve in a complex threefold $X$. How can I see that
$$
\mathrm{ch}(\mathcal{O}_C)=(0,0,[C],\chi(\mathcal{O}_C))\in H^0\oplus H^2 \oplus H^4\oplus H^6,
$$
where $H^0\cong ...
2
votes
0answers
59 views
Is Transversality invariant by losing Dimension?
Setup Let $X$ be a smooth, reduced and irreducible scheme. And let $E$ be a vector bundle of rank $n$ on $X$. Following Eisenbud and Harris we say that the collection $\{\sigma_1,...,\sigma_k\}$ of ...
2
votes
0answers
45 views
Possible restriction on first nonzero Stiefel-Whitney classes?
I've been reading Hatcher's book on vector bundles and I'm just getting into the section of Steifel-Whitney numbers. Naturally, I'm interested in which sequences of such numbers are realizable, and ...
2
votes
0answers
74 views
Image of Thom Class under Sequence of Maps?
So I've been trying to do problems in Milnor & Stasheff's Characteristic Classes as a quick review, not having done anything with them in a while. However, I'm stuck on some parts in attempting ...
2
votes
0answers
63 views
The characteristic class of a fibration is a fibre homtopy invariant
Let $p:E \to B$ be a fibration with fibre $F$ (denote the fibration $\xi$). Let us assume that the fibre is $(n-1)$ connected. There is a fundamental class $\iota_F \in H^n(F;\pi_n(F))$. We can define ...
2
votes
0answers
87 views
Question involving the Chern character from the book “Fibre Bundles”
On page 311 of Dale Husemöller's book Fibre Bundles in Theorem 11.6 he has
the following
commutative diagram
$$\begin{array}
& & K(BG)\\
&\nearrow &\downarrow\\
...
1
vote
1answer
278 views
Vanishing of Chern classes
I finally got the characterization of Chern classes, but i have another question:
Let E be a Complex vector bundle of rank r on M. Given $s_1, \cdots ,s_r$ generic global sections, i can characterize ...
1
vote
1answer
78 views
Existence of a square root of a given line bundle via Chern class?
I come across a statement like
Let $L$ be a complex line bundle on a manifold. $c_1(L)=0$ mod $2$ if and only if there exists a line bundle $K$ such that $L\cong K^{\otimes 2}$.
How can one ...
1
vote
1answer
123 views
Example of a nontrivial fiber bundle with total space compact, spin, and $p_1=0$
I would really appreciate if anyone could provide me with an example of a locally trivial, but globally nontrivial, fiber bundle $Y\hookrightarrow Z \rightarrow X$, where $X$, $Y$, and $Z$ are all ...
1
vote
1answer
45 views
Dropping the orientable condition from the Thom isomorphism theorem.
My first question is: what are some examples of un-oriented n-plane bundles over $ \mathbb{Z}$ where it is easy to see that there is no Thom class?
I would like to know some examples because the real ...
1
vote
1answer
103 views
Stiefel-Whitney numbers
I am reading Milnor's "Characteristic classes", and there are two things about Stiefel-Whitney numbers that made me confused.
The following theorem (due to Pontrjagin) is being proved.
If $B$ ...
