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2
votes
1answer
159 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?
5
votes
0answers
95 views

Steenrod operations on cohomology of grassmannians

Let $G_k(\mathbb{R}^n)$, $n\geq k$ and $G_k(\mathbb{R}^\infty)$ be the finite-dimensional and infinite-dimensional grassmannians respectively. Their cohomology rings are expressed in terms of ...
6
votes
1answer
225 views

Four-dimensional vector bundles over $S^4$, intuition?

I know that $\pi_3(SO(4)) = \mathbb{Z} \oplus \mathbb{Z}$. We can choose an explicit identification as follows: given $(i, j) \in \mathbb{Z}$, we have a map $\phi: S^3 \to SO(4)$ which sends a unit ...
6
votes
0answers
147 views

Bundles over Grassmanian with given top Chern class

So, I have been working on Chern classes for my master thesis and apparently (My proofs could be wrong and a few things are still vague) I was able to give a construction method and exhibit, via ...
13
votes
3answers
2k views

Intuition behind the Kodaira Vanishing Theorem?

As the question suggests, what is the intuition behind the Kodaira Vanishing Theorem? The Kodaira Vanishing Theorem says that the cohomology groups $H^q(M, L \otimes K_M)$ vanish for $q \ge 1$ when ...
3
votes
0answers
109 views

Does there exist a smooth compact manifold whose boundary is $\mathbb{R}P^3$? [closed]

As the question suggests, does there exist a smooth compact manifold whose boundary is $\mathbb{R}P^3$?
1
vote
1answer
134 views

Chern classes of three (two) dimensional complex vector bundles

Let $M$ be a manifold. Let $F(M,3)=\{(m_1,m_2,m_3)\mid m_1, m_2, m_3\in M, m_i\neq m_j, \text{ for any } i\neq j\}$. Let $S_3$ be the symmetric group of order $3$. Let $S_3$ act on $F(M,3)$ by ...
0
votes
0answers
82 views

rational cohomology of finite dimensional real grassmannian

Let $G_k(R^n)$, $n>k$, be the finite dimensional real grassmannian. What is the rational cohomology algebra $H^*(G_k(R^n);Q)$? I have searched out that $H^*(BO_k;Q)=Q[p_1,p_2,...,p_[k/2]]$ is the ...
-5
votes
1answer
215 views

Stiefel-Whitney class of complex projective spaces [closed]

Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$? Let $a_m$ be the maximal integer such that the $a_m$-th dual ...
2
votes
1answer
146 views

rational cohomology of finite real grassmannian

Let $p_j$ to be the $j$-th Pontryagin class of the universal $n$-plane bundle $E_n(\mathbb{R}^\infty)\to G_n(\mathbb{R}^\infty)$. Then according to Theorem 1.6, The Cohomology of BSO n and BO n with ...
0
votes
0answers
67 views

Wedge product of Endomorphism-Valued Forms

To define characteristic classes in smooth vector bundles $E\longrightarrow M$ there is a more or less standard procedure: to choose a connection $\nabla$ and to derive the curvature $\Omega$, which ...
0
votes
1answer
121 views

$\mathbb{Z}_{2}$ -equivariant vector bundles over manifold of rank-$k$ matrices

Edit: I remove the trivial part of the first version, according to comment of Alex Degtyarev Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. There is ...
2
votes
2answers
272 views

Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle?

Let $V\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\pi:\mathbb{P}V ...
0
votes
0answers
63 views

discrete group action on Stiefel manifold

Let $S_3$ be the permutation group of order $3$. Let $V_2(\mathbb{R}^n)$ be the stiefel manifold of $2$-frames in $\mathbb{R}^n$. Let $S_3$ act on $V_2(\mathbb{R}^n)$ by $$ (1,2)(u,v)=(-u,v-u), $$ $$ ...
1
vote
0answers
152 views

Pontryagin class of quaternionic line bundle

Let $\xi^{\mathbb{C}}$ be a complex line bundle over a CW complex $B$. Then $$ VB_{\mathbb{C}^1}(B)\cong [B,BU(1)]=[B,\mathbb{C}P^\infty]=[B,K(\mathbb{Z},2)]\cong H^2(B;\mathbb{Z}). $$ Hence if ...
0
votes
0answers
140 views

obstructions of Chern class and Pontryagin class

Let $\xi$ be a real $n$ dimensional vector bundle over a CW-complex $B$. Then the Stiefel-Whitney class (coefficient in $Z/2$) $$ w_i(\xi)=0$$ if and only if $\xi|_{sk^i(B)}$ has $n-i+1$ linearly ...
1
vote
2answers
331 views

integral or rational cohomology of real grassmannians

I have obtained that the cohomology rings $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]. $$ Also $$ H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
27
votes
2answers
1k views

Unifying Geometry for Characteristic Classes

When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches: Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, ...
4
votes
1answer
311 views

Is the space of real conics with a singular point an orientable manifold?

Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set of such a polynomial gives a real curve ...
2
votes
3answers
432 views

Is true that $\left[\frac{\hat{A}(\mathbb HP^m)} { \hat{M}(\mathbb HP^m) }\right]_{4m} = 0$?

There are two questions: How to prove that in general $[\hat{A}(\mathbb HP^m)]_{4m} = 0$ It is possible to verify it for low values of $m$. How to prove that in general ...
0
votes
0answers
115 views

An integrality theorem for immersions of quaternion projective spaces in the euclidean space

There are three questions: Please let me know your proof of the following theorem: If $\mathbb HP^2$ can be immersed in $\mathbb R^{12}$ with an Euler class $W_{4}(\nu)$ for the normal bundle of ...
0
votes
1answer
176 views

An integrality theorem for immersions of complex projective spaces in the euclidean space

There are three questions: Please let me know your proof of the following theorem: If $CP^3$ can be immersed in $R^8$ with an Euler class $W_{2}(\nu)$ for the normal bundle of $CP^3$ respect to ...
10
votes
1answer
491 views

Is there any relationship between the Euler class and the Vandermonde determinant?

Several Wikipedia articles claim that the relationship between the Euler class $e(V)$ and the top Pontryagin class $p_k(V)$ of an oriented $2k$-dimensional real vector bundle $V$ corresponds, via the ...
-2
votes
1answer
292 views

Computing the Chern class of $S^6$ [closed]

I am trying to calculate the Chern class of the tangent bundle of the sphere $S^6$. I am told that this is an interesting case, since $S^6$ is not a complex manifold, but it has an almost complex ...
1
vote
1answer
220 views

Euler Class of a vector field [closed]

Let M be a closed 3-manifold and let X be a vector field on M. In which conditions might we define a Euler class associated to X? For example, is it possible to define for a rotational Beltrami ...
5
votes
1answer
325 views

Obstructions to the existence of stable (and unstable?) complex structures?

Let $V$ be a real vector bundle on a space $X$, perhaps the tangent bundle of a smooth compact manifold. I'm interested in understanding the obstructions to $V$ admitting a stable complex structure, ...
5
votes
1answer
208 views

Pontryagin number for 4-dim surface bundle

In paper arXiv:math/0701247 "Divisibility of the stable Miller-Morita-Mumford classes" by Soren Galatius, Ib Madsen, Ulrike Tillmann, it was shown that the Pontryagin numbers for a 4-dim surface ...
3
votes
0answers
96 views

Relations between Stiefel-Whitney classes on mapping torus

In question Relations between Stiefel-Whitney classes the relations between Stiefel-Whitney classes on manifold are obtained. My question is that do we have additional relations between ...
1
vote
1answer
248 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 ...
1
vote
1answer
148 views

Relation between Chern characteristic and Pontryagin characteristic

A 2-dim complex manifold can be viewed as a 4-dim real manifold. What is the relation between the Chern characteristic and the Pontryagin characteristic of the tangent bundle? It should be $p_1=n_1 ...
4
votes
0answers
112 views

Concrete almost-complex structures on $3 \#CP^2$

The connect sum $X:=CP^2\# CP^2 \# CP^2$ supposedly supports almost-complex structures, i.e. endomorphisms $J$ of the tangent bundle such that $J^2=-id$. The existence of these almost-complex ...
10
votes
2answers
444 views

Vector bundle for prescribed Stiefel-Whitney classes

I hope this is not trivial. Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice) For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology ...
9
votes
1answer
306 views

How to flow submanifolds?

Motivation We want a consistent way of perturbing a submanifold away from itself. For $0$-dimensional submanifolds, this is the same data as a nowhere-vanishing vector field: we may flow the points ...
1
vote
2answers
249 views

A generalized Thom Isomorphism

The ordinary Thom isomorphism says $H^{*+n}(E,E_{0}) \simeq H^{*}(X)$, where $E$ is a vector bundle over $X$ and $E_{0}$ is $E$ minus the zero section. Now assume that $S$ is a non vanishing section ...
3
votes
2answers
195 views

Canonical n plane bundle over Lagrangian Grassmanian

Recall that the Lagrangian Grassmanian, which is denoted by $\Lambda(n)$, is the subset of the standard Grassmanian $G(n,2n)$, which consists lagrangian sub vector spaces of $\mathbb{R}^{2n}$. Lets ...
13
votes
1answer
738 views

Adams' theorems on the Hopf-Whitehead J-homomorphism

The J-homomorphism is a well-known and classical map $\pi_n (O(k)) \to \pi_{n+k} (S^k)$, or after stabilizing with respect to $k$, a map $J_n:\pi_n (O) \to \pi_{n}^{st}$, from the stable homotopy of ...
6
votes
1answer
212 views

Stiefel-Whitney classes of virtual vector bundles

Let $E=\xi-\eta$ be a virtual vector bundle over a compact base $B$, which we may assume is a CW complex. A quick and dirty way to define the total Stiefel-Whitney class $w(E)\in ...
7
votes
2answers
273 views

Integral versus real (universal) characteristic classes

I'm pretty confused about the precise relation of the integral and the real cohomology of the classifying space $BG$ of a compact Lie group $G$. The natural map $H^n(BG;\mathbb{Z})\to ...
8
votes
1answer
482 views

Pontryagin numbers on manifolds with an $S^1$-action

Let $M$ is a smooth compact manifold with an $S^1$-action with isolated fixed points. Suppose the representation of $S^1$ at tangent spaces at all fixed points is known. Can one then find all ...
1
vote
0answers
101 views

How to show the Euler Characteristic is equal to self-intersection number of zero-section [duplicate]

myThe definition of the Euler characteristic (given in Guillemin and Pollack's "Differential Topology") of a compact oriented manifold $X$ is the self-intersection number of the diagonal $\Delta$ in ...
4
votes
0answers
228 views

Principal bundles and Čech cohomology with non-good open covers

I'm trying to compute characteristic classes of principal bundles by defining transition functions and computing them in Čech cohomology. However, it seems all the constructions are defined in terms ...
4
votes
1answer
298 views

Rigidity of secondary characteristic classes

For a representation $\rho:\pi_1M\rightarrow GL(n,C)$ and the associated flat $GL(n,C)$-bundle $E_\rho\rightarrow M$ one has the Cheeger-Chern-Simons classes $$\hat{c}_k(E_\rho)\in H^{2k-1}(M,R/Z)$$ ...
6
votes
3answers
332 views

Second Stiefel Whitney class of quotients of odd spheres

I don't know much of algebraic topology so the following question could be very silly. Let $G$ a finite subgroup of $U(n)$ that acts linearly (the action induced by the action of $U(n)$ on ...
2
votes
1answer
168 views

Stiefel classes and generic sections

I asked this question in math.stackexchange few days ago. Unfortunately, I haven't seen any simple answer. One can say that the Stiefel-Whitney classes is dual classes to the locus of linearly ...
17
votes
0answers
499 views

Status of the Euler characteristic in characteristic p

In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes: Enfin signalons que la situation en caractéristique positive est loin d'être aussi ...
2
votes
1answer
248 views

Chern Character Isomorphism for non-finite CW complexes, resp. for non-CW complexes

This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/397164/7110/ Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that ...
2
votes
1answer
134 views

Non-(stable)-triviality of the tautological bundles

This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/396217/7110/ The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann manifold ...
0
votes
1answer
180 views

How to characterize this particular kind of bundle?

I am considering the following situation. Let $M_5$ be a 5-dimensional manifold which is an $S^1$ principal bundle over 4-manifold $M_4$. For instance, $M_5 = S^5$ and $M_4 = \mathbb{CP}^2$ with ...
1
vote
1answer
245 views

Proof that the Hodge-de Rham Rank Equals the Euler Characteristic

Can someone please provide a good (online accessible) reference for the well-known identity $$ \text{rank((d + d}^*)^+) = \sum_{i=}^n (-1)^i \dim(H^i(M)), $$ where $M$ is a manifold of dimension $n$, ...
16
votes
3answers
1k views

How we do actually compute the topological index in Atiyah-Singer?

This is migrated by math.stackexchange as I did not receive an answer. I do not know if it is too naive for this site. I am taking a lectured class in Atiyah-Singer this semester. While the class is ...