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0
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1answer
207 views

Computing the Chern class of $S^6$ [on hold]

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
127 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
278 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
158 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 ...
2
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0answers
62 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
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 ...
1
vote
1answer
134 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
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0answers
94 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 ...
9
votes
2answers
370 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
282 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
234 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 ...
2
votes
2answers
169 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
518 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
184 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 ...
0
votes
0answers
94 views

Dimension of the set of polarized sections

Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian, i.e. Maximal isotroic, dim$P_m=n$, ...
7
votes
2answers
252 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
474 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
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0answers
95 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
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0answers
195 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 ...
6
votes
3answers
307 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
160 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 ...
13
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0answers
390 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
159 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 ...
1
vote
1answer
123 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
161 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
222 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 $n$, ...
2
votes
1answer
196 views

Confusions over the definitions of universal bundle and characteristic class

In most references, only a principal G-bundle is called universal (ie every other bundle can be pullbacked from this one, or an equivalent definition). Does it make sense to speak of a universal F-G ...
0
votes
1answer
240 views

Helped needed with some characteristic class / number questions

Suppose M is a $2n$-complex dimensional complex manifold. a) Why is Pontryagin class independent of orientation of the bundle? ...
7
votes
2answers
357 views

Sum of two tangent bundles of $S^{2n}$

I was wondering if the sum $TS^{2n}\oplus TS^{2n}$ is a trivial bundle? The same is true for spheres of odd dimension (one can find a nowhere zero section of the second bundle, add it to the first, ...
12
votes
1answer
382 views

Chern numbers via Euler characteristics?

Let $X$ be a space good enough to have a fundamental class, and $E$ a complex vector bundle on $X$. Let $P$ be some polynomial expression, and say I want to evaluate $P(c_i(E)) \cap [X]$. Is ...
6
votes
0answers
132 views

“Mathai-Quillen-type” form on $M\times M$?

Let $(M,g)$ be a compact, oriented, $(2n)$-dimensional Riemannian manifold. I'm wondering whether there is a "canonical" construction of a $(2n)$-form $\eta_g$ on $M\times M$, such that $\eta_g$ is ...
24
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1answer
740 views

n-categorical description of Chern classes

The Chern classes of a rank $n$ vector bundle on $X$ are obtained from composing the associated classifying map in $[X, BU(n)]$ with the maps $BU(n) \to B^{2i} \mathbb{Z}$ corresponding to the ...
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4answers
1k views

What is geometrically the Pontryagin class?

What does the Pontryagin class detects or is an obstruction to? Please avoid any answer using that it's the even Chern class of the complexified bundle or any interpretation that relies on the ...
5
votes
0answers
382 views

R. Bott's lectures on characteristic classes

I am searching for R. Bott's lectures on characteristic classes and Gel'fand Fuks cohomology (New Mexico State Univ. 1973), apparently there are notes of these lectures taken by Mostow and Perchik ...
5
votes
1answer
197 views

Index theorems and orientability

Given a Dirac operator $D$ acting on some Clifford bundle $\mathcal{E}$ over a compact, even-dimensional, oriented manifold $M$, the Atiyah-Singer index theorem states that its index is given by ...
12
votes
2answers
410 views

Non-stably trivial bundle with trivial charactertic classes

Though it's relatively clear that the characteristic classes do not characterise a vector bundle (and after looking through some books) I could not find an example of a vector bundle which is not ...
5
votes
3answers
442 views

How does one go from Chern--Weil to cohomology classes on BGL(n,C)?

Let's assume we start with Chern--Weil theory in the following form: Given a manifold $M$ and a complex vector bundle $V$ over $M$, we can equip $V$ with a $\mathfrak g\mathfrak l_n(\mathbb C)$ ...
2
votes
2answers
254 views

A Existence Problem of (p,q) metric

My question is: Can we judge a manifold that can admit a (p,q) metric? I only know the case that the existen of a lorentz metric is equivalent to Euler Character is zero
0
votes
1answer
242 views

recurrence formula for *i*-th Chern class of $CP^n$

one can show that the relation between first Chern class and second Chern class of $CP^n$ is $\frac{2(n+1)}{n} c_2 (M)=c_1 (M)^2$ here $c_1 (M)^2=c_1 (M)∧c_1 (M)$. So is there any recurrence ...
4
votes
1answer
184 views

Computation of KO characteristic classes/numbers

How to compute KO characteristic classes/numbers? They were introduced by Anderson/Brown/Peterson to study the structure of the spin cobordism ring. I looked through the literature but I did not find ...
1
vote
1answer
868 views

First chern class

I know some examples that first Chern class has not sign(negative, positive or zero). But I am looking for a necessary and sufficient condition that first Chern class has sign.
16
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2answers
402 views

Characteristic classes for block bundles

Block bundles are PL analogs of vector bundles, see e.g. Rourke-Sanderson's article in Bulletin of AMS, 1966. There is a classifying space $B\widetilde{PL}_q$ for rank $q\ $ block bundles, which is ...
11
votes
1answer
872 views

Is it true that all real projective space $RP^n$ can not be smoothly embedded in $R^{n+1}$ for n >1

So first for n even, $RP^n$ is not orientable, hence can not be embedded in $\mathbb{R}^{n+1}$. For odd n, $RP^{n}$ is orientable, hence the normal bundle is trivial. Now using stiefel-Whitney ...
7
votes
2answers
328 views

References/surveys concerning characteristic classes of flat vector bundles

I'm looking for good surveys about characteristic classes of flat real vector bundles. Letting $G$ be $\text{SL}_n(\mathbb{R})$ with the discrete topology, orientable flat $n$-dimensional real vector ...
4
votes
4answers
512 views

Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles

I am looking for characteristic classes of vector bundles (either real or complex) with values in generalized multiplicative cohomology theories such that: i) they vanish if the bundle of unit ...
26
votes
2answers
2k views

A geometric characterization for arithmetic genus

Let $X$ be a smooth projective variety over $\mathbb{C}$. The following information is all equivalent (any of these numbers can be computed by a linear equation from any of the others): the ...
37
votes
4answers
2k views

Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic

Hello, I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the ...
3
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0answers
307 views

K-theory of differential graded modules over differential graded algebras

Suppose you have a smooth vector bundle $E$ over a smooth manifold $X$. If you consider the algebra $ \Omega^\ast (E)$ of differential forms on $E$, it will be homotopy equivalent to the algebra of ...
15
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0answers
471 views

Characteristic Classes for $E_8$ Bundles

Given a principal $E_8$ bundle $P\rightarrow X$ one can take the adjoint representation $\rho :E_8\rightarrow SU(\mathbb C^{248})$ and form the associated vector bundle $V=P\times_{\rho}\mathbb ...
18
votes
2answers
893 views

If the total Chern class of a vector bundle factors, does it have a sub-bundle?

Motivation: $T_{\mathbb P^2}$ isn't an extension of line bundles Here's a trick to show that the tangent bundle $T$ of $\mathbb P^2$ is not an extension of line bundles. If it were, we would have a ...