The characteristic-classes tag has no wiki summary.

**4**

votes

**1**answer

280 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

**3**answers

416 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

**0**answers

103 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

**1**answer

160 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

**1**answer

436 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 ...

**-1**

votes

**1**answer

248 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

**1**answer

172 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

**1**answer

304 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

**1**answer

174 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**

votes

**0**answers

75 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

**1**answer

144 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

**1**answer

141 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

**0**answers

100 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

**2**answers

380 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

**1**answer

293 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

**2**answers

239 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

**2**answers

181 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

**1**answer

595 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

**1**answer

201 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

**0**answers

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

**2**answers

259 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

**1**answer

477 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

**0**answers

97 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

**0**answers

212 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

**3**answers

316 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

**1**answer

163 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 ...

**14**

votes

**0**answers

423 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

**1**answer

187 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

**1**answer

127 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

**1**answer

164 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

**1**answer

229 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

**1**answer

199 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

**1**answer

252 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

**2**answers

363 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

**1**answer

397 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 ...

**7**

votes

**0**answers

147 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 ...

**25**

votes

**1**answer

763 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 ...

**20**

votes

**4**answers

2k 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

**0**answers

409 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

**1**answer

200 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 ...

**13**

votes

**2**answers

435 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 ...

**6**

votes

**3**answers

461 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

**2**answers

266 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

**1**answer

247 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

**1**answer

186 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

**1**answer

906 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**

votes

**2**answers

411 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

**1**answer

990 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

**2**answers

367 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

**4**answers

520 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 ...