30
votes
What are the possible Stiefel-Whitney numbers of a five-manifold?
Recall that on a closed $n$-manifold $M$, there is a unique class $\nu_k$ such that $\operatorname{Sq}^k(x) = \nu_kx$ for all $x \in H^{n-k}(M; \mathbb{Z}_2)$; this is called the $k^{\text{th}}$ Wu ...
- 19.2k
28
votes
Accepted
A difficult integral for the Chern number
If Stokes' theorem counts as a standard technique, then here's an answer:
Introduce a "vector potential"
\begin{equation}
A_i = \frac{1-\hat n_z}{\hat n_x^2 + \hat n_y^2}(\hat n_x \partial_i ...
- 628
19
votes
Accepted
Is a 4-dimensional submanifold of a spin manifold always spin?
Let $i$ denote an immersion $N \to M$. There is an exact sequence of vector bundles on $N$ given by
$$0 \to TN \to i^*TM \to \nu \to 0$$
where $\nu$ is the normal bundle. As total Stiefel-Whitney ...
- 19.2k
18
votes
Accepted
Is the cohomology ring $H^*(BG,\mathbb{Z})$ generated by Euler classes?
Let $G = \mathbb{Z}/p \times \mathbb{Z}/p$ for $p$ odd. Then $H^3(BG; \mathbb{Z}) \cong \mathbb{Z}/p$ is not in the subring generated by Euler classes, since the non-trivial irreducible ...
- 9,183
18
votes
Accepted
Wu formula for manifolds with boundary
A relative Wu formula for manifolds with boundary is discussed in Section 7 of
Kervaire, Michel A., Relative characteristic classes, Am. J. Math. 79, 517-558 (1957). ZBL0173.51201.
In particular, ...
- 35.6k
17
votes
Accepted
Who discovered this definition of Stiefel-Whitney classes?
For the relation $Sq(U) = \Phi(w)$, where $Sq$ is the total Steenrod squaring operation, $U$ is the Thom class, $\Phi$ is the Thom isomorphism and $w$ is the total Stiefel-Whitney class, I would cite ...
- 9,183
17
votes
Betti numbers as characteristic numbers?
No. The Stiefel-Whitney and Pontryagin numbers of a closed oriented manifold are cobordism invariants, but the Betti numbers are not.
More explicitly, all closed oriented $3$-manifolds are frameable ...
- 118k
17
votes
Accepted
A 4-manifold with Stiefel-Whitney classes $w_1\neq 0$, $w_3 \neq 0$, and $w_1^2=0$?
Let's begin by reformulating the question a bit. Note that any orientable 4-manifold is spin-c, so in particular has $w_3 = 0$. The condition that $w_1 \not= 0 $ is thus redundant.
Wu's theorem gives ...
- 1,936
16
votes
Pin$^+$ and Pin$^−$ structure for manifolds in any dimensions
The condition for having a $Pin^+$-structure is the vanishing of $w_2$, and for having a $Pin^-$-structure is the vanishing of $w_2 +w_1^2$, for manifolds of any dimension. This is because the Lie ...
16
votes
Accepted
A Compact Manifold with odd Euler characteristic whose tangent bundle admits a field of lines
I believe there is no example satisfying all your constraints. If I recall (my memory is a little foggy on this) the result likely goes back to Hopf, and one of his variations on the Poincare-Hopf ...
- 43.9k
15
votes
Accepted
Fourth obstruction, Pontryagin and Euler class
Geometric generators for $\pi_3(SO(4))$ have been identified in §22 of Steenrod's "Topology of fibre bundles", using the identification of $S^3$ as unit quaternions. Conjugation of quaternions induces ...
- 17.4k
14
votes
Accepted
Infinite Grassmannian does not have the homotopy type of a finite-dimensional complex
We have $H^*(BO(k); \mathbb{Z}_2) \cong \mathbb{Z}_2[w_1, \dots, w_k]$ where $\deg w_i = i$. In particular, $H^n(BO(k); \mathbb{Z}_2) \neq 0$ for every $n$ as $w_1^n$ is a non-zero element. Therefore $...
- 19.2k
14
votes
Accepted
Second Stiefel-Whitney class is a square
At least there are quite a lot of such manifolds: up to multiplying by powers of 2, any oriented bordism class contains such a manifold.
Proof: Let $f: X \to BSO$ be the universal map such that $w_2$ ...
- 751
14
votes
Analogy between Stiefel-Whitney and Chern classes
Here is one way I like to think of the analogy.
The maximal torus of diagonal matrices $T^{n} \subset U(n)$ gives a map $BT^n \to BU(n)$ which on integral cohomology gives an isomorphism from $H^...
- 30.4k
13
votes
Chern class on a symplectic manifold
The answer is no.
First of all, if you want $f\omega$ to define a cohomology class, you should ask that $d(f\omega)=0$, and this is equivalent to $df\wedge\omega=0$, since $d\omega=0$. Using Darboux ...
- 7,882
13
votes
Accepted
Is there a closed 5-manifold $M$ with $w_1(M)w_2(M)\ne 0$?
Note that the third Wu class is $\nu_3 = w_1w_2$, so on a closed connected smooth $n$-manifold $M$, $\operatorname{Sq}^3 : H^{n-3}(M; \mathbb{Z}_2) \to H^n(M; \mathbb{Z}_2)$ is given by $\operatorname{...
- 19.2k
13
votes
Why is the first integral Pontryagin class a homeomorphism invariant?
For spin manifolds this is proved in Corollary 1.22 (p.17) of Kammeyer's Diploma. Kreck's claim that the "spin" assumption can be dropped is mentioned after the corollary, with the caveat that "the ...
- 29k
13
votes
Accepted
Conversion formula between "generalized" Stiefel-Whitney class of real vector bundles: O(n) and SO(n)
First I will write up what your question is asking in terms of Arun Debray's comment. I strongly suggest that when discussing questions like this, you use precise notation as in the following; I found ...
- 9,553
13
votes
Why do Chern classes and Stiefel-Whitney classes satisfy the "same" Whitney sum formula?
I prefer this approach which I believe is due to Grothendieck. (I haven't checked how this compares with the sources cited by Nick Kuhn.)
Let $(\mathbb{K},R,d)$ be $(\mathbb{R},\mathbb{Z}/2,1)$ or $(\...
- 55.7k
13
votes
What is the Todd class *really*?
To my knowledge, currently the best "motivation" for the Todd class comes from the so called "orientation theory" and the formal group laws associated to "oriented" ...
- 2,851
13
votes
Computation on characteristic classes
I wrote a blog post that turned into quite a nice exercise in characteristic classes. The goal was to compute the cohomology of a smooth hypersurface of degree $d$ in $\mathbb{CP}^3$, as a ring. This ...
- 118k
12
votes
rational cohomology of finite real grassmannian
I could not find the explicit formulas in the Algebraic models book (they seem to only do infinite Grassmannians and Stiefel varieties) or Mimura-Toda (they do the complex and symplectic case but not ...
- 17.4k
12
votes
Accepted
Generalize Wu formula to integral cohomology classes
I do not think such a cohomology operation can exist, as it would descend to a rational cohomology class in $H^{n+4}(K(\mathbb Z,n);\mathbb Q)$. But for any odd $n$ and also for all even $n >4$, ...
- 11.9k
12
votes
Accepted
Oriented Bordism Group and Un-Oriented Bordism Group of points $pt$
Unoriented cobordism: can be read off from the structure of the unoriented cobordism ring (calculated in Thom's thesis): $\Omega_6^O = (\mathbb Z/2)^3$, $\Omega_7^O = \mathbb Z/2$, $\Omega_8^O = (\...
- 6,831
12
votes
Mathematical/Physical uses of $SO(8)$ and Spin(8) triality
Seiberg and Witten showed that the $\mathcal{N}=2$ supersymmetric SU(2) gauge theory with $N_f=4$ flavor is endowed with SO(8) flavor symmetry, and it enjoys SO(8) triality.
Later, Gaiotto's ...
- 2,273
12
votes
Chern classes of generators of $K(S^{2n})$
Note that $K(S^{2n}) = \mathbb{Z}\times\mathbb{Z}$ has many different sets of generators and these can have different Chern classes, so the question doesn't have a well-defined answer. For one ...
- 19.2k
12
votes
Accepted
Motivation for the definition of complex orientable cohomology theory
As you wrote, complex orientability can be characterized by the cohomology of $\mathbb{C}P^\infty$: $E$ is complex orientable if $E^*(\mathbb{C}P^\infty)$ splits according to the cell structure of $\...
- 10.4k
12
votes
Accepted
Steenrod powers of the Thom class
I don't know a reference, but you can proceed as follows. By the splitting principle, it suffices to give the formula for vector bundles which are sums of complex line bundles, and we may as well then ...
11
votes
Accepted
Vector bundle over an oriented manifold with non-vanishing w_2w_3
As far as I know the Wu manifold $X=SU(3)/SO(3)$ is orientable and has mod 2 cohomology ring $H^*(X;\mathbb{Z}_2)=\Lambda(\omega_2(X),\omega_3(X))$. Thus $\omega_2(X)\cdot\omega_3(X)\neq 0$, and in ...
- 5,296
11
votes
Accepted
What is the first Pontryagin class of the $n$-dimensional representation of $S_n$?
If I am not confused, $p_1(V)=-c_2(V\otimes {\mathbb C})$. According to Theorem 7.1 in the book Characteristic Classes and the Cohomology of Finite Groups by Charles Thomas, $c_2$ of the standard ...
- 10.9k
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