Characteristic classes of spheres

Question

Since an exotic sphere is a differentiable manifold $M$ that is homeomorphic but not diffeomorphic to the standard Euclidean $n$-sphere, we may not be able to distinguish spheres from exotic sphere through characteristic classes.

However, it is still worthwhile to know the basic characteristic class data of spheres:

1. Stiefel–Whitney class $w_i$: Spheres are orientable and non-spin, thus $$ w_1(S^d)=0, $$ $$ w_2(S^d)=0, $$ More generally, what do we have for other $i$: $$ w_i(S^d)=? $$

2. Chern class $c_i$: Even-dimensional spheres have an even-real dimensional tangent bundle $TS^d$, thus we may define the Chern class $$c_i(TS^d)=c_i(S^d)=?$$ One may also consider the frame bundle of spheres $$c_i(FS^d)=?$$

3. Euler class: $$ \chi(S^d)=2, \text{ if $d$ even}; $$
   $$ \chi(S^d)=0, \text{ if $d$ odd.} $$

4. Wu class $u_i$: is related to the Stiefel–Whitney class $w_i$ through Stenrod square, so $$ u_1(S^d)=u_2(S^d)=u_3(S^d)=0 $$ More generally, what do we have for other $i$: $$ u_i(S^d)=? $$

5. Pontryagin class $p_i$: $$ p_i(S^d)=? $$ We can consider all the $d=0 \pmod 4$ dimensions of spheres. We know that $p_1(TS^4)=0$ and $p_1(FS^4)=?$ (The frame bundle of spheres).

6. Are there other powerful/ useful Characteristic classes of spheres?

Answer 1

1. Stiefel-Whitney classes of $X$ live in the cohomology groups of $X$. Spheres don't have cohomology groups except in dimensions $0$ and $n$. Therefore all their Stiefel-Whitney classes vanish except maybe the last one, but the top SW class is the Euler characteristic modulo $2$, therefore $w_i(\Bbb S^n)=0$ for all $i$.

Another point of view is that these classes are obstructions to build linearly independent families of sections over the $k$-skeleton. The sphere doesn't have a $k$-skeleton, so there is no obstruction.

Answer 2

The tangent bundle $TS^n\to S^n$ is stably trivial: Clearly $TS^n \oplus \nu = \theta^{n+1}$, and the normal line bundle $\nu$ admits the nowhere-vanishes section $\nu(x) = x$ and thus is trivial. Besides, $\tilde H^*(S^n)$ vanishes except in the top dimension. Thus all the classes above vanish. For more interesting characteristic classes attached to spheres (or sphere-like spaces), you'll have to look for something more exotic like $K$-theory, the Casson invariant for homology spheres, etc.

If you're looking at characteristic classes to classify vector bundles over spheres, for example, then you can do that directly by considering how the restrictions to each hemisphere are attached via a clutching map. You can write it out explicitly in terms of $\pi_* SO(n)$, and leads to the discussion of $K$-theory and Bott periodicity above.