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Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal

$$ w(TM)=w(TN)? $$ Do the Chern classes of the tangent bundles of $M$ and $N$ equal

$$ c(TM)=c(TN)? $$

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  • $\begingroup$ Manifolds don't have Chern classes! $\endgroup$
    – Qiaochu Yuan
    Sep 4, 2015 at 7:40
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    $\begingroup$ The question as originally stated assumes the two manifolds have different dimensions. If this is what was intended, then the manifolds cannot both be closed, and counterexamples are easy to find. For example, let $M$ be the circle and $N$ the Moebius band (either the compact or the noncompact version). These have different classes $w_1$ since $M$ is an orientable manifold and $N$ is not. Higher-dimensional counterexamples are equally easy to find. $\endgroup$
    – Allen Hatcher
    Sep 4, 2015 at 10:14
  • $\begingroup$ yes. The manifolds may mot be closed. $\endgroup$
    – QSR
    Sep 4, 2015 at 10:57

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Wu's formula for the Stiefel-Whitney classes implies that they are invariants of homotopy type. See for example here.

Chern classes are not even diffeomorphism invariant, and it is possible to have two complex structures on the same manifold with different Chern classes. See this question and its answers.

Also, Pontryagin classes are not invariants of homotopy type. As Danny mentions below, there are examples of fake projective spaces with different Pontryagin classes than $\Bbb C P^n$, but I don't have an explicit example. This paper describes how surgery theory can be used to start with a parallelizable manifold of a certain form and obtain a non-parallelizable manifold which is homotopy equivalent to the original. An interesting question is which rational polynomial expressions in the Pontryagin classes are homotopy invariants. The Novikov conjecture asserts that only the $L$-polynomials (from the Hirzebruch signature theorem) are.

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  • $\begingroup$ It is true that Pontryagin classes are not homotopy invariants, but that's not in Milnor's paper. His exotic spheres are 7-dimensional, and have no non-trivial Pontryagin classes because the groups where they live are 0. The simplest examples I know of are fake complex projective spaces. $\endgroup$
    – Danny Ruberman
    Sep 4, 2015 at 11:51
  • $\begingroup$ @DannyRuberman Thanks for the correction, I tried to answer the question too fast. The $\lambda$-invariant for Milnor's exotic spheres are defined in terms of Pontryagin classes, but of course are not related to the (trivial) Pontryagin classes of the exotic sphere itself but instead those of an $8$-manifold it bounds. $\endgroup$
    – Henry T. Horton
    Sep 4, 2015 at 16:46
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    $\begingroup$ Why doesn't this statement about homotopy invariance of Stiefel-Whitney classes contradict Allen's Mobius band example above? At a quick glance, I guess the issue is that the Wu formula only applies to closed manifolds? $\endgroup$
    – Greg Friedman
    Sep 9, 2015 at 6:27

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