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.

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.

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. An explicit example of this can be seen in Milnor's construction of exotic spheres. There, he constructs homotopy spheres with nonzero Pontryagin classes. 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.

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.