Fundamental class of products of spaces

Question

Let $M$ be smooth oriented manifold where $M=X\times F$, $X$ and $F$ smooth oriented manifolds. We note by $[M]$ the fundamental class of $M$. Is this equality true: $$[X\times F] = [X]\times [F]?$$

Answer 1

The Künneth theorem with real coefficients gives an isomorphism $$H_*(X\times F)\cong H_*(X)\otimes H_*(F).$$ Under this isomorphism, $\left[X\times F\right]$ corresponds to $\left[X\right]\otimes \left[F\right]$.

Answer 2

This statement is proved in Propositions 80.10 and Proposition 80.11 of my lecture notes: https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/1a-uptodate.pdf Getting the signs right was painful.