TITLE: Fundamental class of products of spaces
QUESTION [3 upvotes]: 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: $$[X\times F] = [X]\times [F]?$$

REPLY [3 votes]: 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.
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