For closed smooth manifolds $M$ and $N$, the Hirzebruch $L$ class is multiplicative, i.e. $L(M\times N)=L(M)L(N)$. Is this property still true if $M$ and $N$ are assumed to be closed topological manifolds and one considers combinatorial $\ell$ classes for rational homology manifolds (as defined, for example, in Milnor and Stasheff's book "Characteristic Classes")?
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