Let $X$ and $Y$ be two compact connected oriented manifolds of real dimension $m$ and $n$, respectively. Let us assume that these are also smooth projective complex varieties (the reason to be clear later). Let us assume moreover that the singular cohomologies of $Y$ are torsion-free, so that Kunneth formula holds for both homology and cohomology of $X\times Y$.
If $p: X\times Y\rightarrow X$ denotes the projection, then for any $k\leq m$,
let $p_*:H_k(X\times Y,\mathbb{Z})\rightarrow H_k(X,\mathbb{Z})$ be the pushforward map. I want to understand $p_*$.
Writing $H_k(X\times Y,\mathbb{Z})\cong\oplus_{i+j=k}H_i(X)\otimes H_j(Y),$ is it true that $p_*$ corresponds to the identity map $H_k(X)\otimes H_0(Y)\cong H_k(X)\otimes \mathbb{Z}\xrightarrow{\text{id}}H_k(X)$ in the first summand, and zero in the other summands?
The reason I think this should be true is that if I think of the pushforward as a proper pushforward in algebraic cycles, then I believe that the cycles in $H_i(X)\otimes H_j(Y)$ for any $i<k$ should be given by a submanifold whose image under $p$ should have dimension $i$, so that the pushforward should be zero by definition of proper pushforward.
Is my claim correct? Even if it is correct, I'm not completely satisfied with my argument; can someone provide a better explanation if possible?