I was wondering given a CW-complex $X$, can we by homological tools discern whether $X$ cannot have torus as a subcomplex?
If I am not mistaken, for a $d$-dimensional torus, we have
$$ H_k(\mathbb{T}^d;\mathbb{Z})= \begin{cases} \mathbb{Z}^{\binom{d}{k}} &;0\leq k\leq d\\ 0 & ;k\geq d \end{cases} $$
My Question is if we assume that $X$ contains a realization of $\mathbb{T}^d$ as a subcomplex, does that mean that $\mathbb{Z}^{\binom{d}{k}}$ is a sub-module of $H_k(X;\mathbb{Z})$? It seems to me like a set of independent cycles in the subcomplex, should still be independent cycles in the whole complex. Is this indeed the case?
I saw in another thread, that 'embedding the spaces' does not imply 'embedding' the homology. But I don't see what breaks down in the intuition written above?
I am not strong in my comprehension of Homology, but I thought this might be a well resolved question which I could not find an answer to, by an online search.