Let $(X, x)$ be some 'nice' pointed topological space - e.g. a CW complex - and $[\alpha] \in \pi_n(X,x) $ some nontrivial homotopy class represented by a map $\alpha: S^n \to X$. Then there is a standard way to 'kill' this class $[\alpha]$ by modifying the space $X$ appropriately attatching a $D^{n+1}$-ball to $X$ along $\alpha$ forcing $\alpha$ to contract to a single point.
One obtains then a new space $X':= X \cup_{\alpha} D^{n+1}$ together with natural inslusion $\iota_X: X \to X'$. which induces maps $\iota_k: \pi_k(X,x) \to \pi_k(X',x)$ on homology groups which are isomorphisms for $k < n$ and surjective for $k = n$ with $i_n([\alpha])=0$. That's why this modification can be also called 'controlled killing' a homotopy class. 'Controlled' because up all $n > k$ homotopy groups not change. Higher homotopy groups on the other hand may change in highly nontrivial way, but that's 'the price we have to pay.' This is one of the main ingredients how the trunicated groups of Postnikov tower are constructed.
Question: Is it also possible to perform in similar manner a modification of $X$ but now with the goal to kill a homology class $[\beta] \in H_n(X,x)$ is similar 'controlled' way, ie by not to destroy 'too much' stucture of the homology group like in case above with homology.
More precisely I'm looking for a way to find a construction $m: X \to X'$ modyfying $X$ in the way that the induced maps on $m_k: H_k(X,x) \to H_k(X',x)$ such that $m_k$ are isomorphisms, $m_n$ is surjective and kills $[\beta]$.
Can it be done in a similar manner like for homotopy groups? The problem: Every homotopy class $[\alpha]$ can be represented by a map $\alpha: S^k \to X$ which can be killed by attatching geometrically a $D^{n+1}$ ball to $X$. In contract a class $[\beta] \in H_n(X,x)$ is a less geometrical object; it is represented by a formal sum of $n$-simplices $\sigma: D^n \to X$ with $\mathbb{Z}$-coefficients, modulo boundaries. So I not see how previous strategy could be applied here.