Kill a Homology class $[\beta] \in H_k(X)$ only by attaching $(k+1)$-cells

Question

Let $X$ be a $n$-dimensional dimensional CW-complex build of skeleta $(X_0,X_1,...,X_k ,..., X_n) $. Let $[\beta] \in H_k(X)$ a non zero homology class which by definition can be represented by a formal sum $\beta= \sum_i a_i \sigma_i^k$ which is not contained in the image of the boundary map $\partial_{k+1}: C_{k+1}(X) \to C_k(X)$ and where only finitely many $a_i$ are non zero and $\sigma_i^k $ represent the $k$-cells of $X$, which freely generate by construction $C_k(X)$ as $\mathbb{Z}$-module.

Question: How to show that $X$ can be modified to another CW complex $X'$ obtained from $X$ by only glueing some additional $k+1$-cells to it in the way that $\beta$ lies in the image of the boundary map $\partial_{k+1}: C_{k+1}(X') \to C_k(X')= C_k(X)$? Is this procedure always possible? In other words to modify $X$ in the way such that

1. the homology class $[\beta] $ get killed

2. the homology below $k$ and $2$-dimensions above $k$ not changes, ie the canonical inclusion $\iota: X \to X'$ induces identity map $\iota_k= \text{id}: H_m(X) \to H_m(X')$ for $k > m $ or $m \ge k+2$

Note that by construction of the associated cell complex if $X$ is obtained from $X$ by gluing only some $k+1$-cells, the free groups $C_m(X')$ and differentials $\partial_m$ not change as long as $k > m $ or $m \ge k+2$. So the question reduces to the problem if it's always possible to attach $k+1$-cells $e_1^{k+1}, e_2^{k+1}, ..., e_d^{k+1}$ to $X$ in the way that $\beta$ lies in the differential

$$\partial'_{k+1}: C_{k+1}(X'):=C_{k+1}(X) \bigoplus \oplus_{i=1}^d \mathbb{Z} e_i^{k+1} \to C_k(X')=C_k(X) $$

induced by attaching maps. Note that not every map $\phi: C_{k+1}(X) \to C_k(X)$ of $\mathbb{Z}$-modules comes from geometry, i.e. is induced by attaching map of cells. That makes it (at least for me) hard, to decide if such pure geometric modification on $X$ is always realizable.

Answer 1

If $\beta$ is the fundamental class of a torus $X = S^1 \times S^1$ then any map $S^2 \to X$ is nullhomotopic so attaching $3$-cells is the same as taking the wedge sum with $3$-spheres, which does not affect $H_2$.

On the other hand it is possible to kill $\beta$ by attaching $k$-cells if $X$ is $(k-1)$-connected, since then the map $\pi_k(X) \to H_k(X)$ is surjective.