Attaching 1-cells to CW-complex affects homotopy groups?

Question

Is it true that attaching 1-cells to a CW-complex doesn‘t change it’s higher homotopy groups $\pi_n$ for $n\ge 2$?

(I am aware that a corresponding result for cells of higher dimension is far from being true.)

Answer 1

This is not true. Consider attaching a one-cell to $S^2$ via a constant attaching map. The resulting space is $S^2\vee S^1$. The universal cover of $S^2\vee S^1$ is $\mathbb{R}$ with an $S^2$ wedged at every integer, which is homotopy equivalent to $\bigvee_{\mathbb{Z}}S^2$. Now

$$\pi_2(S^2\vee S^1) \cong \pi_2(\bigvee_{\mathbb{Z}}S^2) \cong H_2(\bigvee_{\mathbb{Z}}S^2; \mathbb{Z}) \cong \bigoplus_{\mathbb{Z}}H_2(S^2; \mathbb{Z}) \cong \bigoplus_{\mathbb{Z}}\mathbb{Z}$$

while $\pi_2(S^2) \cong \mathbb{Z}$.