This was an exercise from last week in our topology class. Since there are no solutions I wanted to ask here.
Let $h:X \times [0,1] \to S^n$ be an embedding. Where $X$ is connected. Let $\gamma \in H_n(S^n \setminus h(X \times [0,1]))$ be nonzero.
Show that then if we understand $\gamma_1 \in H_n(S^n \setminus h(X \times [0,\frac{1}{2}]))$ as the image under the map $H_n(S^n \setminus h(X \times [0,1])) \to H_n(S^n \setminus h(X \times [0,\frac{1}{2}]))$ induced by inclusion, and the same for $\gamma_2 \in H_n(S^n \setminus h(X \times [\frac{1}{2},1])$. Then either $\gamma_1 \neq 0$ or $\gamma_2 \neq 0$.
My ideas: I managed to show the statement in the case where $g$ is a closed embedding. Using Mayer-Vietoris I got the L.E.S:
$0=H_{n+1}(S^n) \to H_n(S^n \setminus h(X \times [0,1]) \to H_n(S^n \setminus h(X \times [0,\frac{1}{2}]) \oplus H_n(S^n \setminus h(X \times [\frac{1}{2},1]))$
which is exactly the statement. Can this proof somehow be extended to all embeddings ?