# Homology of $X_{\infty}$ space for a given Seifert surface of an oriented link $L$.

I am reading the proof of Theorem 6.5 given in the chapter "The Alexander Polynomial" of the book "An introduction to knot theory" by "Lickorish". I have some doubts related to the homology of spaces used in the proof.

Suppose $$F$$ is a Seifert surface for an oriented link $$L$$ in $$\mathbb{S}^3$$. Let $$X$$ be the closure of $$\mathbb{S}^3 -N$$ where $$N$$ is a regular neighborhood of $$L$$. Since $$F \cap X$$ is homeomorphic to $$F$$, we will call it $$F$$. Consider space $$Y$$ to be complement of $$F \times (-1, 1)$$ in $$X$$. Let $$X_{\infty}$$ be the quotient space of $$Y \times \mathbb{Z}$$, identifying points of $$F \times \{-1\} \subset Y_i$$ with points of $$F \times \{+1\} \subset Y_{i+1}$$ under natural homeomorphism. Consider homeomorphism $$t: X_{\infty} \rightarrow X_{\infty}$$ defined by natural homeomorphism of $$Y_i$$ with $$Y_{i+1}$$, i.e, translating $$Y_i$$ to $$Y_{i+1}$$. Now space $$X_{\infty}= Y' \cup Y''$$, where $$Y'= \cup_{i} ~Y_{2i+1}$$ and $$Y'' = \cup_{i}~ Y_{2i}$$. Clearly $$Y' \cap Y''$$ is collection of countably many copies of $$F$$.

Now in the proof, it is written "$$H_0(Y' \cap Y'', \mathbb{Z})$$ consists of one copy of $$\mathbb{Z}$$ for every power of $$t$$ and so can be identified, as a module, with $$\mathbb{Z}[t^{-1}, t] \otimes_{\mathbb{Z}}H_0(F; \mathbb{Z})$$ $$\ldots$$ ."

I understand $$H_0(Y' \cap Y'';\mathbb{Z})= \oplus_{\mathbb{Z}} \mathbb{Z}$$, because $$Y' \cap Y''$$ is countably many copies of $$F$$ and $$F$$ is path connected. Moreover, it is also clear to me that $$\oplus_{\mathbb{Z}} \mathbb{Z}$$ is isomorphic (as $$\mathbb{Z}$$-module) to $$\mathbb{Z}[t^{-1}, t] \otimes_{\mathbb{Z}} \mathbb{Z}= \mathbb{Z}[t^{-1}, t]$$. Also, it is clear to me that $$t^n$$ maps $$F \times \{+1\} \subset Y_0$$ (which is same as $$F \times \{-1\} \subset Y_{-1}$$ in $$X$$) to $$F \times \{+1\} \subset Y_n$$. Thus, I can say that I understand the phrase "$$H_0(Y' \cap Y'', \mathbb{Z})$$ consists of one copy of $$\mathbb{Z}$$ for every power of $$t$$".

But I don't understand how author come up with the conclusion $$H_0(Y' \cap Y''; \mathbb{Z})=\mathbb{Z}[t^{-1}, t] \otimes_{\mathbb{Z}}H_0(F; \mathbb{Z})$$. Even homologies (zero and first homology) of other spaces in the proof are expressed as the tensor product of "some" modules with $$\mathbb{Z}[t^{-1}, t]$$. So, I think I can say that I want to understand how the geometrical interpretation of the action of group $$\langle t \rangle$$ on $$Y' \cap Y''$$ is written in the tensor product form?

Can someone explain it to me, please? Any help is appreciated.