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.



You must log in to answer this question.

Browse other questions tagged .