I'm currently reading the book "Algebraic Topology" by Tammo Tom Dieck: https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf
I don't understand the proof he gives of the theorem of Eilenberg-Zilber. He wants to use the acyclic model theorem:
and he proves that the functors $S_{\bullet}(-) \otimes S_{\bullet}(-) $ and $S_{\bullet}(-\times-) $ are both free and acyclic.
He then defines the Eilenberg-Zilber morphisms as $P:S_0(X)\otimes S_0(Y) \to S_0(X\times Y)$ such that $x \otimes y \mapsto (x, y)$ and $Q:S_0(X\times Y) \to S_0(X)\otimes S_0(Y)$ such that $(x,y) \mapsto x \otimes y$.
In order to conclude we should prove that $P$ and $Q$ induce natural transformation on the zeroth homology group so that we can apply the acyclic model theorem and extend $P$ and $Q$ to their chain complexes. Their composition is homotopic to the identity using again the acyclic model theorem.
The problem is that the map $P:S_0(X)\otimes S_0(Y) \to S_0(X\times Y)$ doesn't seem to induce a well defined map $\bar{P}:H_0(X\otimes Y) \to H_0(X\times Y)$. In fact let $a \in S_1(X)$ and $b \in S_0(Y)\setminus Im(\partial_1^Y)$, then $\partial_1^X(a)\otimes b \in Im(\partial_1^{X\otimes Y})$ because $\partial_1^X(a)\otimes b=\partial_1^{X\otimes Y}(a\otimes b)$, on the other hand $(\partial_1^X(a),b) \notin Im(\partial_1^{X\times Y}) $ because i chose $b \notin Im(\partial_1^Y)$.
What is the problem? What am I missing?
Best Answer
For every continuous map $a:\Delta_1\to X$ and every $b\in Y,$ $$(\partial_1^X(a),b)=(a(1),b)-(a(0),b)=(a,b)(1)-(a,b)(0)=\partial_1^{X\times Y}(a,b).$$