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3 edited body

Some preliminary definitions:

Let $\Pi = \langle \alpha | \alpha^2 = 1\rangle$ be the cyclic group of order $2$ and let $\mathbb{Z}\Pi$ denote the group ring of $\Pi$ over $\mathbb{Z}$. Embed $\Pi$ in $\mathbb{Z}\Pi$ by letting $1:\Pi \rightarrow \mathbb{Z}$ be the map defined by $1(1) = 1$ and $1(\alpha) = 0$ and letting $\alpha:\Pi \rightarrow \mathbb{Z}$ be the map defined by $\alpha(1) = 0$ and $\alpha(\alpha) = 1$.

Now let $(W,\partial)$ be the chain complex defined as follows. Set $W_n$ as the free $\mathbb{Z}\Pi$-module on the single generator $e_n$ if $n \geq 0$ and set $W_n = 0$ otherwise. Let $\partial_n$ be the zero homomorphism for all $n \leq 0$ and let $\partial_n$ be the homomorphism defined by $\partial_n(r) = (\alpha+(-1)^n)(r)$ for all $n > 0$. Lastly, let $\varepsilon:W_0 \rightarrow \mathbb{Z}$ be the augmentation map defined by setting $\varepsilon(1) = \varepsilon(\alpha) = 1$.

The problem:

I have managed to show that the augmented complex $(W,\partial,\varepsilon)$ is acyclic and am currently trying to find a chain map $W \nabla:W \rightarrow W \otimes_{\mathbb{Z}} W$ which lifts the identity $\nabla:\mathbb{Z} \mathbb{Z} \rightarrow \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}$. I have been told that it is possible to take $\nabla(e_0) = e_0 \otimes e_0$ and $\nabla(e_1) = e_1 \otimes \alpha e_0 + e_0 \otimes e_1$ and so on, but I can't seem to figure out how this works. Alexander-Whitney maps and Eilenberg-Zilber maps seem like they might be relevant, but I am uncertain how to use them here. Would someone help orient me in the right direction here?

2 added 26 characters in body; edited title

# Existence of a chain map lifting the identity; Alexander-WhitneyAlexander-Whitney/Eilenberg-Zilber maps

Some preliminary definitions:

Let $\Pi = \langle \alpha | \alpha^2 = 1\rangle$ be the cyclic group of order $2$ and let $\mathbb{Z}\Pi$ denote the group ring of $\Pi$ over $\mathbb{Z}$. Embed $\Pi$ in $\mathbb{Z}\Pi$ by letting $1:\Pi \rightarrow \mathbb{Z}$ be the map defined by $1(1) = 1$ and $1(\alpha) = 0$ and letting $\alpha:\Pi \rightarrow \mathbb{Z}$ be the map defined by $\alpha(1) = 0$ and $\alpha(\alpha) = 1$.

Now let $(W,\partial)$ be the chain complex defined as follows. Set $W_n$ as the free $\mathbb{Z}\Pi$-module on the single generator $e_n$ if $n \geq 0$ and set $W_n = 0$ otherwise. Let $\partial_n$ be the zero homomorphism for all $n \leq 0$ and let $\partial_n$ be the homomorphism defined by $\partial_n(r) = (\alpha+(-1)^n)(r)$ for all $n > 0$. Lastly, let $\varepsilon:W_0 \rightarrow \mathbb{Z}$ be the augmentation map defined by setting $\varepsilon(1) = \varepsilon(\alpha) = 1$.

The problem:

I have managed to show that the augmented complex $(W,\partial,\varepsilon)$ is acyclic and am currently trying to find a chain map $W \rightarrow W \otimes_{\mathbb{Z}} W$ which lifts the identity $\nabla:\mathbb{Z} \rightarrow \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}$. I have been told that it is possible to take $\nabla(e_0) = e_0 \otimes e_0$ and $\nabla(e_1) = e_1 \otimes \alpha e_0 + e_0 \otimes e_1$ and so on, but I can't seem to figure out how this works. Alexander-Whitney maps and Eilenberg-Zilber maps seem like they might be relevant, but I am uncertain how to use them here. Would someone help orient me in the right direction here?

1

# Existence of a chain map lifting the identity; Alexander-Whitney maps

Some preliminary definitions:

Let $\Pi = \langle \alpha | \alpha^2 = 1\rangle$ be the cyclic group of order $2$ and let $\mathbb{Z}\Pi$ denote the group ring of $\Pi$ over $\mathbb{Z}$. Embed $\Pi$ in $\mathbb{Z}\Pi$ by letting $1:\Pi \rightarrow \mathbb{Z}$ be the map defined by $1(1) = 1$ and $1(\alpha) = 0$ and letting $\alpha:\Pi \rightarrow \mathbb{Z}$ be the map defined by $\alpha(1) = 0$ and $\alpha(\alpha) = 1$.

Now let $(W,\partial)$ be the chain complex defined as follows. Set $W_n$ as the free $\mathbb{Z}\Pi$-module on the single generator $e_n$ if $n \geq 0$ and set $W_n = 0$ otherwise. Let $\partial_n$ be the zero homomorphism for all $n \leq 0$ and let $\partial_n$ be the homomorphism defined by $\partial_n(r) = (\alpha+(-1)^n)(r)$ for all $n > 0$. Lastly, let $\varepsilon:W_0 \rightarrow \mathbb{Z}$ be the augmentation map defined by setting $\varepsilon(1) = \varepsilon(\alpha) = 1$.

The problem:

I have managed to show that the augmented complex $(W,\partial,\varepsilon)$ is acyclic and am currently trying to find a chain map $W \rightarrow W \otimes_{\mathbb{Z}} W$ which lifts the identity $\nabla:\mathbb{Z} \rightarrow \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}$. I have been told that it is possible to take $\nabla(e_0) = e_0 \otimes e_0$ and $\nabla(e_1) = e_1 \otimes \alpha e_0 + e_0 \otimes e_1$ and so on, but I can't seem to figure out how this works. Alexander-Whitney maps seem like they might be relevant, but I am uncertain how to use them here. Would someone help orient me in the right direction here?