Let $G$ be a discrete group and let $(X,x\_0)$ be a based Eilenberg-MacLane space for $G$, so there is a fixed isomorphism $\pi\_1(X,x\_0) = G$ and the universal cover $\widetilde{X}$ is contractible. The homology of $X$ is thus the same as the homology of $G$.


Choose some $\gamma \in G$, and let $c\_{\gamma} \in \text{Aut}(G)$ be the inner automorphism that conjugates elements of $G$ by $\gamma$. It is standard that $c\_{\gamma}$ acts as the identity on $H\_k(G)$ for all $k$.


Since $(X,x\_0)$ is an Eilenberg-MacLane space for $G$, there is a based map $\psi\_{\gamma}\colon (X,x\_0) \rightarrow (X,x\_0)$ that induces $c\_{\gamma}$ on $\pi\_1(X,x\_0)$, and in fact $\psi\_{\gamma}$ is unique up to based homotopy.


**Question**: Is it possible to write down $\psi\_{\gamma}$ in some natural way where in particular it is obvious that it acts as the identity on $H\_k(X)$? All the proofs I know that inner automorphisms act trivially on homology are either very algebraic or are based on very specific models of Eilenberg--MacLane spaces that mimic the algebraic constructions, and I'd like to see this fact topologically/geometrically.



Realizing inner automorphisms on Eilenberg-MacLane spaces
Let's assume that the space $X$ is reasonable in the sense that the inclusion $x\_0 \hookrightarrow X$ is a cofibration, i.e. has the homotopy extension property. This will hold, for instance, if $X$ is a CW complex and $x\_0$ is a vertex.


Represent $\gamma$ by a path $\rho\colon [0,1] \rightarrow X$ with $\rho(0)=\rho(1)=x\_0$. We can then use the homotopy extension property to extend $\rho$ to a homotopy $\phi\_t\colon X \rightarrow X$ such that $\phi\_0 = \text{id}$ and $\phi\_t(x\_0) = \rho(t)$ for all $t \in [0,1]$. Since $\phi\_{1}$ is homotopic to the identity (but through a homotopy where the basepoint moves!), it clearly induces the identity on homology.


So it is enough to prove that $\phi\_1\colon (X,x\_0) \rightarrow (X,x\_0)$ induces $c\_\gamma$ on $\pi\_1$. Let $(\widetilde{X},\widetilde{x}\_0) \rightarrow (X,x\_0)$ be the based universal cover of $(X,x\_0)$. There is a unique lift $\Phi\colon (\widetilde{X},\widetilde{x}\_0) \rightarrow (\widetilde{X},\widetilde{x}\_0)$ of $\phi\_1$. Identify the orbit of $\widetilde{x}\_0$ under the deck group with $G$, so $1 = \widetilde{x}\_0$. It is enough to prove that $\Phi(g) = \gamma^{-1} g \gamma$ for all $g \in G$. 


The path $\rho$ lifts to a collection of paths that connect $g \in G$ to $g \gamma$ for all $g \in G$. Lifting the homotopy $\phi\_t$, we get a homotopy $\widetilde{\phi}\_t\colon \widetilde{X} \rightarrow \widetilde{X}$ starting at the identity that slides each $g \in G$ along these paths. You might think that $\widetilde{\phi}\_1$ is $\Phi$, but while it is a lift of $\phi\_1$, it is not $\Phi$ since it takes the basepoint $1 = \widetilde{x}\_0$ to $\gamma$. We have to correct this by multiplying $\widetilde{\phi}\_1$ by the element $\gamma^{-1}$ of the deck group, so $\Phi(x) = \gamma^{-1} \cdot \widetilde{\phi}\_1(x)$ for all $x \in \widetilde{X}$. In particular, for $g \in G$ we have $\Phi(g) = \gamma^{-1} g \gamma$ as desired.


