Let $G$ be an abelian group.

It seems to be a well-known fact (for example here) that $B\text{Aut}(K(G,1))$, the classifying space of the topological monoid of (unbased) self homotopy equivalences of the Eilenberg-Maclane space $K(G,1)$, has nontrivial homotopy groups $\pi_1 = \text{Aut}(G),\pi_2 = G$.

There is a fibration of topological monoids $$\text{Aut}_*(K(G,1)) \to \text{Aut}(K(G,1)) \to K(G,1)$$

where the first map is the inclusion of the fiber, and the second map is evaluating at the basepoint $e$ ($\text{Aut}_*$ is the topological monoid of based homotopy equivalences).

One can readily show that $\text{Aut}_*(K(G,1))$ is homotopy equivalent to $\text{Aut}(G)$ endowed with the discrete topology. Taking classifying spaces we get a fibration sequence

$$K(\text{Aut}(G),1) \to B\text{Aut}(K(G,1))\to K(G,2)$$

which induces the LES in homotopy

$$\cdots \to 0 \to \pi_2(X) \to G \overset{\phi}{\to}\text{Aut}(G) \to \pi_1(X) \to 0 \to \cdots$$

where $X$ is $B\text{Aut}(K(G,1))$.

To get the stated result, it suffices to show that the map $\phi: G \to \text{Aut}(G)$ sends $g$ to conjugation by $g$. Why is this true? I appreciate any references.

$\newcommand{\Aut}{\operatorname{Aut}}$Let $G$ be an abelian group.

It seems to be a well-known fact (for example here) that $B\Aut(K(G,1))$, the classifying space of the topological monoid of (unbased) self homotopy equivalences of the Eilenberg-Maclane space $K(G,1)$, has nontrivial homotopy groups $\pi_1 = \Aut(G),\pi_2 = G$.

There is a fibration of topological monoids $$\Aut_*(K(G,1)) \to \Aut(K(G,1)) \to K(G,1)$$

where the first map is the inclusion of the fiber, and the second map is evaluating at the basepoint $e$ ($\Aut_*$ is the topological monoid of based homotopy equivalences).

One can readily show that $\Aut_*(K(G,1))$ is homotopy equivalent to $\Aut(G)$ endowed with the discrete topology. Taking classifying spaces we get a fibration sequence

$$K(\Aut(G),1) \to B\Aut(K(G,1))\to K(G,2)$$

which induces the LES in homotopy

$$\cdots \to 0 \to \pi_2(X) \to G \overset{\phi}{\to} \Aut(G) \to \pi_1(X) \to 0 \to \cdots$$

where $X$ is $B\Aut(K(G,1))$.

To get the stated result, it suffices to show that the map $\phi: G \to \Aut(G)$ sends $g$ to conjugation by $g$. Why is this true? I appreciate any references.

It seems to be a well-known fact (for example here) that $B\text{Aut}(K(G,1))$, the classifying space of the topological monoid of (unbased) self homotopy equivalences of the Eilenberg-Maclane space $K(G,1)$, has nontrivial homotopy groups $\pi_1 = \text{Out}(G),\pi_2 = Z(G)$.

There is a fibration of topological monoids $$\text{Aut}_*(K(G,1)) \to \text{Aut}(K(G,1)) \to K(G,1)$$

where the first map is the inclusion of the fiber, and the second map is evaluating at the basepoint $e$ ($\text{Aut}_*$ is the topological monoid of based homotopy equivalences).

One can readily show that $\text{Aut}_*(K(G,1))$ is homotopy equivalent to $\text{Aut}(G)$ endowed with the discrete topology. Taking classifying spaces we get a fibration sequence

$$K(\text{Aut}(G),1) \to B\text{Aut}(K(G,1))\to K(G,2)$$

which induces the LES in homotopy

$$\cdots \to 0 \to \pi_2(X) \to G \overset{\phi}{\to}\text{Aut}(G) \to \pi_1(X) \to 0 \to \cdots$$

where $X$ is $B\text{Aut}(K(G,1))$.

To get the stated result, it suffices to show that the map $\phi: G \to \text{Aut}(G)$ sends $g$ to conjugation by $g$. Why is this true? I appreciate any references.

Let $G$ be an abelian group.

It seems to be a well-known fact (for example here) that $B\text{Aut}(K(G,1))$, the classifying space of the topological monoid of (unbased) self homotopy equivalences of the Eilenberg-Maclane space $K(G,1)$, has nontrivial homotopy groups $\pi_1 = \text{Aut}(G),\pi_2 = G$.

There is a fibration of topological monoids $$\text{Aut}_*(K(G,1)) \to \text{Aut}(K(G,1)) \to K(G,1)$$

where the first map is the inclusion of the fiber, and the second map is evaluating at the basepoint $e$ ($\text{Aut}_*$ is the topological monoid of based homotopy equivalences).

One can readily show that $\text{Aut}_*(K(G,1))$ is homotopy equivalent to $\text{Aut}(G)$ endowed with the discrete topology. Taking classifying spaces we get a fibration sequence

$$K(\text{Aut}(G),1) \to B\text{Aut}(K(G,1))\to K(G,2)$$

which induces the LES in homotopy

$$\cdots \to 0 \to \pi_2(X) \to G \overset{\phi}{\to}\text{Aut}(G) \to \pi_1(X) \to 0 \to \cdots$$

where $X$ is $B\text{Aut}(K(G,1))$.

To get the stated result, it suffices to show that the map $\phi: G \to \text{Aut}(G)$ sends $g$ to conjugation by $g$. Why is this true? I appreciate any references.