# Why are principal crossed homomorphisms coboundaries?

According to Wikipedia (and to many other sources):

The first cohomology group is the quotient of the so-called crossed homomorphisms, i.e. maps (of sets) $$f : G \to M$$ satisfying $$f(ab)=f(a)+af(b)$$ for all $$a$$, $$b$$ in G, modulo the so-called principal crossed homomorphisms, i.e. maps $$f : G \to M$$ given by $$f(a) = am−m$$ for some fixed $$m\in M$$.

Coboundaries are generally such cocycles which "doesn't matter", that is, adding them to a cocycle doesn't change some property of the cocycle. For example, in group cohomology, 2-coboundaries doesn't change the group extension defined by a 2-cocycle (see here).

My question: what is the property of a crossed homomorphism that remains unchanged by adding a principal crossed homomorphism to it?

Edit

The closest to my wish is May's Proposition 4.3:

Proposition 4.3. The groups $$Z^1(G;M)$$ and $$Aut(G;M)$$ are isomorphic, and the isomorphism restricts to an isomorphism $$B^1(G;M)\cong Inn(G;M)$$. Therefore $$H^1(G;M)\cong Out(G;M)$$.

Here $$Out(G;M):=Aut(G;M)/Inn(G;M)$$, and $$Aut(G;M)$$ is the automorphisms group of the canonical split extension of $$M$$ by $$G$$, and $$Inn(G;M)$$ is the inner automorphism group of it, that is the group of conjugations by the elements of $$G$$.

So, if we call two automorphisms essentially the same, if they differ by a conjugation, and we identify the crossed homomorphisms with $$Aut(G;M)$$ and the principal crossed homomorphisms to $$Inn(G;M)$$, then we can say that an a principal crossed homomorphism essentially doesn't change the crossed homomorphism. But this seems a bit far-fetched to me.

• You say it seems far-fetched, but just note that higher (co)homology groups are always harder to interpret than the lower dimensional ones – Max Feb 17 at 15:08