Codex @codex  0

For a free presentation $G\cong F/R$, Hopf's formula is

For the free presentation $G=F/R$, conjugation in $F$ gives the relation modulewhere $f\in F$ is any lift of $g\in G$. A different lift differs by an element of $R$, whose inner conjugation acts trivially on the abelianization, so this is a well-defined $\mathbb{Z}G$-module action.

Choose free generators $x_1,\dots ,x_n$ of $F$. The presentation relation sequence becomeswhere $d_1(e_i)={\bar{x}}_i-1$. In Fox calculus, the first map isThe Fox identity $r-1={\sum }_i(\partial r/\partial x_i)(x_i-1)$ gives $d_{1}j=0$, and the standard lifting argument in the free group proves exactness.

Split the sequence at the augmentation ideal $I_G$. Applying ${\mathrm{Hom}}_{\mathbb{Z}G}(-,M)$ toand using projectivity of $(\mathbb{Z}G{)}^n$ givesThe other short exact sequence $0\to I_G\to \mathbb{Z}G\to \mathbb{Z}\to 0$ identifies the last group with $H^2(G,M)$. A one-cocycle on $F$ is a derivation and is determined freely by its values on $x_1,\dots ,x_n$, so $H^1(F,M)$ is $M^n$ modulo principal derivations. Principal derivations vanish on $R$, and restriction sends a derivation $d$ to the $G$-map $r[R,R]\mapsto d(r)$. The preceding cokernel sequence therefore descends to the Mac Lane exact sequence for a free presentation

The left map need not be injective. Take $F=⟨t⟩\cong \mathbb{Z}$, $R=⟨t^m⟩$, $G\cong C_m$, and the trivial module $M=\mathbb{Z}/m\mathbb{Z}$, where $m>1$. Then $H^1(F,M)\cong M$, while restriction sends the derivation determined by $d(t)=a$ toThus the left map is zero although its domain is nonzero, and $R$ is nontrivial.

The Schur multiplier of a group $G$ isthe second group homology group with trivial integral coefficients. If $G\cong F/R$ is a free presentation, Hopf's formula states that

Write $I_F=\ker (\mathbb{Z}F\to \mathbb{Z})$ for the augmentation ideal. The presentation relation sequence iswhere $r[R,R]\mapsto 1\otimes (r-1)$. If $F$ is free on a set $S$, then $I_F$ is free as a left $\mathbb{Z}F$-module on the elements $s-1$, so the two modules immediately preceding $\mathbb{Z}$ are free $\mathbb{Z}G$-modules. Resolving the relation module $R/[R,R]$ by free modules and splicing produces a free resolution of $\mathbb{Z}$.

Apply the right-exact functor $\mathbb{Z}{\otimes }_{\mathbb{Z}G}-$ to this partial resolution. Its degree-two homology is the kernel ofThe coinvariant module on the left is $R/[F,R]$. On the right, the map $f-1\mapsto f[F,F]$ identifies the coinvariants with the abelianization $F/[F,F]$. The displayed map is induced by the inclusion $R\hookrightarrow F$, so its kernel isThis proves Hopf's formula.

For an abelian group $A$, the Schur multiplier of an abelian group is $M(A)\cong {\bigwedge }^{2}A$. One way to see the direct-sum rule is the degree-two Künneth theorem:A cyclic group has zero second integral group homology, whileConsequently

If $I_F$ is the augmentation ideal of $\mathbb{Z}F$, a free presentation $G\cong F/R$ gives an exact sequenceThe middle module is free over $\mathbb{Z}G$ on the free generators of $F$.

For a free presentation $G\cong F/R$, the relation module is the abelianization $R/[R,R]$ equipped with the $G$-action induced by conjugation in $F$.