---
tag: S31M1ZYJRK
kind: statement
role: theorem
name: Hopf's formula
msc: [20J06, "20F05"]
uses: [0KTW26C80J, 0CB0DX5BVV, 0KYX2N69J0, 0ASGSTXSBW, 0PXW0BN7DW, CFAG3VRXEA, 0JGEQXQ4AC]
about: [CFAG3VRXEA]
sources:
  - url: https://en.wikipedia.org/wiki/Schur_multiplier
---

## Statement

Let $G$ be a group, let $F$ be a [[free group|0KTW26C80J]], and let $R$ be a [[normal subgroup|0CB0DX5BVV]] of $F$ with $G \cong F/R$.
Write $[A, B]$ for the subgroup generated by the [[commutators|0PXW0BN7DW]] $aba^{-1}b^{-1}$ with $a \in A$ and $b \in B$.
Then the second integral [[homology group|CFAG3VRXEA]] of $G$ is
$$H_2(G, \Z) \cong \frac{R \cap [F, F]}{[F, R]}.$$

## Remarks

Idea of the proof: the short exact sequence $1 \to R \to F \to G \to 1$ gives the exact sequence of low-degree terms
$$H_2(F, \Z) \to H_2(G, \Z) \to R / [F, R] \to F / [F, F] \to G / [G, G] \to 0.$$
Since $F$ is free, $H_2(F, \Z) = 0$, so $H_2(G, \Z)$ is the kernel of $R / [F, R] \to F / [F, F]$, which is $(R \cap [F, F]) / [F, R]$.

In particular, the right side does not depend on the chosen [[presentation|0JGEQXQ4AC]] of $G$.

Example: $G = \Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle$, with $F$ free on $a, b$.
Here $R = [F, F]$, because the normal closure of $aba^{-1}b^{-1}$ in $F$ is the commutator subgroup $[F, F]$.
So $H_2(\Z^2, \Z) \cong [F, F] / [F, [F, F]]$, which is infinite cyclic, generated by the image of $aba^{-1}b^{-1}$.
This matches the homology of the torus.

Example: $G = \Z / m\Z$, with $F = \Z$ and $R = m\Z$.
Then $[F, F]$ is trivial, so $H_2(\Z/m\Z, \Z) = 0$.

## Explanations

### L1 human

Hopf's formula computes a subtle invariant of a group from any description of the group by generators and relations.
It looks at the relations that are products of commutators, and it divides out the ones that every presentation has automatically.
The answer is the same for every presentation, which is surprising at first sight.

### L2 human

For a finite group $G$, the group $H_2(G, \Z)$ is the Schur multiplier, which controls the central extensions of $G$ and its projective representations.
Hopf's formula gives a purely algebraic way to compute it.
The exact sequence in the proof is the five-term sequence of the [[Lyndon–Hochschild–Serre spectral sequence|VC3QNG6TKR]].