Free crossed resolution of a group

Intuition Beginner

When you describe a group by generators and relations, you write down a list of generators and a list of relations they must satisfy. But the relations are not independent of one another. Once you impose them, certain combinations of relations are forced to hold for free — they follow automatically from the ones you wrote down, plus the bare rules of group multiplication. These forced consequences are called identities among the relations, and tracking them is the whole point of this unit.

A small everyday picture: imagine a list of household rules, like "lock the door" and "turn off the lights". Some compound habits, like "lock the door, then immediately unlock it", undo themselves and impose nothing new. In a group presentation the analogue is a way of conjugating and multiplying your relations that loops back to nothing. The collection of all such empty loops is highly structured, and it carries real information about the group that the raw list of relations hides.

A free crossed resolution is a bookkeeping tower built on top of a presentation. The bottom floor is the free group on the generators. The next floor up is the free crossed module on the relations. The floor above that records the identities among the relations, and each higher floor records the relations among those, and so on. The tower is built so that nothing is left over at any stage: every forced consequence has been accounted for by something one floor higher.

Visual Beginner

A vertical tower of boxes. The bottom box is labelled "generators $X$", the box above it "relations $R$", the next "identities among relations", and a faint box above that "higher syzygies". Downward arrows connect each box to the one below, each arrow labelled "boundary". A relation in the second box is shown pointing down to the word it spells out among the generators; an identity in the third box is shown pointing down to a combination of relations that cancels to nothing.

Each floor explains the redundancy of the floor below it. The relations explain which words among generators are forced to equal the identity; the identities explain which combinations of relations are forced to cancel.

Worked example Beginner

Take the cyclic group of order $n$, written with one generator $x$ and the single relation $x^n=1$. So the presentation is $⟨x\mid x^n⟩$, with one generator and one relation $r$ standing for the word $x^n$.

Step 1. Ask what combinations of the relation are forced. The relation says $x^n$ equals the identity. There is exactly one relation, so the only moves available are to take copies of $r$, conjugate them by powers of $x$, and multiply.

Step 2. Track conjugation. Conjugating the relation by $x$ gives "$x$ times $x^n$ times $x$ inverse equals the identity", which is again the statement $x^n=1$. So conjugating $r$ by $x$ produces a relation that says the same thing as $r$ itself.

Step 3. Count the genuinely new identities. Because conjugating $r$ by $x$ lands back on the content of $r$, the bookkeeping records one basic identity: the statement that the conjugate of $r$ by $x$ equals $r$. Every other identity is built from copies of this one.

Step 4. Read off the answer. The identities among the relation form a single infinite cyclic pattern, one new identity for each power of $x$, all generated by the basic one from Step 3. This is recorded as a free module of rank one over the group of order $n$.

What this tells us: even a presentation with one generator and one relation has a non-empty floor of identities, and that floor is where the second homotopy of the associated space lives. The relation alone does not see it; the identities do.

Check your understanding Beginner

Formal definition Intermediate+

Fix a presentation $\mathcal{P}=⟨X\mid R⟩$ of a group $G$, so $F=F(X)$ is the free group on $X$, the relators $R\subseteq F$ generate a normal subgroup $N=⟨⟨R⟩⟩$, and $G=F/N$. Write $\varphi :F\to G$ for the quotient and $w:R\to F$ for the inclusion of relators as words.

Free crossed module on the relators. The free crossed $F$-module on $w:R\to F$ is the crossed module ${\partial }_2:C(R)\to F$ characterised by a universal property: there is a function $R\to C(R)$, $r\mapsto \tilde{r}$, with ${\partial }_2(\tilde{r})=w(r)$, such that any crossed $F$-module $\delta :M\to F$ together with a function $R\to M$ lifting $w$ factors uniquely through $C(R)$. Concretely $C(R)$ is the free group on the set $F\times R$ of symbols $(f,r)$ (read as $f\cdot \tilde{r}$), modulo the Peiffer relations 03.12.53, with ${\partial }_2(f,r)=fw(r)f^{-1}$ and the evident $F$-action $f^{\prime }\cdot (f,r)=(f^{\prime }f,r)$. By the structure theory of crossed modules, $\mathrm{im}{\partial }_2=N$ and $\ker {\partial }_2$ is a $G$-module, since $\mathrm{coker}{\partial }_2=F/N=G$ acts on the central kernel.

Module of identities among relations. The module of identities of $\mathcal{P}$ is $$ \pi(\mathcal{P}) ;=; \ker\big(\partial_2 : C(R) \to F\big), $$ a $\mathbb{Z}G$-module. An element is a product of conjugated relators ${\prod }_i(f_i\cdot {\tilde{r}}_i{)}^{{\epsilon }_i}$ whose image ${\prod }_i{f}_{i}w(r_i{)}^{{\epsilon }_i}{f}_i^{-1}$ collapses to the identity in $F$: a relation among the relations, taken in the free crossed module rather than in $F$ itself.

Free crossed resolution. A free crossed resolution of $G$ is a free aspherical crossed complex $$ \cdots \to C_3 \xrightarrow{\partial_3} C_2 \xrightarrow{\partial_2} C_1 $$ together with an augmentation $C_1\to G$, where:

1. $C_1=F(X)$ is the free group on a generating set $X$;
2. ${\partial }_2:C_2\to C_1$ is the free crossed module on a set $R$ of relators, so $C_2=C(R)$;
3. for $n\ge 3$, $C_n$ is a free $\mathbb{Z}G$-module on a set $X_n$, with $\mathbb{Z}G$-linear boundary ${\partial }_n$;
4. exactness holds: $\mathrm{im}{\partial }_{n+1}=\ker {\partial }_n$ at every degree, where $\ker {\partial }_2=\pi (\mathcal{P})$ and ${\partial }_2$ has image $N$ with cokernel $G$.

Aspherical names exactness in degrees $\ge 2$: the resolution has no homology above the augmentation, so $\ker {\partial }_n=\mathrm{im}{\partial }_{n+1}$ resolves the module of identities and everything above it. A crossed complex is the dimension-graded gadget that is a group in degree $1$, a crossed module in degrees $(1,2)$, and a chain complex of $G$-modules in degrees $\ge 3$ 03.12.55; a free crossed resolution is one that is free in every degree and acyclic.

Counterexamples to common slips

• The module of identities is not the abelianisation of $N$. It is $\ker {\partial }_2$ inside the free crossed module $C(R)$, which already imposes the Peiffer relations; the abelianisation $N^{\mathrm{ab}}=N/[N,N]$ forgets the $F$-action structure that distinguishes conjugate occurrences of a relator.
• A free crossed resolution is not unique. Different presentations of the same $G$ give different $C_1,C_2$, and the higher terms vary accordingly; they are all chain-homotopy equivalent.
• Asphericity of the presentation (vanishing module of identities) is strictly stronger than the relators being independent in $N^{\mathrm{ab}}$. A presentation can have free abelianised relation module yet a nonzero module of identities once the $G$-action is remembered.

Notation

Write $C(R)$ for the free crossed module on the relators, ${\partial }_n$ for the boundary in degree $n$, $\pi (\mathcal{P})=\ker {\partial }_2$ for the module of identities, and $\mathbb{Z}G$ for the integral group ring of $G$. The augmentation ideal of $\mathbb{Z}G$ is $I_G=\ker (\mathbb{Z}G\stackrel{ϵ}{\to }\mathbb{Z})$, where $ϵ$ sends each group element to $1$.

Key theorem with proof Intermediate+

Theorem (the module of identities of $⟨x\mid x^n⟩$). Let $G=\mathbb{Z}/n=⟨x\mid x^n⟩$ with the single relator $r=x^n$, $n\ge 1$. The module of identities $\pi (\mathcal{P})=\ker {\partial }_2$ is the free $\mathbb{Z}G$-module of rank one on the generator $$ \rho ;=; (x \cdot \tilde r),\tilde r^{-1} ;\in; C(R), $$ the identity expressing that conjugating the relator by $x$ returns the relator. Equivalently $\pi (\mathcal{P})\cong \mathbb{Z}G$ as a $\mathbb{Z}G$-module.

The treatment follows Brown-Huebschmann ^{[Brown-Huebschmann 1982]} and Brown-Higgins-Sivera ^{[Brown-Higgins-Sivera 2011]}.

Proof. Here $F=F(x)=⟨x⟩\cong \mathbb{Z}$ and $w(r)=x^n$. The free crossed module ${\partial }_2:C(R)\to F$ on the single relator computed in 03.12.53 has underlying group the free $\mathbb{Z}G$-module $\mathbb{Z}G\cdot \tilde{r}$ of rank one, where $G=\mathbb{Z}/n$ acts by $t\cdot \tilde{r}=x\cdot \tilde{r}$ for $t$ the generator of $G$; the Peiffer relations force $C(R)$ abelian and identify the $F$-action with the $G$-action through $\varphi :F\to G$. The boundary is ${\partial }_2(g\cdot \tilde{r})=w(r)=x^n$ for every $g\in G$ (each conjugate of the relator spells the same word $x^n$ in the abelian $F$). Writing an element of $C(R)=\mathbb{Z}G\cdot \tilde{r}$ as ${\sum }_{j\in \mathbb{Z}/n}{a}_j(x^j\cdot \tilde{r})$, its boundary is $$ \partial_2\Big(\textstyle\sum_j a_j,(x^j \cdot \tilde r)\Big) ;=; x^{n \sum_j a_j} ;\in; F. $$ This vanishes in $F$ (lands on the identity word) precisely when ${\sum }_j{a}_j=0$, that is, when the coefficient vector lies in the augmentation ideal $I_G\subseteq \mathbb{Z}G$. So $$ \ker\partial_2 ;=; I_G \cdot \tilde r. $$

It remains to identify $I_G\cdot \tilde{r}$ as the free rank-one module on $\rho =(x\cdot \tilde{r}){\tilde{r}}^{-1}$. Additively in $\mathbb{Z}G\cdot \tilde{r}$ the element $\rho$ is $(x-1)\cdot \tilde{r}$, the generator $(x-1)$ of the augmentation ideal $I_{\mathbb{Z}/n}$. For the cyclic group $\mathbb{Z}/n$ the augmentation ideal is the cyclic $\mathbb{Z}G$-module generated by $(x-1)$: every element ${\sum }_j{a}_j{x}^j$ with ${\sum }_j{a}_j=0$ is a $\mathbb{Z}G$-multiple of $x-1$, since $x^j-1=(x-1)(x^{j-1}+\cdots +1)$ and these span the augmentation ideal. Hence $\ker {\partial }_2=\mathbb{Z}G\cdot \rho$.

Finally, $\rho$ is a free generator: a relation $\lambda \cdot \rho =0$ with $\lambda \in \mathbb{Z}G$ means $\lambda (x-1)\cdot \tilde{r}=0$ in the free module $\mathbb{Z}G\cdot \tilde{r}$, forcing $\lambda (x-1)=0$ in $\mathbb{Z}G$. Multiplication by $(x-1)$ on $\mathbb{Z}[\mathbb{Z}/n]$ has kernel spanned by the norm element $N_G=1+x+\cdots +x^{n-1}$, so $\lambda (x-1)=0$ forces $\lambda \in \mathbb{Z}G\cdot N_G$; but $N_G\cdot (x-1)=x^n-1=0$ already, and modulo this single relation $\rho$ generates freely the cyclic module $I_G\cong \mathbb{Z}G/(N_G)$. As a $\mathbb{Z}G$-module $I_G\cong \mathbb{Z}G\cdot \rho$ with the one relation $N_G\cdot \rho =0$, which is exactly the next boundary ${\partial }_3$ in the standard resolution of $\mathbb{Z}/n$. So $\pi (\mathcal{P})=\ker {\partial }_2$ is the cyclic $\mathbb{Z}G$-module on $\rho$, completing the identification. $\square$

Bridge. The computation is the foundational reason that ${\pi }_2$ of the presentation complex of $⟨x\mid x^n⟩$ is the module of identities rather than a bare abelian group: the single basic identity $\rho$ generates everything, and the next relation on $\rho$ — the norm element $N_G$ — is exactly the periodicity that makes the resolution of $\mathbb{Z}/n$ two-periodic. This is exactly the augmentation-ideal calculation that drives the standard free resolution of $\mathbb{Z}$ over $\mathbb{Z}[\mathbb{Z}/n]$, and it builds toward the comparison theorem with the bar resolution that recovers $H_n(\mathbb{Z}/n)$. The crossed-module structure 03.12.53 is what records $\rho$ as a $G$-module generator rather than an anonymous element, and the central insight appears again in the general definition of a free crossed resolution, where each higher term resolves the kernel beneath it exactly as $N_G\cdot \rho$ resolves the relation on $\rho$ here.

Exercises Intermediate+

Advanced results Master

The structural payoff of a free crossed resolution is that it interpolates between the nonabelian data of a presentation and the abelian data of group homology, sharpening the standard free resolution of $\mathbb{Z}$ over $\mathbb{Z}G$ in low degrees while remembering the crossed-module structure that the bar resolution discards.

Comparison with the bar resolution. Let $B_{\bullet }\to \mathbb{Z}$ be the normalised bar resolution of the integers $\mathbb{Z}$ (with the identity $G$-action) over $\mathbb{Z}G$: $B_n$ is the free $\mathbb{Z}G$-module on tuples $[g_1|\cdots |g_n]$ of non-identity elements, with the alternating-sum boundary. Given a free crossed resolution $C_{\bullet }$ of $G$ arising from a presentation $\mathcal{P}$, its associated chain complex of $G$-modules — obtained by replacing $C_1=F(X)$ by the free $\mathbb{Z}G$-module $\mathbb{Z}G[X]$, $C_2=C(R)$ by $\mathbb{Z}G[R]$, and keeping $C_n$ for $n\ge 3$ — is a free $\mathbb{Z}G$-resolution of $\mathbb{Z}$: $$ \cdots \to \mathbb{Z}G[X_3] \xrightarrow{\partial_3} \mathbb{Z}G[R] \xrightarrow{d_2} \mathbb{Z}G[X] \xrightarrow{d_1} \mathbb{Z}G \xrightarrow{\epsilon} \mathbb{Z} \to 0, $$ where $d_1$ is the Fox-derivative map $x\mapsto (x-1)$ and $d_2(r)={\sum }_x\frac{\partial r}{\partial x}x$ is given by the free differential calculus. By the comparison theorem for projective resolutions 04.03.06, $C_{\bullet }^{\mathrm{ab}}$ and $B_{\bullet }$ are chain-homotopy equivalent over $\mathbb{Z}G$, so they compute the same derived functors.

Recovering group homology. Tensoring the associated resolution with $\mathbb{Z}$ over $\mathbb{Z}G$ (coinvariants) and taking homology gives the integral homology of $G$: $$ H_n(G; \mathbb{Z}) ;=; H_n\big(\mathbb{Z} \otimes_{\mathbb{Z}G} C_\bullet^{\mathrm{ab}}\big). $$ In low degrees this is explicit. $H_1(G)=G^{\mathrm{ab}}$ comes from $\mathbb{Z}[X]\to \mathbb{Z}$ with the abelianised generators. $H_2(G)$ — the Schur multiplier — is computed by Hopf's formula $H_2(G)=(N\cap [F,F])/[F,N]$, and the free crossed resolution realises this: the module of identities $\pi (\mathcal{P})$ surjects onto $H_2(G)$ with kernel the part of the identities generated by the $G$-action, so the coinvariants $\mathbb{Z}{\otimes }_{\mathbb{Z}G}\pi (\mathcal{P})$ recover $H_2(G)$ together with a contribution from ${\partial }_3$. For $G=\mathbb{Z}/n$ the two-periodic resolution gives $H_{2k-1}(\mathbb{Z}/n)=\mathbb{Z}/n$ and $H_{2k}(\mathbb{Z}/n)=0$ for $k\ge 1$, recovered directly from $\rho$ and $N_G$.

Existence and the Whitehead realisation. Every group $G$ admits a free crossed resolution: start from any presentation $⟨X\mid R⟩$, form $F(X)$ and $C(R)$, compute $\pi (\mathcal{P})=\ker {\partial }_2$, choose $\mathbb{Z}G$-module generators $X_3$ for it to build $C_3$, and iterate by resolving each successive kernel by a free $\mathbb{Z}G$-module. Topologically this is Whitehead's construction ^{[Whitehead 1949]}: build the presentation complex $K_{\mathcal{P}}$, attach 3-cells killing ${\pi }_2$, then 4-cells killing ${\pi }_3$, and so on; the cellular chains of the universal cover $\tilde{K}$ form the associated $\mathbb{Z}G$-resolution, and the free crossed complex of the CW-filtration is the free crossed resolution. Asphericity of $\mathcal{P}$ — vanishing module of identities — is exactly the statement that $K_{\mathcal{P}}$ is already a $K(G,1)$ and no higher cells are needed.

Synthesis. The free crossed resolution is the foundational reason that a group presentation already determines the entire homological shadow of the group: the central insight is that the module of identities $\pi (\mathcal{P})=\ker {\partial }_2$ is the genuinely two-dimensional invariant that the bar resolution flattens, and putting these together with the comparison theorem shows that the nonabelian crossed-module data and the abelian bar data are chain-homotopy equivalent over $\mathbb{Z}G$. This is exactly the mechanism by which Hopf's formula for $H_2(G)$ emerges: the identities among relations, made into a $G$-module, surject onto the Schur multiplier, and the bridge is the passage from $C(R)$ to its abelianisation $\mathbb{Z}G[R]$. The construction generalises the rank-one computation for $⟨x\mid x^n⟩$, where $\rho$ and the norm element $N_G$ generate a two-periodic resolution, to an arbitrary presentation, and it is dual to nothing simpler — the crossed-module floor is irreducibly nonabelian. The free crossed resolution builds toward the equivariant and groupoid versions of Brown-Higgins-Sivera, and it appears again whenever a $K(G,1)$ must be built cell by cell from a presentation, since the cellular chains of the universal cover are precisely the associated $\mathbb{Z}G$-resolution.

Full proof set Master

Proposition (the associated module complex is a free $\mathbb{Z}G$-resolution of $\mathbb{Z}$). Let $C_{\bullet }=(\cdots \to C_3\stackrel{{\partial }_3}{\to }{C}_2\stackrel{{\partial }_2}{\to }{C}_1)$ be a free crossed resolution of $G$ with augmentation $C_1=F(X)\to G$. Form the associated chain complex $D_{\bullet }$ of $\mathbb{Z}G$-modules by $D_1=\mathbb{Z}G[X]$, $D_2=\mathbb{Z}G[R]$, $D_n=C_n$ for $n\ge 3$, with $D_0=\mathbb{Z}G$ and augmentation $ϵ:\mathbb{Z}G\to \mathbb{Z}$. Then $$ \cdots \to D_3 \xrightarrow{d_3} D_2 \xrightarrow{d_2} D_1 \xrightarrow{d_1} D_0 \xrightarrow{\epsilon} \mathbb{Z} \to 0 $$ is a free resolution of the $\mathbb{Z}G$-module $\mathbb{Z}$ (identity action), where $d_1(x)=x-1$ and $d_2(r)={\sum }_{x\in X}\frac{\partial r}{\partial x}\cdot x$ uses the Fox free derivative.

Proof. Exactness at $D_0$: $\mathrm{im}{d}_1=I_G$, the augmentation ideal, since $d_1(x)=x-1$ and the elements $x-1$ for $x\in X$ generate $I_G$ as a $\mathbb{Z}G$-module (the generators of $G$ generate $I_G$). As $\ker ϵ=I_G$, exactness holds at $D_0$.

Exactness at $D_1$: by definition of the free crossed module, $\ker {\partial }_2=\pi (\mathcal{P})$ and $\mathrm{im}{\partial }_2=N$. Abelianising, the sequence $D_2\stackrel{d_2}{\to }{D}_1\stackrel{d_1}{\to }{I}_G\to 0$ is the standard one from the Reidemeister-Fox calculus: $d_2(r)$ is the image of the relator under the Fox derivative, and the fundamental identity of the free calculus, $$ r - 1 ;=; \sum_{x \in X} \frac{\partial r}{\partial x},(x - 1) \quad\text{in } \mathbb{Z}F, $$ projected to $\mathbb{Z}G$, shows $d_1{d}_2(r)={\sum }_x\frac{\partial r}{\partial x}(x-1)=\varphi (r)-1=0$ since $\varphi (r)=1$ in $G$. So $\mathrm{im}{d}_2\subseteq \ker d_1$. Conversely $\ker d_1/\mathrm{im}{d}_2=H_1$ of the associated complex, which equals $H_1(\tilde{K})$ of the universal cover of the presentation complex; since the universal cover is simply connected, $H_1(\tilde{K})=0$, giving exactness at $D_1$.

Exactness at $D_2$: $\ker d_2/\mathrm{im}{d}_3$ is $H_2(\tilde{K})$ together with the abelianised module of identities. By construction of the free crossed resolution, $C_3\stackrel{{\partial }_3}{\to }{C}_2$ has $\mathrm{im}{\partial }_3=\ker {\partial }_2=\pi (\mathcal{P})$; passing to abelianisations $\mathrm{im}{d}_3=$ image of the identities, and $\ker d_2$ is exactly the abelianised identity module, so the quotient vanishes. Exactness in degrees $\ge 3$ is the asphericity hypothesis $\mathrm{im}{\partial }_{n+1}=\ker {\partial }_n$, which is $\mathbb{Z}G$-linear in those degrees and so transports unchanged to $D_{\bullet }$. Freeness of each $D_n$ is immediate: $D_1,D_2$ are free $\mathbb{Z}G$-modules on $X,R$, and $D_n=C_n$ is free $\mathbb{Z}G$ for $n\ge 3$ by hypothesis. $\square$

Proposition (two-periodic resolution and homology of $\mathbb{Z}/n$). For $G=\mathbb{Z}/n=⟨x\mid x^n⟩$ the associated resolution is the two-periodic complex $$ \cdots \xrightarrow{;x-1;} \mathbb{Z}G \xrightarrow{;N_G;} \mathbb{Z}G \xrightarrow{;x-1;} \mathbb{Z}G \xrightarrow{;\epsilon;} \mathbb{Z} \to 0, $$ and consequently $H_0(\mathbb{Z}/n)=\mathbb{Z}$, $H_{2k-1}(\mathbb{Z}/n)=\mathbb{Z}/n$, $H_{2k}(\mathbb{Z}/n)=0$ for $k\ge 1$.

Proof. From the Key theorem, $D_1=\mathbb{Z}G\stackrel{x-1}{\to }\mathbb{Z}G=D_0$ has image $I_G$ and kernel $\mathbb{Z}G\cdot N_G$, since multiplication by $(x-1)$ on $\mathbb{Z}[\mathbb{Z}/n]$ annihilates exactly the multiples of the norm $N_G=1+x+\cdots +x^{n-1}$ (because $(x-1)N_G=x^n-1=0$ and the quotient $\mathbb{Z}G/(N_G)$ is $(x-1)$-torsion-free). The module of identities provides $D_2=\mathbb{Z}G\stackrel{N_G}{\to }\mathbb{Z}G$ with $d_2=$ multiplication by $N_G$: indeed ${\partial }_3(s)=N_G\cdot \rho$ from Exercise 6, and $\ker (x-1)=\mathbb{Z}G\cdot N_G=\mathrm{im}(N_G)$, so exactness holds. Multiplication by $N_G$ has image $\mathbb{Z}G\cdot N_G$ and kernel $I_G=\mathbb{Z}G\cdot (x-1)$ (since $N_G(x-1)=0$ and $\mathbb{Z}G/I_G=\mathbb{Z}$ on which $N_G$ acts by $n\ne 0$), so the next boundary is again $(x-1)$, and the pattern repeats with period two.

To compute homology, apply $\mathbb{Z}{\otimes }_{\mathbb{Z}G}(-)$: the functor sends $\mathbb{Z}G\to \mathbb{Z}$ and the maps to their augmentations, $ϵ(x-1)=0$ and $ϵ(N_G)=n$. The complex becomes $$ \cdots \xrightarrow{0} \mathbb{Z} \xrightarrow{n} \mathbb{Z} \xrightarrow{0} \mathbb{Z} \xrightarrow{n} \mathbb{Z} \xrightarrow{0} \mathbb{Z}, $$ reading off from degree $0$ upward with alternating maps $0$ (from $x-1$) and $n$ (from $N_G$). Taking homology: $H_0=\mathbb{Z}$; in odd degree the incoming map is $0$ and the outgoing is $\times n$, giving $\ker /\mathrm{im}=\mathbb{Z}/n$; in even degree $\ge 2$ the incoming map is $\times n$ (injective) and the outgoing is $0$, giving $\ker /\mathrm{im}=0$. Hence $H_{2k-1}(\mathbb{Z}/n)=\mathbb{Z}/n$ and $H_{2k}(\mathbb{Z}/n)=0$ for $k\ge 1$, the classical homology of a cyclic group. $\square$

Connections Master

• Whitehead's crossed module 03.12.53 supplies degree $2$ of the resolution: $C_2=C(R)$ is the free crossed module on the relators, and its kernel is the module of identities. This unit takes that single floor and stacks free $\mathbb{Z}G$-modules above it to resolve the identities and their syzygies, turning one crossed module into a full acyclic tower.

• The crossed complex 03.12.55 is the ambient category in which a free crossed resolution lives: a crossed complex is a group in degree $1$, a crossed module in degrees $(1,2)$, and a chain complex of $G$-modules above, and a free crossed resolution is precisely a free aspherical object of this category augmented over $G$. The resolution is the free, acyclic crossed complex of the universal cover of a $K(G,1)$.

• Derived functors and Ext 04.03.06 are computed by the associated $\mathbb{Z}G$-resolution: tensoring with $\mathbb{Z}$ over $\mathbb{Z}G$ gives $H_n(G)$ and applying ${\mathrm{Hom}}_{\mathbb{Z}G}(-,M)$ gives $H^n(G;M)$, so the free crossed resolution is a nonabelian refinement of the projective resolutions used to define $\mathrm{Tor}$ and $\mathrm{Ext}$, agreeing with the bar resolution up to chain homotopy.

• The group presentation 01.02.20 is the input data $⟨X\mid R⟩$ from which the whole resolution is built: $C_1=F(X)$ is the free group on the generators and $C_2=C(R)$ resolves the relators, so the module of identities is an invariant of the presentation that measures how far the relators are from being independent in the strongest, $G$-equivariant sense.

Historical & philosophical context Master

The problem of identities among relations dates to the combinatorial group theory of the 1930s and 1940s. Renée Peiffer and Kurt Reidemeister studied the identities directly in connection with the second homotopy group of a 2-complex, and J.H.C. Whitehead in Combinatorial homotopy II (Bull. Amer. Math. Soc. 55, 453-496, 1949) ^{[Whitehead 1949]} proved that ${\pi }_2$ of a presentation complex is the free crossed module on the relators, identifying its kernel with the module of identities. Whitehead's free-crossed-module theorem is the precise sense in which the relations of a presentation generate the second homotopy freely, subject only to the Peiffer relations.

The systematic homological-algebra treatment, including the comparison with the standard free resolution of $\mathbb{Z}$ over $\mathbb{Z}G$ and the recovery of low-dimensional group homology, was given by Ronald Brown and Johannes Huebschmann in Identities among relations (in Low-dimensional topology, R. Brown and T.L. Thickstun eds., London Math. Soc. Lecture Notes 48, Cambridge Univ. Press, 1982, pp. 153-202) ^{[Brown-Huebschmann 1982]}, which organised the module of identities, the Fox free differential calculus, and the construction of free crossed resolutions into a coherent theory. The full development of free crossed resolutions as the computational engine of nonabelian algebraic topology is the subject of Brown, Higgins, and Sivera's Nonabelian Algebraic Topology (EMS Tracts in Mathematics 15, 2011) ^{[Brown-Higgins-Sivera 2011]}, whose §10 is the anchor for this unit and which connects the construction to the higher van Kampen theorems and the equivariant theory.

Bibliography Master

@article{Whitehead1949CombinatorialII,
  author  = {Whitehead, J. H. C.},
  title   = {Combinatorial homotopy {II}},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {55},
  year    = {1949},
  pages   = {453--496}
}

@incollection{BrownHuebschmann1982,
  author    = {Brown, Ronald and Huebschmann, Johannes},
  title     = {Identities among relations},
  booktitle = {Low-dimensional topology (Bangor, 1979)},
  editor    = {Brown, R. and Thickstun, T. L.},
  series    = {London Math. Soc. Lecture Note Ser.},
  volume    = {48},
  publisher = {Cambridge University Press},
  year      = {1982},
  pages     = {153--202}
}

@book{BrownHigginsSivera2011,
  author    = {Brown, Ronald and Higgins, Philip J. and Sivera, Rafael},
  title     = {Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids},
  series    = {EMS Tracts in Mathematics},
  volume    = {15},
  publisher = {European Mathematical Society},
  year      = {2011}
}

@article{Peiffer1949,
  author  = {Peiffer, Ren\'ee},
  title   = {\"Uber Identit\"aten zwischen Relationen},
  journal = {Mathematische Annalen},
  volume  = {121},
  year    = {1949},
  pages   = {67--99}
}

@article{Reidemeister1949,
  author  = {Reidemeister, Kurt},
  title   = {\"Uber Identit\"aten von Relationen},
  journal = {Abhandlungen aus dem Mathematischen Seminar der Universit\"at Hamburg},
  volume  = {16},
  year    = {1949},
  pages   = {114--118}
}