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In group theory, and specifically in group cohomology-theory, the notion of crossed homomorphisms (Def. below) is a generalization of that of group homomorphisms where a group action – of the domain-group $G$ by automorphisms on the codomain group $\Gamma$ – is incorporated.
Crossed homomorphisms are equivalently homomorphic sections of the projection to $G$ out of the corresponding semidirect product group $\Gamma \rtimes G$ (Prop. below).
This largely explains their relevance in all contexts of $G$-equivariant algebraic topology and equivariant differential topology, such as in equivariant principal bundle-theory (Prop. below) where semidirect product groups appear as $G$-equivariant groups.
In particular (Prop. below, which is not made explicit in traditional literature):
the local coefficient bundle for non-abelian group cohomology in degree-1 is the image of the projection out of the semidirect product group under forming delooping groupoids,
the strict (2,1)-category-theoretic sections of this bundle form precisely the groupoids of crossed homomorphisms with “crossed conjugations” (Def. below) between them;
and the connected components of these groupoids is the non-abelian group cohomology $H^1(G,\Gamma)$ (Rem. , ).
This last fact is widely appreciated (see Rem. below) in the special case that $\Gamma$ is an abelian group, where it serves to define ordinary group cohomology with coefficients in degree 1.
Beware that, while the notion of crossed homomorphisms makes sense for group objects internal to any ambient category $\mathcal{C}$, it may not need to strictly coincide with that of cocycles in first group cohomology beyond $\mathcal{C} =$ Set (for reasons discussed at Lie group cohomology, or, for the case of profinite groups, in NSW 2008, p. 24).
In all of the following, let $G \,\in\,$ Grp be a group (or group object internal to an ambient category with finite products) and let $\Gamma \,\in\, G Act$ be an $G$-equivariant group, hence a group equipped with a group action
of $G$ on $\Gamma$ by group automorphisms, hence equivalently a semidirect product group, to be denoted:
A crossed homomorphism from $G$ to $\Gamma$ is a function (morphism in the ambient category)
satisfying the following “$G$-crossed” homomorphism property:
Def. appears, already with the non-abelian generality in mind, back in Whitehead 1949, (3.1) (together with the notion of crossed modules, both as tools for analyzing homotopy types with non-trivial fundamental group). For abelian $\Gamma$ the definition is in Eilenberg & MacLane (1947) §3.1, MacLane (1975) IV (2.1) and many following references on group cohomology, e.g. Brown 1982, p. 45. For general non-abelian $\Gamma$ the notion is later used in Murayama & Shimakawa 1995 for discussion of equivariant principal bundles. In the context of nonabelian group cohomology it is reviewed in Gille & Szamuely, 2006, (1) on p. 25. Another textbook account is in Milne 2017, 15.a (16.a) in the pdf, aimed at application to algebraic groups.
For $\alpha$ the trivial action, the notion of a crossed homomorphism (Def. ) reduces to that of an ordinary group homomorphism.
(as 1-cocycles in group cohomology)
In the special case that $\Gamma$ in Def. is an abelian group, crossed homomorphisms are also known as cocycles in group cohomology in degree 1 (e.g. Brown 1982, p. 45). In general they may be understood as 1-cocyles in non-abelian group cohomology (e.g. NSW 2008, p. 16).
The constant function $const_{\mathrm{e}} \,\colon\, G \to \Gamma$ on the neutral element $\mathrm{e} \in \Gamma$ is always a crossed homomorphism (Def. ), being the trivial cocycle under the interpretation of Rem. .
(as splittings/homomorphic sections of the semidirect product group-projection)
Crossed homomorphisms $G \to \Gamma$ (Def. ) are equivalently homomorphic sections of the projection out of the semidirect product group $\Gamma \rtimes_\alpha G \xrightarrow{pr_2} G$ (see also at split group extensions):
By immediate unwinding of the definition of the semidirect product group, such a section is an assignment
such that
(graphs of crossed homomorphisms)
The graph of a function of a crossed homomorphism (Def. ) is a subgroup of the semidirect product group
and every subgroup of this form arises as the graph of a unique crossed homomorphism.
The first statement is immediate from the definition.
For the converse, consider a subgroup $\widehat{G}$ as in (1). Then the subgroup property implies that
From this the second condition in (1) implies that
Together with the first condition in (1) this implies that $\widehat{G}$ is the graph of a function $\phi \;\colon\; G \to \Gamma$. From this the claim follows by Prop. .
Subgroups of the form (1) are considered throughout articles by Peter May on equivariant principal bundles (e.g. in Lashof & May 1986, Thm. 10, May 1990, Thm. 7). That these are equivalently (graphs of) crossed homomorphisms may have been one the key observations that lead to Murayama & Shimakawa 1995, though the statement of Prop. was still not explicit there.
(crossed conjugation)
Two crossed homomorphisms $\phi_1, \phi_2$ (Def. ) are adjoint $\phi_1 \xrightarrow{\;\gamma\;} \phi_2$ if in their incarnation as homomorphisms to $\Gamma \rtimes G$ (Prop. ) they are related by conjugation with an element
$\gamma \in \Gamma \xhookrightarrow{\;} \Gamma \rtimes G$, in that:
(as 1-coboundaries in group cohomology)
When $\Gamma$ is an abelian group, the conjugates according to Def. of the constant function $const_{0}$ (Exp. ) are of the form
(writing now in additive notation as befits an abelian group).
These (3) are also known as “principal crossed homomorphisms”.
With crossed homomorphisms understood as 1-cocycles in group cohomology (Rem. ), these elements (3) are the 1-coboundaries.
The definitions here apply verbatim also for non-abelian $\Gamma$, where we get non-abelian group cohomology (e.g. NSW 2008, p. 16).
All the formulas above may conceptually be understood as follows:
(groupoid of crossed homomorphisms is sliced functor groupoid of delooping groupoids)
The groupoid
of crossed homomorphisms (Def. ) with conjugations between them (Def. ) is isomorphic to the sliced functor groupoid of sections of the delooping groupoid $\mathbf{B}(\Gamma \rtimes G)$ of the semidirect product group:
By definition, a morphism in the groupoid on the right is a commuting diagram of functors and natural transformations as shown on the left of the following:
Unwinding the definitions, it is clear that functors fitting into this diagram (indicated by the dashed single arrows on the left) are bijective (under passage to their component functions shown on the right) to group homomorphisms $G \to \Gamma \rtimes G$ whose composition with $\Gamma \rtimes G \xrightarrow{pr_2} G$ is the identity morphisms – hence are bijective to crossed homomorphisms, by Prop. .
Moreover, natural transformations $F \Rightarrow F'$ fitting into this diagram (indicated by the dashed double arrow on the left) must be such that their whiskering with $\mathbf{B}pr_2$ is the identity natural transformation, which means that their single component (as shown on the right) is of the form
where $\mathrm{e} \in G$ denotes the neutral element.
In conclusion, the naturality square in $\mathbf{B}(\Gamma \rtimes G)$ that all these components must make commute is of the form
Since composition in the delooping groupoid is given by the group product, this is equivalently the condition that
which is evidently equivalent to the defining relation from Def. .
(conjugacy classes of crossed homomorphisms are non-abelian first group cohomology)
In view of the identification of crossed homomorphisms with 1-cocycles in non-abelian group cohomology, we may identify their conjugacy classes (Def. ), hence, by Prop. , the connected components of the sliced functor groupoid (5) with the group cohomology sets (at least for discrete groups):
But we also get higher homotopy information, beyond the cohomology set:
(automorphism group of crossed homomorphisms)
For $\phi \colon G \xrightarrow{\;} \Gamma$ a crossed homomorphism (Def. ), its automorphism group in the conjugation groupoid (4), hence the subgroup
of crossed conugations (Def. ) that fix it, is, equivalently, the intersection of $i(\Gamma) = \{ (\gamma, \mathrm{e}) \,\vert\, \gamma \in \Gamma \}$ with
the centralizer $C_{{}_{\Gamma \rtimes G}}\big(Graph(\phi)\big)$
the normalizer $N_{{}_{\Gamma \rtimes G}}\big(Graph(\phi)\big)$
of the graph of $\phi$ inside the semidirect product group:
The first statement is immediate from the re-formulation (6) of crossed conjugation (2).
From this, the second statement follows by observing that the action in the form (6) manifestly fixes the $G$-component under projection along $pr_2$, while the graph of a function $G \to \Gamma$ has at most one element with a given $G$-component.
(Weyl group acts on non-abelian group 1-cohomology of subgroup)
For $H \subset G$ a subgroup, the set of crossed homomorphisms $H \to \Gamma$ (Def. ), with respect to the restricted action of $H$ on $\Gamma$, carries a group action of the normalizer subgroup $N_G(H) \,\subset\, G$, given by
Moreover, on crossed-conjugation classes (Def. ), hence on first non-abelian group cohomology, this action descends to an action of the Weyl group $W_G(H) \coloneqq N_G(H)/H$:
1. Action property. It is clear that (7) is a group action if only $\phi_n$ is indeed a crossed homomorphism. This follows by a direct computation:
2. Descent to crossed-conjugation classes. To see that the action (7) descends to group cohomology, we need to show for
a crossed conjugation between some $\phi'$ and $\phi$, that there also exists a crossed conjugation between $\phi'_n$ and $\phi_n$. The following direct computation shows that this is given by crossed conjugation with $\alpha(n)(\gamma)$:
Alternative argument for 1. & 2. Alternatively, the previous two statements also follow more immediately, by using the identification from Prop. of the conjugation groupoid of crossed homomorphisms $\phi \colon H \to \Gamma$ with that of homomorphic sections $\big( \phi(-),\, (-) \big) \,\colon\, H \to \Gamma \rtimes H$. In this latter incarnation, the action by $n$ is simply the “conjugation action by the adjoint action”, in that:
In this formulation it is manifest that homomorphisms and conjugation are preserved, and the only point to check is that the section-property is also respected, which is immediate.
3. Descent to action of Weyl group. To conclude, we need to show that the action of $H \subset B_G(H)$ is trivial on conjugacy classes, hence that for $n \in H \subset N(H)$ there is a crossed conjugation between $\phi_n$ and $\phi$. The following direct computation shows that this is given by crossed conjugation with $\phi(n)$ (which is well-defined, by the assumption that $n \in H$):
(a relation to crossed modules)
Let $(\Gamma \xrightarrow{\delta} G)$ be a crossed module, with the corresponding strict 2-group
Then in the strict (2,1)-category of strict 2-groups, the 2-morphisms out of the identity 1-morphism on $\mathcal{G}$ are in bijection to the crossed homomorphisms $G \to \Gamma$:
Namely, such a 2-morphism is a natural transformation $\eta \colon id_{\mathcal{G}} \Rightarrow F$ of endofunctors of the underlying action groupoid, whose component function
is a group homomorphism
such that
where
is the source map of the groupoid $\mathcal{G}$.
This means that the admissible $\eta_0$ are precisely the homomorphic sections of $\Gamma \rtimes G \xrightarrow{\;} G$. Conversely, by the invertiblity of all morphisms involved, every such $\eta_0$ is the component homomorphism of some 2-morphism $\eta$ out of $id_{\mathcal{G}}$.
Therefore the statement (8) follows by Prop. .
The analogous statement for general 2-morphisms is indicated in Noohi 07, p. 12.
Let $G \,\in\, Grp(Top)$ and $\Gamma \,\in\, Grp\big( G Act(Top) \big)$ be topological groups which are compact Lie groups (to be on the safe side). Then:
(equivariant connected components of equivariant classifying spaces)
A G-equivariant classifying space $B_G \Gamma$ for $G$-equivariant $\Gamma$-principal bundles exists, and the connected components of its $H$-fixed loci, for compact subgroups $H \subset G$, are in bijection to the conjugacy classes (Def. ) of crossed homomorphisms (Def. ) from $H$ to $\Gamma$ (with respect to the restricted action of $H$ on $\Gamma$):
After a little reformulation via Prop. and Prop. , this is the statement of Lashof & May 1986, Thm. 10, May 1990, Thm. 7.
In view of the explicit construction of universal equivariant principal bundles in Murayama & Shimakawa 1995, one may understand the statement of Prop. on elementary grounds, as follows:
Consider the following $G$-action objects internal to Groupoids:
via the $G$-action on $\Gamma$ by group automorphisms;
the pair groupoid
via the left multiplication action of $G$ on all three copies of $G$;
the functor groupoid
with the induced conjugation action on component functions of functors and natural transformations:
(crossed homomorphisms are fixed loci in functor groupoid from $\mathbf{E}G$ to $\mathbf{B}\Gamma$)
If $G$ a discrete group, then for each subgroup $H \subset G$, there is an equivalence of groupoids (in fact of topological groupoids)
between
By Prop. the statement is equivalently that
In the special case $H = G$ there is in fact an isomorphism, evidently exhibited by the following functor:
For general $H \subset G$, choose a section of the coset space-projection
which exists and is continuous by the assumption that $G$ is discrete.
Observe that then $\mathbf{E}G$ is generated, under
composition,
taking inverses,
acting with elements of $H$
by the following two classes of morphisms:
Using this, consider the following expression for a pair of comparison functors:
Here
$L$ is the restriction along $(H \times H \rightrightarrows H) \xhookrightarrow{\;} (G \times G \rightrightarrows G)$ of the previous isomorphism for $H =$G;
$R$ is given in terms of the above generating morphisms (12) as follows:
One readily sees that this is well-defined, and that $L \circ R \,=\, id$.
Therefore it is now sufficient to give a natural transformation $id \xRightarrow{\eta} R \circ L$, hence for each functor $F$ a natural transformation
This may be taken as follows, again stated in terms of the generating morphisms (12):
It just remains to check that this is indeed natural in the functors $F$, which amounts, for each $H$-equivariant natural transformation $G \xRightarrow{\beta} F'$, to the commutativity of the two types of squares shown on the right here:
Indeed, the top square commutes by the $H$-equivariance of $\beta$, while the bottom square commutes by the naturality of $\beta$.
(residual Weyl group-action on fixed locus of equivariant classifying space)
Transported through the equivalence of Prop. , the canonical group action (see this Prop.) of the Weyl group $W_G(H)$ on the $H$-fixed locus $Fnctr\big(\mathbf{E}G ,\, \mathbf{B}\Gamma\big)^H$ becomes, on connected components $\pi_0
\big(
CrsHom(H,\,\Gamma) \sslash_{\!\!ad} \Gamma
\big)
\;\;
\simeq
\;\;
H^1_{Grp}(H,\,\Gamma)$, the $W_G(H)$-action on the non-abelian group 1-cohomology of $H$ from Prop. .
We make explicit use of the functors $L, R$ constructed in the proof of Prop. . Noticing that $R$ is a section of $L$, we need to (1) send a crossed homomorphism up with $R$, (2) there act on it with $n$, (3) send the result back with $L$. The result is the desired induced action.
Explicitly, by the definition of $L$ in the proof of Prop. , this way a crossed homomorpism $\phi \,\colon\, H \to \Gamma$ is sent by $n \in N_G(H)$ to the assignment
It just remains to evaluate the right hand side.
Notice that the definition of $L$ is independent of the choice of $\sigma \,\colon\, G/H \xrightarrow{\;} G$ (11), and that $R$ (whose definition does depend on this choice ) is a section for each choice. Hence we may choose $\sigma$ in a way convenient way for any given $n$.
Now if $n \in H \subset N_G(H)$ then its canonical action on the $H$-fixed locus is trivial, and also the claimed induced action is trivial, so that in this case there is nothing further to be proven. Therefore we assume now that $n$ is not in $H$, and then we choose $\sigma$ such as to pick $n^{-1}$ as the representative in its $H$-coset:
Notice that with this choice, $R\phi$ (13) assigns the neutral element to the morphism between the neutral element and $n^{-1}$ in the pair groupoid:
This way, the right hand side of (14) is evaluated as follows:
This is indeed the claimed formula (7).
Discussion for the special case that $\Gamma$ is an abelian group:
in the context of homological algebra:
in the context of group cohomology:
Samuel Eilenberg, Saunders MacLane, §3.1 in: Cohomology Theory in Abstract Groups. I, Annals of Mathematics, Second Series 48 1 (1947) 51-78 [doi:10.2307/1969215]
Kenneth Brown, p. 45 of: Cohomology of Groups, Graduate Texts in Mathematics, 87, Springer 1982 (doi:10.1007/978-1-4684-9327-6)
The general non-abelian notion:
In discussion of homotopy theory (together with crossed modules):
A brief textbook account in this generality
As 1-cocycles in non-abelian group cohomology:
Philippe Gille, Tamás Szamuely, Def. 2.3.2 in: Central Simple Algebras and Galois Cohomology, Cambridge University Press 2006 (doi:10.1017/CBO9780511607219, pdf)
James Milne, Sec. 15.a-b & 3.k (16.a-b & 27.a in the pdf) of: Algebraic Groups, Cambridge University Press 2017 (doi:10.1017/9781316711736, webpage, pdf)
Groupprops, First cohomology set with coefficients in a non-abelian group
In view of homotopy fixed points:
In discussion of equivariant principal bundles:
Tammo tom Dieck, Sec. 2.1 in: Faserbündel mit Gruppenoperation, Arch. Math 20, 136–143 (1969) (doi:10.1007/BF01899003)
(not using the terminology “crossed homomorphism”, but already characterizing their graphs, as in Lem. )
Mitutaka Murayama, Kazuhisa Shimakawa, p. 2 of: Universal equivariant bundles, Proc. Amer. Math. Soc. 123 (1995), 1289-1295 (doi:10.1090/S0002-9939-1995-1231040-9)
Bertrand Guillou, Peter May, Mona Merling, around Def. 4.1 in: Categorical models for equivariant classifying spaces, Algebr. Geom. Topol. 17 (2017) 2565-2602 (arXiv:1201.5178, doi:10.2140/agt.2017.17.2565)
Discussion for finite groups:
Discussion for Lie groups:
Discussion in relation to crossed modules:
Last revised on June 22, 2023 at 16:06:19. See the history of this page for a list of all contributions to it.