In my last post I mentioned that first Čech cohomology classes of a sheaf \(\mathcal{G}\) of (maybe non-abelian) groups admit a geometric interpretation in terms of \(\mathcal{G}\)-torsors. In this post I am going to introduce the notion of a \(\mathcal{G}\)-torsor over a topological space \(X\), and show how the set of equivalence classes of \(\mathcal{G}\)-torsors on \(X\) can be identified with \(H^1(X,\mathcal{G})\).

The action groupoid

Before I talk about torsors, let me introduce a very simple concept, but that will be very useful in this and other posts. Take any left action \(G\times S\rightarrow S\) of a group \(G\) on a set \(S\). To this action we can associate its action groupoid, which is the category \([S,G]\) whose objects are precisely the elements of the set \(S\) and, for every \(x,y \in S\), the set of morphisms from \(x\) to \(y\) is

\[ \mathrm{Mor}_{[S,G]} (x,y)= \{g \in G: g\cdot x=y\}. \]

This is clearly a groupoid since every morphism \(g\) is an isomorphism, with inverse given by \(g^{-1}\). The moduli set of this category (that is, its set of isomorphism classes) is simply the set of orbits \(S/G\).

In the last post we saw an example of a group action. Recall that for any open covering \(\mathfrak{U}\) of a topological space \(X\) and for any sheaf of groups \(\mathcal{G}\) over \(X\), we had that an action of \(0\)-cochains on \(1\)-cochains by “conjugation”:

\begin{align*} C^0(\mathfrak{U},\mathcal{G}) \times C^1(\mathfrak{U},\mathcal{G})&\longrightarrow C^1(\mathfrak{U},\mathcal{G})\\ ((f_U)_{U\in \mathfrak{U}}, (g_{UV})_{U<V \in \mathfrak{U}}) &\longmapsto (f_U g_{UV} f_V^{-1})_{U<V \in \mathfrak{U}}. \end{align*}

Moreover, we saw that this action respects cocycles. Now, as in the last post, we can use refinement maps to define the sets \(C^0(X,\mathcal{G})\), \(C^1(X,\mathcal{G})\) and \(Z^1(X,\mathcal{G})\) as limits by refinement. Suppose that we have some element of \(Z^1(X,\mathcal{G})\) represented by a pair \((\mathfrak{U},g)\), with \(g\in Z^1(\mathfrak{U},\mathcal{G})\) and some element of \(C^0(X,\mathcal{G})\) represented by another pair \((\mathfrak{V},f)\). We can take a common refinement of both open covers by defining

\[ \mathfrak{W}=\{ U\cap V: U \in \mathfrak{U}, V \in \mathfrak{V} \} \]

(which is of course an open cover, since every point is in some \(U\) and in some \(V\)) with refinement maps given by

\begin{align*} \mathfrak{W}&\longrightarrow \mathfrak{U}\\ U\cap V &\longmapsto U \end{align*}

(analogously for \(\mathfrak{W} \rightarrow \mathfrak{V}\)). Thus, for the previously chosen elements we can take representatives \(g \in Z^1(\mathfrak{W},\mathcal{G})\) and \(f \in C^0(\mathfrak{W},\mathcal{G})\) and define

\[ f \cdot g = (f_U g_{UV} f_V^{-1})_{U < V \in \mathfrak{W}}. \]

In conclusion, we have just defined an action of the group \(C^0(X,\mathcal{G})\) in the set \(Z^1(X,\mathcal{G})\) (of course, in the same way we can define an action on \(C^1\), but we are particularly interested in this one). Moreover, the good properties of the direct limit guarantee that the set of orbits is precisely \(H^1(X,\mathcal{G})\).

We can now consider the action groupoid \([Z^1(X,\mathcal{G}),C^0(X,\mathcal{G})]\) associated to this action, whose moduli set is the Čech cohomology set \(H^1(X,\mathcal{G})\). What we are going to do now is to give an interpretation of this action groupoid in terms of \(\mathcal{G}\)-torsors.

Torsors

As above, let \(X\) be a topological space and \(\mathcal{G}\) a sheaf of groups over \(X\).

Definition 1. A \(\mathcal{G}\)-torsor is a sheaf of sets \(\mathcal{F}\) on \(X\) endowed with an action \(\mathcal{G} \times \mathcal{F} \rightarrow \mathcal{F}\) such that

  1. whenever \(\mathcal{F}(U)\) is nonempty, the action \(\mathcal{G}(U) \times \mathcal{F}(U) \rightarrow \mathcal{F}(U)\) is free and transitive, and

  2. for every \(x\in X\), the stalk \(\mathcal{F}_x\) is nonempty.

A morphism of \(\mathcal{G}\)-torsors \(\mathcal{F} \rightarrow \mathcal{F}'\) is simply a morphism of sheaves compatible with the \(\mathcal{G}\)-actions (we say that it is \(\mathcal{G}\)-equivariant).

More precisely, given a morphism of \(\mathcal{G}\)-torsors \(\varphi:\mathcal{F} \rightarrow \mathcal{F}'\), being \(\mathcal{G}\)-equivariant means that, if \(\mathcal{F}(U)\) is nonempty, for every \(p\in \mathcal{F}(U)\) we have

\[ \varphi_U(g\cdot p)=g\cdot \varphi_U(p). \]

The best way to unravel this definition is by looking at examples. The simplest example of a \(\mathcal{G}\)-torsor is the trivial \(\mathcal{G}\)-torsor, which is \(\mathcal{F}=\mathcal{G}\) with the natural action given by the group operation. A key fact is now the following:

Proposition 1. Let \(\mathcal{F}\) be a \(\mathcal{G}\)-torsor. If \(\mathcal{F}\) admits a global section, that is, if \(\mathcal{F}(X)\) is nonempty, then it is isomorphic to the trivial \(\mathcal{G}\)-torsor.

Proof. Choose \(f\in \mathcal{F}(X)\). Since \(f|_U \in \mathcal{F}(U)\) for every open subset \(U\subset X\), the action \(\mathcal{G}(U)\times \mathcal{F}(U) \rightarrow \mathcal{F}(U)\) is free and transitive. Therefore, every \(h_U \in \mathcal{F}(U)\) can be written in a unique way as \(h_U=g_U \cdot f|_U\), for \(g_U \in \mathcal{G}(U)\). Thus, the map

\begin{align*} \mathcal{F}(U)&\longrightarrow \mathcal{G}(U)\\ h_U &\longmapsto g_U, \end{align*}

which is clearly equivariant, defines a sheaf isomorphism. \(\blacksquare\)

Note now that since for every \(x\in X\), we have that \(\mathcal{F}_x \neq \varnothing\), there is an open cover \(\mathfrak{U}\) of \(X\) such that, for every \(U\in \mathfrak{U}\), the set \(\mathcal{F}(U)\) is nonempty. Therefore, on every \(U \in \mathfrak{U}\), the sheaf \(\mathcal{F}|_U\) is isomorphic to the trivial \(\mathcal{G}|_U\)-torsor. In conclusion, what property 2 in the definition of \(\mathcal{G}\)-torsor actually means is that every \(\mathcal{G}\)-torsor is, in some way, “locally trivial”. The open cover \(\mathfrak{U}\) is called a trivializing cover.

There are other examples of of \(\mathcal{G}\)-torsors that the reader could be familiar with. To introduce these examples, first consider a topological group \(G\). Associated to this group we can define two different sheaves. One is the sheaf of \(G\)-valued functions, which we denote simply by \(G\), and is defined by

\[ G(U)= \{ \text{Continuous maps } U\rightarrow G \}. \]

The other one is the sheaf of locally constant \(G\)-valued functions, denoted by \(\underline{G}\), and defined by

\[ \underline{G}(U)= \{ \text{Continuous maps } U\rightarrow G \text{ that are locally constant} \}. \] Note that these two sheaves are essentially different, although they coincide if the group \(G\) is endowed with the discrete topology. Now, \(G\)-torsors are best known as principal \(G\)-bundles (or rather, as their sheaves of sections). On the other hand \(\underline{G}\)-torsors can be identified with \(G\)-covering spaces. I will say a lot about these two examples in future posts. If the reader is familiar with principal bundles, maybe it is useful for them to think of a \(\mathcal{G}\)-torsor as a generalization of a principal bundle in the sense that the structure group depends continuously on the base point.

Now, let us see how there is a groupoid naturally associated with \(\mathcal{G}\)-torsors:

Proposition 2. Every morphism of \(\mathcal{G}\)-torsors is an isomorphism.

Proof. Consider \(\varphi:\mathcal{F} \rightarrow \mathcal{F}'\) a morhpism of \(\mathcal{G}\)-torsors. First, we will see that the map is injective. Suppose that there are \(p_1,p_2 \in \mathcal{F}(U)\) such that \(\varphi_U(p_1)=\varphi_U(p_2)\). Since the action on \(\mathcal{F}(U)\) is transitive and free, there exists a unique \(g \in \mathcal{G}(U)\) such that \(p_1=g\cdot p_2\) and, since \(\varphi\) is equivariant, \(\varphi_U(p_1)=g\cdot \varphi_U(p_2)\). But the group \(\mathcal{G}(U)\) also acts freely and transitively on \(\mathcal{F}'(U)\), so \(g=1\) and \(p_1=p_2\). On the other hand, to see that it is surjective take any element \(p\in \mathcal{F}(U)\). Since the action is transitive we can write any other element \(p'\in \mathcal{F}(U)\) as \(p'=g\cdot \varphi_U(p)\), for some \(g\in \mathcal{G}(U)\). Therefore, since \(\varphi\) is equivariant, \(p'=\varphi_U(g\cdot p)\). \(\blacksquare\)

What we have just shown is that if we consider the category whose objects are \(\mathcal{G}\)-torsors and whose morphisms are morphisms of \(\mathcal{G}\)-torsors, this category is in fact a groupoid. The main purpose of this post is to show that this groupoid is equivalent to the action groupoid \([Z^1(X,\mathcal{G}),C^0(X,\mathcal{G})]\). In particular, this equivalence will yield a bijective correspondence between isomorphism classes of \(\mathcal{G}\)-torsors and cohomology classes in \(H^1(X,\mathcal{G})\).

Transition functions

The way to obtain a Čech cocycle from a \(\mathcal{G}\)-torsor is by considering transition functions. Consider a \(\mathcal{G}\)-torsor \(\mathcal{F}\) and a trivializing cover \(\mathfrak{U}\) of \(\mathcal{F}\). Now, pick a section \(s_U \in \mathcal{F}(U)\) on each \(U\in \mathfrak{U}\) (I guess you need to use the Axiom of Choice here, but who cares –besides, we already used it to define cochains–). Now, for every two open sets \(U,V \in \mathfrak{U}\), since the action of \(\mathcal{G}(U\cap V)\) on \(\mathcal{F}(U\cap V)\) is transitive, there must exist some cochain \(g=(g_{UV})_{U<V \in \mathfrak{U}} \in C^1(\mathfrak{U},\mathcal{G})\) such that

\[ s_U|_{U\cap V} = g_{UV} s_V|_{U \cap V}. \]

Moreover, this cochain is a cocycle since

\[ g_{UV}g_{VW} s_W = g_{UV} s_V = s_U = g_{UW} s_W. \]

Thus, to any \(\mathcal{G}\) torsor \(\mathcal{F}\) we can associate a cocycle \(g\in Z^1(\mathfrak{U},\mathcal{G})\) for some open cover \(\mathfrak{U}\) of \(X\). This cocycle is called a set of transition functions of \(\mathcal{F}\).

The choice of transition functions is not canonical, since it depends on the choice of the sections \(s_U\). However, if we pick other sections \(s'_U \in \mathcal{F}(U)\) on each \(U\in \mathfrak{U}\), since the action is transitive, we can write each \(s'_U\) as \(s'_U=f_U s_U\), for some \(f_U \in \mathcal{G}(U)\). Therefore, if we consider the cocycle \(g'\) defined by \(s'_U=g'_{UV} s'_V\), we have

\[ f_U s_U = s'_U = g'_{UV} s'_V = g'_{UV} f_V s_V, \]

so \(g'_{UV} = f_U g_{UV} f_V^{-1}\). The same argument shows that if \(\varphi:\mathcal{F} \rightarrow \mathcal{F}'\) is a morphism of \(\mathcal{G}\)-torsors, and given a choice of the \(s_U\) and thus of the cocycle \(g\), this cocycle and the cocycle \(g'\) determined by the \(\varphi_U(s_U)\) are related by a cochain \(f\in C^0(\mathfrak{U},\mathcal{F})\) in the same way, \(g'_{UV}=f_U g_{UV} f_V^{-1}\).

By choosing a trivializing cover for any \(\mathcal{G}\)-torsor and a set of transition functions, after taking the equivalence class in the direct limit we can define a morphism of groupoids by the following functor

\begin{align*} \{\text{$\mathcal{G}$-torsors} \} &\longrightarrow [Z^1(X,\mathcal{G}),C^0(X,\mathcal{G})]\\ \mathcal{F} &\longmapsto \{ \text{Transition functions of $\mathcal{F}$} \}, \end{align*}

which maps any morphism of \(\mathcal{G}\)-torsors to the \(0\)-cochain defined above.

Proposition 3. This functor is an equivalence of categories. In particular, the set \(H^1(X,\mathcal{G})\) classifies isomorphism classes of \(\mathcal{G}\)-torsors.

Proof. Clearly, the functor is fully faithful since the choice of open covering \(\mathfrak{U}\) and of \(f_U \in \mathcal{G}(U)\), for \(U\in \mathfrak{U}\) determines \(\varphi\) as \(\varphi_U(s_U)=f_U s_U\), for \(s_U \in \mathcal{F}(U)\). Thus, it suffices to see that the functor is essentially surjective. This means that what we have to show is that given a cocycle in \(Z^1(X,\mathcal{G})\), we can construct a \(\mathcal{G}\)-torsor whose transition functions are given by this cocycle. The way of doing this is a standard procedure which appears in a lot of places. The idea is to define the torsor locally as \(\mathcal{G}\) and then use the cocycle to “glue” the different patches. More precisely, we choose a representative \((\mathfrak{U},g)\), with \(g\in Z^1(\mathfrak{U},\mathcal{G})\), of the chosen cocycle and define a presheaf

\[ \mathcal{F}(U)= \coprod_{V \in \mathfrak{U}} \mathcal{G}(U\cap V)/\sim, \] with the equivalence relation \(\sim\) given as follows. We say that two sections \(f\in \mathcal{G}(U\cap V)\) and \(f'\in \mathcal{G}(U\cap V')\), with \(V\cap V'\neq \varnothing\), are related if

\[ f|_{U\cap V \cap V'} = g_{VV'} f'|_{U\cap V \cap V'}. \]

This presheaf verifies the sheaf condition by construction and it is a \(\mathcal{G}\)-torsor since on every \(U\in \mathfrak{U}\) it is the trivial \(\mathcal{G}\)-torsor. Again by construction, the cocycle \(g\) gives the transition functions of \(\mathcal{F}\). \(\blacksquare\)

A nice application

For the well known cases associated to \(G\) topological group, the above result is telling us that (isomorphism classes of) prinicipal \(G\)-bundles are classified by the cohomology set \(H^1(X,G)\) and that \(G\)-covering spaces are classified by \(H^1(X,\underline{G})\).

In a future post, I will explain how the correspondence between \(G\)-covering spaces and \(H^1(X,\underline{G})\) gives a nice and maybe “non-standard” approach at the basic results of Algebraic Topology. As for now, I am going to show how the correspondence between principal \(G\)-bundles and \(H^1(X,G)\) can be combined with the results of my last post to prove a nice fact of principal bundle theory.

What we are going to consider now is the problem of lifting the structure group to a group extension. In general, for any group \(G\) we say that another group \(\hat{G}\) is an extension of \(G\) if there is a surjective homomorphism \(\hat{G} \rightarrow G\). More generally, if \(1\rightarrow A \rightarrow \hat{G} \rightarrow G \rightarrow 1\) is a short exact sequence of groups, we say that \(\hat{G}\) is an extension of \(G\) by \(A\). Moreover, if the homomorphism \(A \rightarrow \hat{G}\) factors through the centre of \(\hat{G}\), we say that the extension \(\hat{G}\) is a central extension. In particular, if \(\hat{G}\) is a central extension, the group \(A\) is abelian.

The lifting problem consists on, given a central extension \(1\rightarrow A \rightarrow \hat{G} \rightarrow G \rightarrow 1\) and a principal \(G\)-bundle \(E\) over a topological space \(X\), constructing a principal \(\hat{G}\)-bundle \(\hat{E}\) “lifting” \(E\). In our terms, we can regard \(\hat{G} \rightarrow G\) as a morphism of sheaves, that induces a map \(H^1(X,\hat{G}) \rightarrow H^1(X,G)\). What we want to know is when this map is surjective. Recall now from my last post that, since \(A\) is abelian, \(H^2(X,A)\) is defined and the short exact sequence \(1\rightarrow A \rightarrow \hat{G} \rightarrow G \rightarrow 1\) induces in cohomology the exact sequence

\[ H^1(X,\hat{G}) \rightarrow H^1(X,G) \rightarrow H^2(X,A). \] Therefore, the map \(H^1(X,\hat{G}) \rightarrow H^1(X,G)\) is surjective if and only if \(H^2(X,A)\) is trivial.

Example. A nice example where this lifting problem is interesting is given by spin structures. Let \(X\) be an \(n\)-dimensional Riemannian manifold. Its tangent bundle \(TX\) is a vector bundle and, by considering its frame bundle we can regard it as a principal \(\mathrm{GL}(n,\mathbb{R})\)-bundle. Now, the Riemannian metric gives a reduction of the structure group to a principal \(\mathrm{SO}(n)\)-bundle. A spin structure on \(X\) is a lift of the structure group from this principal \(\mathrm{SO}(n)\)-bundle to the universal covering space \(\mathrm{Spin}(n) \rightarrow \mathrm{SO}(n)\). It is well known that \(\mathrm{SO}(n)\) is doubly connected. For example, \(\mathrm{SO}(3)\) is diffeomorphic to the real projective space \(\mathbb{RP}^3\) and \(\mathrm{Spin}(3)=\mathrm{SU}(2)\) is diffeomorphic to the \(3\)-sphere \(\mathbb{S}^3\). Therefore, the covering homomorphism \(\mathrm{Spin}(n) \rightarrow \mathrm{SO}(n)\) is in fact a central extension

\[ 1 \rightarrow \mathbb{Z}/(2) \rightarrow \mathrm{Spin}(n) \rightarrow \mathrm{SO}(n) \rightarrow 1. \]

We conclude from this that the obstruction for defining spin structures n \(X\) will be given by its Čech cohomology set \(H^2(X,\mathbb{Z}/(2))\). If \(g\in H^1(X,\mathrm{SO}(n))\) denotes the cocycle associated to the tangent bundle, the element \(\delta(g) \in H^2(X,\mathbb{Z}/(2))\) is called the second Stiefel-Whitney class of \(X\), denoted \(\omega_2(X)\). We will be able to define a spin structure on \(X\) whenever this class vanishes, \(\omega_2(X)=0\).

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