#groupoids

Conférence “Index theory and singular structures”

The French grant ANR SINGSTAR (“C*-algebras and Analysis on Singular Manifolds”), is organizing a conference named

Index Theory and Singular Structures

to be held in the Mathematics institute of Toulouse, France, from Monday 29 may 2017 to Friday 2 june 2017.

The plan of the conference is to gather specialists from different horizons related to index theory understood in a broad sense…

Hyperbolic Groupoids And Duality (memoirs Of The American Mathematical Society) Download

[ad_1]

Read Online 1 MB Download

The author introduces a notion of hyperbolic groupoids, generalizing the notion of a Gromov hyperbolic group. Examples of hyperbolic groupoids include actions of Gromov hyperbolic groups on their boundaries, pseudogroups generated by expanding self-coverings, natural pseudogroups acting on leaves of stable (or unstable) foliation of an Anosov diffeomorphism,…

Since Grp is a full subcategory of Grpd, this exponentiation pretty much has all the properties you’d expect it to have. The trivial group raised to any power is itself, and any group raised to the trivial-group-th power is itself. A natural transformation between group homomorphisms A,B: G -> H is an element of H which, acting by conjugation, interchanges the images of A,B (and natural transformations compose according to the group multiplication); we can use this fact to construct exponentials explicitly. And we can use the definition of exponentiation as right adjoint to the product to interpret them in terms of group homomorphisms.

Observation: Consider the simplest nontrivial group exponential, X = (C_2)^(C_2). This should be a groupoid such that groupoid homomorphisms G -> X are in bijection with group homomorphisms G × (C_2) -> (C_2). Since (C_2) has just two endomorphisms, and since neither of its inner automorphisms do anything (just like in any abelian group), we can see that X is two disjoint copies of (C_2), i.e. X = (C_2) + (C_2), where “+” denotes the coproduct in Grpd. A homomorphism G -> X thus consists of a single bit of information that determines which copy of (C_2) is in its image, plus an arbitrary homomorphism G -> (C_2). We can see that any homomorphism f: G × (C_2) -> (C_2) is determined by the same data: the bit is b = f(e,1), where “e” is the identity of G, and the homomorphism f`: G -> (C_2) is f` = f(g,0).

Observation: Since any abelian group has no nontrivial inner automorphisms, A^G for any abelian group A consists of a disjoint copy of A for each homomorphism G -> A. The above observation is an example of this, and similarly we can see that, for example, T^(C_n) = n*T, where T is the circle group and n*T is the coproduct of n copies of T.

Observation: For any group G, the groupoid G^G has a connected component for each outer automorphism of G.

The material in this post combined with the previous one comprises the first half of Chapter 3 of May’s A Concise Course in Algebraic Topology. As there, I assume all spaces are connected and locally path-connected.

Recall that our proof that \(\pi_1(S^1, 1) \simeq \mathbb{Z}\) hinges on the following fact: given a loop based at \(1\) (working in the complex plane), we can lift it onto a \(0\)-based loop on the real line while remembering orientation. The lift occurs along the map \(\mathbb{R} \overset{p}{\to} S^1\) given by wrapping successive unit-length intervals around the circle once. That we can lift locally depends on this map being a surjective local homeomorphism. That we can glue these local lifts together depends on the compactness of \([0,1]\) and the fact that each point \(x \in S^1\) has a neighborhood \(V_x\) such that each of \(p^{-1}(V_x)\)’s (path-, in this case) connected components are taken homeomorphically to \(V_x\) by \(p\).

These key properties of \(p\) are precisely those of a covering space.

Definition. A map \(E \overset{p}{\to} B\) of topological spaces is a covering if it is surjective, and for each \(b \in B\) there exists a neighborhood \(V_b\) such that each connected component of \(p^{-1}(V_b)\) is open in \(E\) and is taken homeomorphically to \(V_b\) by \(p\). \(E\) is said to be the total space, \(B\) the base space, and we denote the fiber \(p^{-1}(b)\) by \(F_b\) for \(b \in B\) (adding superscripts to distinguish between coverings if necessary).

Theorem. (Unique path lifting) Let \(E \overset{p}{\to} B\) be a covering. Fix \(b\) a basepoint for \(B\). Let \(e, e’ \in F_b\).

(Lifts exist): A path \(f : I \to B\) with \(f(0) = b\) lifts uniquely to a path \(g : I \to E\) such that \(g(0) = e\) and \(p \circ g = f\).

(Lifts exist uniquely): Equivalent paths starting at \(b\) lift to equivalent paths starting at \(e\).

\(p_* : \pi_1(E,e) \to \pi_1(B,b)\) is a monomorphism.

\(p_*(\pi_1(E,e’))\) is conjugate to \(p_*(\pi_1(E,e))\).

In fact, as \(e’\) runs through \(F_b\), the groups \(p_*(\pi_1(E,e’))\) run through all conjugates of \(p_*(\pi_1(E,e))\) in \(\pi_1(B,b)\).

Proof. By our computation of \(\pi_1(S^1,1)\) (in fact, we’ve seen the same argument about five times already), we know items \(1\) and \(2\). Next, we introduce the language of coverings of groupoids, which will make the formality of items \(3 \to 5\) apparent.

Recall that the category of objects over \(X \in \mathbf{C}\) comprises objects morphisms in \(\mathbf{C}\) into \(X\) and morphisms morphisms between the domains of those morphisms making the triangle commute, denoted \(\mathbf{C}/X\). Dualizing this yields the category of objects under \(X\), denoted \(X \backslash \mathbf{C}\). They are also called the slice and co-slice categories over and under \(X\), respectively.

Definition. Let \(\mathbf{E} \overset{p}{\to} \mathbf{B}\) be a morphism of connected groupoids. It is said to be a covering if it is surjective on objects and for each \(e \in \mathbf{E}\) the induced functor \(\mathbf{E}/e \to \mathbf{B}/p(e)\) (equivalently, \(e \backslash \mathbf{E} \to p(e) \backslash \mathbf{B}\)) is an isomorphism of categories. (This follows just from bijectivity on objects, because all morphisms of objects over/under \(e\) correspond to objects over/under \(e\)).

Since morphisms in the fundamental groupoid are equivalence classes of paths, knowing \(1\) and \(2\) gives that whenever \(E \overset{p}{\to} B\) is a covering of spaces, then \(\Pi(E) \overset{\Pi(p)}{\to} \Pi(B)\) is a covering of groupoids.

So, item \(3\) follows from the injectivity of the induced maps between slice categories and the fact that fundamental groups are automorphism groups, which live in those slice categories. Similarly, surjectivity of the induced maps gives that any \(f \in \pi_1(\Pi(B),b)\) arises as \(p_*(g)\) for some \(e’ \overset{g}{\to} e\) in \(\Pi(E)/e\), so that the mapping \[\left(e \overset{\sigma}{\to} e  \right) \mapsto \left(e’ \overset{g}{\to} e \overset{\sigma}{\to} e \overset{g^{-1}}{\to} e’ \right) \mapsto \left(b \overset{f}{\to} b \overset{p_* \sigma}{\to} b \overset{f^{-1}}{\to} b \right)\] yields (since \(\sigma\) was arbitrary) \(p_*\left(\pi_1(E,e’)\right)\) as the conjugate of \(p_*\left(\pi_1(E,e) \right)\) by \(f\) in \(\pi_1(B,b)\). That gives item \(5\). To see \(4\), connectivity of \(E\) yields a path \(e \overset{g}{\to} e’\), which is taken by \(p_*\) to something by which we can conjugate \(\pi_1(E,e’)\) to \(\pi_1(E,e)\), as displayed above. This completes the proof.