Topology, Geometry & Dynamics

March 16, 2012

Holonomy and H-stiffenings of G-structures

Suppose that we have a group $G$ of analytic diffeomorphisms acting on $\mathbb{R}^n$ and that $H$ is a subgroup of $G$ . Notice that any $H$ -structure on a manifold $M$ induces a $G$ -structure; all $H$ -manifolds are naturally $G$ -manifolds. If we have a $G$ -manifold which is not a priori an $H$ -manifold, we might like to know when the $G$ -structure can be stiffened into an $H$ -structure. This is called an $H$ -stiffening of the $G$ -structure and occurs if the $G$ -structure contains an $H$ -atlas. To determine if such an atlas exists we look at the holonomy group of the manifold. The following proof is paraphrased from Ratcliffe’s Foundations of Hyperbolic Manifolds.

Lemma

If $M$ is a $G$ -manifold and $\mathrm{Hol}(M)$ is contained in $H$ , then an $H$ -stiffening of the $G$ -structure of $M$ exists.

Proof

Suppose $\mathrm{Hol}(M) = \mathrm{hol}(\pi_1(M)) \subseteq H$ and pick an atlas whose chart domains are evenly covered by the universal covering $p : \widetilde{M} \to M$ . Assign $\widetilde{M}$ the pullback $G$ -structure. By assumption the deck transformations look like elements of $H$ . For each evenly covered neighboord $U_i$ in $M$ , pick some deck in the preimage $U_i$ , call it $U_{ij}$ . We associate cooordinates to $U_i$ now by applying the developing map to the chosen deck, $U_{ij}$ .

If two such domains, say $U_{i}$ and $U_\ell$ , then we have two preimages $U_{ij}$ and $U_{\ell k}$ . To do a chart change we have to perform a deck transformation, and so chart changes look like $\mathrm{dev} \circ \gamma \circ \mathrm{dev}^{-1}$ where $\gamma$ is a deck transformation. But $\mathrm{dev} \circ \gamma \circ \mathrm{dev}^{-1}$ is by assumption in $H$ . So all chart changes are accomplished by applying an element of $H$ , and we have an $H$ -atlas.

March 14, 2012

The Developing Map and Holonomy

In the last post we talked about $G$ -manifolds, which were just manifolds where chart changes were required to be restrictions of elements of some group $G$ acting on $\mathbb{R}^n$ . Now we want to impose some more conditions on $G$ so that we can get a sort of geometric version of the fundamental group of the manifold. In particular, we want our group $G$ to consist of maps which have a unique continuation. That is, if two elements $g, h \in G$ are equal on any open subset of $\mathbb{R}^n$ , then $g = h$ . Similarities and analytic diffeomorphisms, for example, have this property (see Ratcliffe’s Foundations of Hyperbolic Manifolds or Thurston’s Three-Dimensional Topology and Geometry for details).

The first step in constructing this “geometric” fundamental group is to try to extend a chart on a $G$ -manifold $M$ to give global coordinates. In general this isn’t possible, but we can do it if we assume the unique continuation property mentioned above and if we assume that the space $M$ is simply connected. In this case we can pick a chart $\phi_0 : U_0 \to \mathbb{R}^n$ and then do analytic continuation to extend this chart to a global map, which we call the developing map of $M$ with respect to $\phi_0 : U_0 \to \mathbb{R}^n$ .

Let $x_0$ be a fixed point in $U_0$ . For any point $x \in M$ , let $\gamma : [0, 1] \to M$ be any path connecting $x_0$ to $x$ . Partition the interval $[0, 1]$ by $0 = t_0 < t_1 < t_2 < ... < t_n = 1$ and let $\gamma_i = \gamma|_{[t_{i-1}, t_i]}$ . We can pick a partition so that each $\gamma_i$ has its image contained in some chart $\phi_i : U_i \to \mathbb{R}^n$ . Now notice that the images of $\phi_i(\gamma_i([t_{i-1}, t_i]))$ may not form a continuous path in $\mathbb{R}^n$ , but we can compose with elements of $G$ to get a continuous path. The initial point of this path will be $\phi_0(x_0)$ , while the terminal point will be used as the image of the map $\mathrm{dev} : M \to \mathbb{R}^n$ . This is the developing map and by its construction depends on our choice of $\phi_0 : U_0 \to \mathbb{R}^n$ .

We could have picked another chart $\psi_0 : V_0 \to \mathbb{R}^n$ to use in constructing the developing map, but this won’t change things so much. If we’d picked another chart around $x_0$ , then we could just compose with an element of $G$ to get to $\phi_0$ and go from there. So changing the chart just composes the developing map with an element of $G$ .

For any $G$ -map from $M$ to itself, $f : M \to M$ , there exists a unique element of $G$ , called the holonomy of $f$ and denoted $\mathrm{hol}(f)$ , which makes the following diagram commute.

For a general $G$ -manifold $M$ that may not be simply connected, consider the developing map on the universal cover of $M$ with the pulled back $G$ -structure, $\mathrm{dev}:\widetilde{M} \to \mathbb{R}^n$ . Each element of the fundamental group of $M$ , $\pi_1(M)$ , determines a deck transformation $[\gamma] : \widetilde{M} \to \widetilde{M}$ . Each of these deck transformations is a $G$ -map from $\widetilde{M}$ to itself, so we can associate its holonomy, $\mathrm{hol}([\gamma])$ . The collection of all of the holonomies of each deck transformation gives us the holonomy group of the manifold, $\mathrm{Hol}(M) := \mathrm{hol}(\pi_1(M))$ .

There are two main reasons we’ll care about the holonomy group of a manifold. The first is that the holonomy group will tell us when stiffenings of the $G$ -structure are possible (the topic of the next post), and the other reason is that in some special situations the holonomy group determines the $G$ -structure of the manifold.

March 12, 2012

Pseudogroups and G-manifolds

Recall that a manifold is essentially a topological space that locally looks like another, better understood space. You can impose different types of structures on a manifold by making the chart changes look like certain well understood maps. For a general topological manifold, we only require that chart changes are homeomorphisms between open subsets of $\mathbb{R}^n$ . For smooth manifolds we require that the chart changes are smooth diffeomorphisms between open subsets of $\mathbb{R}^n$ . For Riemann surfaces we require that the chart changes are holomorphic maps between open subsets of $\mathbb{C}$ . We’d like to somehow wrap all of these different cases up in a general theory, and that’s where pseudogroups come in.

A pseudogroup is a collection $\mathscr{G}$ of homeomorphisms between open subsets of $\mathbb{R}^n$ that satisfy the following conditions.

If $g \in \mathscr{G}$ is defined on an open set $U$ , and $V \subseteq U$ is an open set, then the restriction $g|_V$ is also in $\mathscr{G}$ .
If $g \in \mathscr{G}$ , then $g^{-1} \in \mathscr{G}$ .
For any two $g_1, g_2 \in \mathscr{G}$ , if $g_1 \circ g_2$ is defined, then $g_1 \circ g_2 \in \mathscr{G}$ .
If $U \subseteq \mathbb{R}^n$ is open and covered by the open sets $U_\alpha$ , for all $\alpha \in A$ , then a map $g : U \to \mathbb{R}^n$ is an element of $\mathscr{G}$ provided each restriction $g|_{U_\alpha} \in \mathscr{G}$ .

Given such a pseudogroup $\mathscr{G}$ , we say that a topological space $M$ (which we’ll always assume is Hausdorff, second countable, and connected) has a $\mathscr{G}$ -atlas $\mathcal{A}$ if there is a collection of maps $\phi_i : U_i \to \mathbb{R}^n$ where the domains $U_i$ form an open cover of $M$ and any chart change $\phi_{ij} = \phi_i \circ \phi_j^{-1}$ , when defined, is an element of $\mathscr{G}$ . A maximal $\mathscr{G}$ -atlas is called a $\mathscr{G}$ -structure, and a manifold $M$ with a $\mathscr{G}$ -structure is called a $\mathscr{G}$ -manifold.

Now suppose that $G$ is a group acting by homeomorphisms on $\mathbb{R}^n$ . We can associate a pseudogroup to $G$ by considering the restrictions of elements of $G$ to open subsets of $\mathbb{R}^n$ . By a $G$ -manifold we mean a $\mathscr{G}$ -manifold where $\mathscr{G}$ is this pseudogroup generated by restrictions of elements of $\mathscr{G}$ . That is, a $G$ -manifold is a Hausdorff, second countable, connected space $M$ together with an atlas $\{ \phi_i : U_i \to \mathbb{R}^n \}_{i \in I}$ where the chart changes $\phi_{ij} = \phi_i \circ \phi_j^{-1}$ look like restrictions of elements of $G$ .

(There is a slightly more general theory where we allow our charts to map into a space $X$ , but for our purposes we’re only going to consider the case when $X = \mathbb{R}^n$ .)

Given two $G$ -manifolds $M$ and $N$ , we say that a map $f : M \to N$ is a $G$ -map if it locally looks like an element of $G$ . That is, if for each $P \in M$ there are charts $\phi : U \to \mathbb{R}^n$ around $P$ and $\psi : V \to \mathbb{R}^n$ around $f(P)$ in the $G$ -atlases of $M$ and $N$ , respectively, and a $g \in G$ such that $\psi \circ f = g \circ \phi$ .

Suppose that $\rho : M \to N$ is a covering map where $N$ is a $G$ -manifold with $G$ -structure $\nu$ . Then there is a unique $G$ -structure on $M$ , which we’ll denote $\rho^*(\nu)$ and call the pullback of $\nu$ via $\rho$ , which makes $\rho$ a $G$ -map. To see this we simply consider a $G$ -atlas for $\nu$ where the domains of the charts are evenly covered. Looking at the preimages of these chart domains gives an open cover of $M$ , and we assign coordinates to these preimages by projecting down to $N$ and using the coordinates in those charts. It’s easy to check that the chart changes on $M$ are then (restrictions of) elements of $G$ , as the corresponding chart changes on $N$ are.

If $\mu$ is another $G$ -structure on $M$ such that $\rho : M \to N$ is a $G$ -map, then we need to show that $\mu = \rho^*(\nu)$ . Suppose $\mathcal{A}$ is the above-described atlas for $\rho^*(\nu)$ , and $\mathcal{B}$ is an atlas for $\mu$ . Say $\phi : U \to \mathbb{R}^n$ is an element of $\mathcal{A}$ and $\psi : V \to \mathbb{R}^n$ is an overlapping element of $\mathcal{B}$ . We need to show that the coordinate change $\phi \circ \psi^{-1}$ is an element of $G$ . This is easy to see, though, because $\phi$ is by construction just a restriction of $\rho$ , so locally $\phi \circ \psi^{-1} = \rho \circ \psi^{-1}$ . By assumption $\rho$ is a $G$ -map, so $\rho \circ \psi^{-1} : \mathbb{R}^n \to \mathbb{R}^n$ is an element of $G$ (really a restriction of some $g \in G$ ). Thus the chart changes are compatible, so the two atlases determine the same $G$ -structure.

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