The suspension foliation of a representation of the fundamental group

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $B$ be a connected smooth manifold with base point $b_0$ (Smooth manifolds and their smooth charts), let $F$ be a smooth manifold, let $p:\tilde{B}\to B$ be a universal cover with a fixed point ${\tilde{b}}_0$ over $b_0$ (Universal covering spaces), and let $\rho :{\pi }_1(B,b_0)\to \mathrm{Diff}(F)$ be a homomorphism into the diffeomorphism group of $F$ (Diffeomorphisms and local diffeomorphisms of manifolds). The universal cover has a canonical smooth manifold structure with $p$ a local diffeomorphism. To include the second-countability prerequisite, choose a countable cover of $B$ by coordinate balls: for each nonempty member of an enumerated basis that is contained in some coordinate ball, use ${\mathrm{A}\mathrm{C}}_{\omega }$ to select one such ball. These selected balls cover $B$ because its coordinate balls form a neighborhood basis. Each is simply connected by its convex coordinates (Every nonempty convex subset of ${\mathbb{R}}^n$ is contractible; a zero-dimensional connected base is a single point). Each pairwise intersection has at most countably many connected components, and they are path connected (Components of a topological manifold are open and at most countable). Using ${\mathrm{A}\mathrm{C}}_{\omega }$, choose a point in each nonempty overlap component and a path from it to a chosen center in each of the two balls. A subdivided loop can move every junction to the selected overlap point along a path in that component; paths with the resulting fixed endpoints inside a ball are homotopic by simple connectivity. Thus each loop class is represented by a finite word in countably many selected connecting paths. This proves that ${\pi }_1(B,b_0)$ is at most countable (finite words over a countable set are countable under Countable unions of at most countable sets, assuming ${\mathrm{A}\mathrm{C}}_{\omega }$). The sheets over each ball are indexed by this countable fibre. Lifting the countable coordinate cover gives a countable smooth atlas on $\tilde{B}$: transitions are restrictions of base-chart transitions. The total space is Hausdorff, since points over different base points are separated downstairs, and points in one fibre lie in disjoint sheets. The lifted atlas also makes every deck transformation smooth with smooth inverse. The cover exists because $B$ is path connected, locally path connected and semilocally simply connected (Every nonempty path-connected locally path-connected semilocally simply connected space has a universal cover).

Using the isomorphism ${\pi }_1(B,b_0)\cong \mathrm{Deck}(\tilde{B}\to B)$ (For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group), take the deck identification defined by lifted endpoints starting at ${\tilde{b}}_0$, and let ${\pi }_1(B,b_0)$ act on the product $\tilde{B}\times F$ (Products of smooth manifolds have a canonical product smooth structure) diagonally by

$\gamma \cdot (x,y):=(\gamma \cdot x,\rho (\gamma )(y))\mathrm{.}$

This is a free and properly discontinuous action by diffeomorphisms, so that the quotient-foliation proposition applies in the form recorded below. It is free: $\gamma \cdot (x,y)=(x,y)$ forces $\gamma x=x$ on the connected total space $\tilde{B}$, and a deck transformation fixing a point is the identity (On a connected covering space, a deck transformation is determined by one point and the deck action is free). It is properly discontinuous: projections of compact sets are compact (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism), and if a compact $K\subseteq \tilde{B}\times F$ meets $\gamma K$ then the compact projection $C\subseteq \tilde{B}$ of $K$ meets $\gamma C$, so it suffices to show that $\{\gamma :\gamma C\cap C\mathrm{\ne }\mathrm{\varnothing }\}$ is finite for a compact $C\subseteq \tilde{B}$. Suppose ${\gamma }_1,{\gamma }_2,\dots$ are distinct with points $x_n,{\gamma }_n{x}_n\in C$ for every $n$. The manifold $\tilde{B}$ is locally Euclidean, hence first countable, and a sequence in a compact first-countable space has a convergent subsequence: the closed tails have the finite intersection property, so they have a common point, and a nested neighbourhood basis at that point produces the subsequence. Passing to subsequences twice we may therefore assume $x_n\to x$ and ${\gamma }_n{x}_n\to y$ with $x,y\in C$ (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right). Let $W$ be a connected evenly covered coordinate neighbourhood of $p(x)$ and let $U$ be the sheet of $p^{-1}(W)$ containing $x$ (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings); choose a connected open neighbourhood $\mathrm{\Omega }$ of $y$ with $p(\mathrm{\Omega })\subseteq W$ on which $p$ is injective, which exists because $p$ is a local homeomorphism. For all large $n$ one has $x_n\in U$ and ${\gamma }_n{x}_n\in \mathrm{\Omega }$. Now ${\gamma }_n(U)$ is connected with $p({\gamma }_n(U))=p(U)=W$, so it is a sheet over $W$; and $\mathrm{\Omega }$ is connected with $p(\mathrm{\Omega })\subseteq W$, so $\mathrm{\Omega }$ lies in a single sheet over $W$, which must be ${\gamma }_n(U)$ because ${\gamma }_n{x}_n$ lies in both. Hence ${\gamma }_n(U)$ is the same sheet $S$ over $W$ for all large $n$, and ${\gamma }_n{\mathrm{\mid }}_U=(p{\mathrm{\mid }}_S{)}^{-1}\circ (p{\mathrm{\mid }}_U)$ for all large $n$. Two deck transformations of the connected cover $\tilde{B}$ agreeing on the nonempty open set $U$ are equal (On a connected covering space, a deck transformation is determined by one point and the deck action is free), so ${\gamma }_m={\gamma }_n$ for all large $m,n$, contradicting distinctness. Hence only finitely many $\gamma$ meet $C$, and therefore only finitely many meet $K$. The action is in particular a covering-space action (Covering-space actions by disjoint translates of neighbourhoods, The deck group of a connected covering acts by a covering-space action).

The product foliation of $\tilde{B}\times F$ by the leaves $\tilde{B}\times \{y\}$ ($y\in F$) is a regular foliation of codimension $\dim F$, and it is invariant under the action, since $\gamma \cdot (\tilde{B}\times \{y\})=\tilde{B}\times \{\rho (\gamma )y\}$. By The quotient foliation under a free and properly discontinuous foliated action the quotient $M_{\rho }:=(\tilde{B}\times F)\mathrm{/}{\pi }_1(B,b_0)$ is therefore a smooth manifold, the orbit map $\pi :\tilde{B}\times F\to M_{\rho }$ is a covering map, and the product foliation descends to a regular foliation $F_{\rho }$ of $M_{\rho }$ of codimension $\dim F$. This foliation is the suspension foliation of the representation $\rho$.