Leafwise paths and leafwise homotopy relative to endpoints

Let $F$ be a regular foliation of a smooth manifold $M$ with its leaves (Leaves of a regular foliation). A leafwise path for $F$ is a continuous map $a:[0,1]\to M$ whose image is contained in a single leaf of $F$. Thus a leafwise path from $x$ to $y$ has both endpoints in one leaf, and continuity is required in the topology of $M$.

A leafwise homotopy relative to endpoints between leafwise paths $a,b$ with the same endpoints $x,y$ is a continuous map $H:[0,1]\times [0,1]\to M$ on the product space (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space) with

$H(0,t)=a(t),H(1,t)=b(t),H(s,0)=x,H(s,1)=y$

for all $s,t\in [0,1]$, such that $t\mapsto H(s,t)$ is a leafwise path for every $s$. This is a path homotopy relative to the endpoints in the sense of Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints carrying one extra condition: every time slice lies in a single leaf. Leafwise paths $a,b$ are leafwise homotopic relative to endpoints, written $a\simeq b$, when such an $H$ exists.

Leafwise homotopy relative to endpoints is an equivalence relation on leafwise paths with fixed endpoints. Reflexivity and symmetry are Homotopy relative to a subspace is reflexive and symmetric applied to the constant and reversed deformations, which keep every time slice leafwise when the original map does; transitivity is the concatenation of homotopies supplied by Two homotopies relative to the same subspace concatenate after piecewise-linear reparametrisation, whose piecewise-linear reparametrisation again keeps every time slice leafwise. Two leafwise paths are homotopic relative to endpoints when they are equivalent in this relation.

Under ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), every continuous map from a locally connected space into $M$ whose image lies in one leaf is continuous into that leaf's intrinsic manifold topology. Here is the needed local argument. The leaf $L$ is a second-countable injectively immersed manifold with plaque charts (Regular foliations and integrable distributions correspond, Existence and uniqueness of maximal connected integral manifolds). In any foliation chart $U$ its distinct plaques in $L$ are disjoint nonempty open subsets of $L$: plaque inclusions and intrinsic leaf charts are locally diffeomorphic since their tangent images both equal $TF$ and the smooth inverse function theorem applies (The smooth inverse function theorem on manifolds). An enumerated basis of $L$ (Second countability: an at most countable basis for the topology) assigns to each such plaque the least index of a nonempty basic open set contained in it, so there are at most countably many plaques and at most countably many transverse coordinate values. If $f:C\to U\cap L$ is continuous and $C$ is connected, every transverse coordinate of $f$ is constant: two distinct values would force all intermediate values by connectedness (otherwise the two open half-lines at a missing value separate $C$), contradicting Every nondegenerate interval of $\mathbb{R}$ is uncountable. The connected image then lies in one connected component of that level set, hence in one plaque. For a general locally connected domain, take connected open neighborhoods inside $f^{-1}(U)$. On each such neighborhood $f$ maps continuously into the embedded plaque, whose topology is its intrinsic leaf-chart topology. This proves the assertion. It applies to intervals and squares, so the paths and homotopies above agree with intrinsic leaf paths and homotopies; in particular ${\pi }_1(L,x)$ uses the intrinsic leaf topology.