Manifold pathologies and Baire-1 functions as cohomotopy groups

A slight extension of a construction due to Calabi-Rosenlicht (and later Gabard, Baillif and others) produces a typically non-metrizable $n$-manifold $\mathbb{P}$ by gluing two copies of the open upper half-space $\mathbb{H}_{++}$ in $\mathbb{R}^n$ along the disjoint union of the spaces of rays within $\mathbb{H}_{++}$ originating at points ranging over a subset $S\subseteq \mathbb{R}^{n-1}$ of the boundary $\mathbb{R}^{n-1}=\partial\overline{\mathbb{H}_{++}}$. The fundamental group $\pi_1(\mathbb{P})$ is free on the complement $S^{\times}$ of any singleton in $S\ne\emptyset$, and the main result below is that the first cohomotopy group $\pi^1(\mathbb{P})$, regarded as a space of functions $S^{\times}\to \mathbb{Z}$, is precisely the additive group of integer-valued Baire-1 functions on $S^{\times}$. This occasions a detour on characterizations (perhaps of independent interest) of Baire-1 real-valued functions on a metric space $(B,d)$ as various types of non-tangential boundary limits of continuous functions on $B\times \mathbb{R}_{>0}$.