Part I: The Theorem🔗

Throughout, X and T are topological spaces with X compact, \mathcal U = \{U_\alpha\}_{\alpha \in \iota} is an open cover of X indexed by an arbitrary set \iota, and \{r_\alpha\} is a partition of unity subordinate to \mathcal U: each r_\alpha : X \to [0,1] is continuous, the family of supports is locally finite, \sum_\alpha r_\alpha = 1, and the closure of \{x : r_\alpha(x) \ne 0\} is contained in U_\alpha for every \alpha. (We do not assume that all but finitely many r_\alpha vanish identically; this is deduced from compactness in Proposition 3.1.)

P \subseteq \mathbb R^k is a convex linear polyhedron: the convex hull of a finite set. We write \operatorname{conv}(A) for convex hulls. The same integer k serves both as the ambient dimension of P and as the adaptation constant ("at most k of the U_\alpha"); this coincidence is faithful to the statement being proved (P is a k-parameter family) and is load-bearing: the counting argument of Section 5 closes precisely because a simplex in \mathbb R^k has at most k+1 vertices.

Every homotopy F we construct has the special form

F(t,p,x) = f\bigl(u(t,p,x),\, x\bigr),

where u : [0,1] \times P \times X \to P is a continuous reparametrization of the parameter polyhedron: at each time t and point x \in X, the parameter p is moved to a new parameter u(t,p,x) \in P, and f is evaluated there. The proof splits into:

Section 2 (reduction): conclusions (1)–(4) of Theorem A follow formally from three properties of u;
Sections 3–5 (the geometric core, Part II): construction of u from a "handle family", and construction of the handle family itself, via a general-position vertex labelling on a fine triangulation of P;
Section 9 (analysis, Part III): the bi-Lipschitz refinement, where the reparametrization is additionally made Lipschitz in x with a small constant, and a fixed-point argument shows each slice of F remains a bi-Lipschitz homeomorphism.

Two pieces of general-purpose geometric infrastructure are developed on the way and used throughout: the existence of triangulations of polytopes (Section 7) and the barycentric subdivision with its mesh estimate (Section 8); both are in the Appendix.