On the theorems of Borsuk-Ulman and Ljusternik-Shnirelman-Borsuk (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 1993)
The main result: Every homotopically trivial map $f:X → M^n$ from the $(n-1)$-connected space $X$ with involution $φ$ to the $n$-dimensional topological manifold $M^n$ identifies a pair of points $(x,φ(x))$