Completion preserves homotopy fibre squares of connected nilpotent spaces

TL;DR

This paper proves that p-completion preserves homotopy fibre squares in connected nilpotent spaces, leading to a new proof of the Hasse fracture square and clarifying conditions for CW complex homotopy types.

Contribution

It establishes that completion at a set of primes preserves homotopy fibre squares in connected nilpotent spaces, and provides a concise proof regarding CW complex homotopy types in fibre sequences.

Findings

Completion preserves homotopy fibre squares.

Derivation of the Hasse fracture square for nilpotent spaces.

Clarification of CW complex homotopy type conditions.

Abstract

We prove that completion at a set of primes preserves homotopy fibre squares of connected nilpotent spaces. As a consequence, we deduce the Hasse fracture square associated to a connected nilpotent space. Along the way, we give a quick proof of the well-known result that if the base of a fibre sequence has the homotopy type of a CW complex, then the total space has the homotopy type of a CW complex iff each fibre does.