Uniqueness of BP<n>

TL;DR

This paper proves a uniqueness theorem for p-complete spectra with the same mod p cohomology as BP<n>, establishing that such spectra are weakly homotopy equivalent to the p-completion of BP<n> under certain conditions.

Contribution

It demonstrates that the mod p cohomology module structure uniquely determines the p-complete spectrum BP<n> up to weak homotopy equivalence.

Findings

Uniqueness of p-complete spectra with given cohomology

Identification of spectra via mod p cohomology

Conditions under which spectra are equivalent

Abstract

Fix a prime number p and a non-negative integer n. We prove that if a p-complete spectrum X satisfying a mild finiteness condition has the same mod p cohomology as BP<n> as a module over the Steenrod algebra, then X is weak homotopy equivalent to the p-completion of BP<n>.