The Gromov-Lawson-Rosenberg Conjecture for Z/4xZ/4

TL;DR

This paper proves the Gromov-Lawson-Rosenberg Conjecture for the group Z/4xZ/4 by computing connective real k-homology using spectral sequences and detection theorems, advancing understanding of positive scalar curvature obstructions.

Contribution

It establishes the conjecture for Z/4xZ/4 through novel spectral sequence analysis and detection methods, including eta-invariants and homological techniques.

Findings

Computed the connective real k-homology of the classifying space.

Determined differentials in the Adams spectral sequence for relevant spaces.

Developed detection theorems based on eta-invariants and homological methods.

Abstract

We prove the Gromov-Lawson-Rosenberg Conjecture for the group Z/4xZ/4 by computing the connective real k-homology of the classifying space with the Adams spectral sequence and two types of detection theorems for the kernel of the alpha invariant: one based on eta-invariants, closely following work of Botvinnik-Gilkey-Stolz, and a second one based on homological methods. Along the way, we determine differentials of the Adams spectral sequence for classifying spaces involved in the computation, and we study the cap structure of the Adams spectral sequence for sub-hopf algebras of the Steenrod algebra relevant to the computation of connective real and complex k-homology.