# Involution on pseudoisotopy spaces and the space of nonnegatively curved metrics

## TL;DR

This paper establishes the equivalence of certain involutions in algebraic K-theory, enabling the computation of eigenspaces in pseudoisotopy spaces and applying this to analyze spaces of nonnegatively curved metrics on open manifolds.

## Contribution

It demonstrates the coincidence of involutions on rational algebraic K-theory of spaces and uses this to compute eigenspaces in pseudoisotopy spaces, linking to nonnegatively curved metrics.

## Key findings

- Involutions on algebraic K-theory coincide.
- Explicit dimensions of manifolds with nontrivial rational homotopy groups.
- Connection between involutions and spaces of nonnegatively curved metrics.

## Abstract

We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic $K$-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational $S^{1}$-equivariant homology group of the free loop space of a simply-connected manifold. As an application, we give explicit dimensions of the open manifolds $V$ that appear in Belegradek-Farrell-Kapovitch's work for which the spaces of complete nonnegatively curved metrics on $V$ have nontrivial rational homotopy groups.