Some stable homology calculations and Occam's razor for Hodge structures

TL;DR

This paper proves two conjectures about the topology of subspaces of symmetric products motivated by motivic zeta functions, and discredits a stronger version of one conjecture, advancing understanding in algebraic topology and Hodge structures.

Contribution

It confirms two conjectures and refutes a strengthened version, providing new insights into the topology of symmetric product subspaces related to motivic zeta functions.

Findings

Proved two conjectures on the topology of symmetric product subspaces.

Disproved a strengthened conjecture, refining previous hypotheses.

Enhanced understanding of the relationship between motivic zeta functions and Hodge structures.

Abstract

Motivated by motivic zeta function calculations, Vakil and Wood in [VMW12] made several conjectures regarding the topology of subspaces of symmetric products. The purpose of this note is to prove two of these conjectures and disprove a strengthening of one of them.