On Non-zero Degree Maps between Quasitoric 4-Manifolds

TL;DR

This paper investigates the possible degrees of continuous maps between quasitoric 4-manifolds, revealing connections with number theory and quadratic forms, and explicitly determining the sets of feasible degrees for certain manifold pairs.

Contribution

It determines the sets of possible map degrees between specific quasitoric 4-manifolds, linking topological, algebraic, and number-theoretic methods in a novel way.

Findings

Sets of degrees are characterized as sums of squares.

Explicit degrees for certain manifold pairs are computed.

Connections between topology, quadratic forms, and number theory are established.

Abstract

We study the map degrees between quasitoric 4-manifolds. Our results rely on Theorems proved by Duan and Wang. We determine the set D (M, N) of all possible map degrees from M to N when M and N are certain quasitoric 4-manifolds. The obtained sets of integers are interesting, e. g. those representable as the sum of two squares D (C P^2#C P^2, C P^2) or the sum of three squares D (C P^2 # C P^2 # C P^2, C P^2). Beside the general results about the map degrees between quasitoric 4-manifolds, the connections among Duan-Wang's approach, the quadratic forms, the number theory and the lattices is established.