On Snake cones, Alternating cones and related constructions

TL;DR

This paper explores the homotopy equivalences among various cone constructions over the circle, introduces new related constructions, and analyzes their homotopical properties, including the second homology group of a Hawaiian tori wedge.

Contribution

It establishes homotopy equivalences between Snake and alternating cones over the circle and introduces new constructions with detailed homotopical analysis.

Findings

Snake cone over a square is homotopy equivalent to the alternating cone.

Introduces and investigates properties of CSC(-) and CAC(-) constructions.

Explicitly describes the second homology group of the Hawaiian tori wedge.

Abstract

We show that the Snake on a square $SC(S^1)$ is homotopy equivalent to the space $AC(S^1)$ which was investigated in the previous work by Eda, Karimov and Repov\vs. We also introduce related constructions $CSC(-)$ and $CAC(-)$ and investigate homotopical differences between these four constructions. Finally, we explicitly describe the second homology group of the Hawaiian tori wedge.