On 2-dimensional nonaspherical cell-like Peano continua: A simplified approach

TL;DR

This paper introduces a functor that transforms path connected spaces into simply connected, cell-like continua, revealing new relationships between fundamental groups and higher homotopy groups, especially in the context of 2-dimensional nonaspherical continua.

Contribution

It constructs a functor from path connected spaces to simply connected spaces that preserves certain topological properties and relates fundamental and higher homotopy groups in a novel way.

Findings

$AC(X,x)$ is a cell-like Peano continuum for Peano $X$

Dimension of $AC(X,x)$ increases by one relative to $X$

Triviality of $ ext{pi}_1$ corresponds to triviality of $ ext{pi}_2$ in $AC(X,x)$

Abstract

We construct a functor $AC(-,-)$ from the category of path connected spaces $X$ with a base point $x$ to the category of simply connected spaces. The following are the main results of the paper: (i) If $X$ is a Peano continuum then $AC(X,x)$ is a cell-like Peano continuum; (ii) If $X$ is $n-$dimensional then $AC(X, x)$ is $(n+1)-$dimensional; and (iii) For a path connected space $X$, $\pi_1(X,x)$ is trivial if and only if $\pi_2(AC(X, x))$ is trivial. As a corollary, $AC(S^1, x)$ is a 2-dimensional nonaspherical cell-like Peano continuum.