A Theorem of Van Kampen Type for Pseudo Peano Continuum Spaces

TL;DR

This paper develops new categorical frameworks for spaces constructed from groups, establishing a Van Kampen type theorem and analyzing the fundamental group functor's properties in these contexts.

Contribution

It introduces two new categories where the fundamental group functor has a right adjoint, is right exact, and preserves direct limits, extending Van Kampen's theorem.

Findings

Constructed categories where π is right adjoint

Proved π preserves direct limits in these categories

Established a new Van Kampen theorem for joins of spaces

Abstract

G. Conner and K. Eda (Topology and its Applications, 146, (2005), 317-328.) introduced a new construction of spaces from groups. They remarked that the construction is not categorical. In this paper, based on the work of Conner and Eda, we construct two new categories for which the functor of the fundamental group $\pi$ has a right adjoint and consequently is right exact and preseves direct limits. Also, we study the behavior of the functor $\pi$ on quotient spaces and give a new version of Van Kampen theorem for join of spaces in the new categories.