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June 13, 2026Open Access

The Group of Self-Homotopy Equivalences of an F0-Space and the Whitehead Exact Sequence

Key Points

This research aims to explore the group of self-homotopy equivalences of an F0-space using Whitehead's exact sequence.
Studied self-homotopy equivalences of simply connected rational CW complex X.
Utilized Whitehead's exact sequence to analyze the group ℰ(X) and its subgroup ℰ*(X) in relation to homology.
Proved that the subgroup ℰ*(X) is trivial and described the structure of ℰ(X).
The subgroup ℰ*(X) is proven to be trivial.
The group ℰ(X) is found to be isomorphic to a specific subgroup of ℚ* × ℰ(X m-1 ).
F0-spaces maintain finitedimensional homotopy and positive Euler characteristic.

Abstract

A simply connected rational CW complex X is called an F 0 -space if it has finitedimensional homotopy, finite-dimensional cohomology, and positive Euler characteristic. Making use of Whitehead's certain exact sequence, this paper aims to study the group ℰ (X) of self-homotopy equivalences of X and its subgroup ℰ * (X) constituted by elements inducing the identity on the homology groups. Specifically, we prove that ℰ * (X) is trivial and ℰ (X) is isomorphic to a certain subgroup of ℚ * × ℰ (X m-1), where m is the formal dimension of X, X m-1 is its (m -1) -skeleton, and ℚ * = ℚ\0.

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