[Paper Review] A geometric decomposition of spaces into cells of different types

Gabriel Minian, Miguel Ottina|ArXiv.org|Dec 10, 2006

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper introduces CW(A)-complexes, a geometric generalization of classical CW-complexes that replace spheres with an arbitrary base space A to construct spaces from cells of different topological types. It establishes a homotopy-theoretic framework by generalizing Whitehead's Theorem, showing that a map between CW(A)-complexes is a homotopy equivalence if and only if it is an A-weak equivalence, thereby extending classical results to a broader class of cell-structured spaces.

ABSTRACT

We develop the theory of CW(A)-complexes, which generalizes the classical theory of CW-complexes, keeping the geometric intuition of J.H.C. Whitehead's original theory. We obtain this way generalizations of classical results, such as Whitehead Theorem, which allow a deeper insight in the homotopy properties of these spaces.

Motivation & Objective

• To generalize the classical theory of CW-complexes by replacing the 0-sphere S⁰ with an arbitrary pointed space A as the foundational building block.
• To preserve the geometric intuition of Whitehead's original construction while extending homotopical results to spaces built from cells of different types.
• To investigate the homotopy properties of these generalized cell complexes, particularly the conditions under which homotopy equivalences are detected via A-weak equivalences.
• To analyze the behavior of dimension and connectivity in CW(A)-complexes, especially when the core space A is l-connected or finite-dimensional.
• To study the relationship between different decompositions of a space into A-cells and the effect of changing the core space A via maps to another space B.

Proposed method

• Define CW(A)-complexes via a constructive approach: attach n-cells of type A using the reduced suspension Σⁿ⁻¹A as the attaching space and the reduced cone CΣⁿ⁻¹A as the characteristic map.
• Use pushout diagrams to formalize the cell attachment process, ensuring compatibility with pointed topology and base-point preserving maps.
• Apply the homotopy extension property by leveraging that inclusions of the form Σⁿ⁻¹A → CΣⁿ⁻¹A are closed cofibrations when A is a CW-complex.
• Generalize connectivity results by analyzing the connectivity of pairs (Xⁿ, Xⁿ⁻¹) using the connectivity of Σⁿ⁻¹A and the dimension of A.
• Introduce the notion of A-weak equivalence and use it to formulate a generalized Whitehead Theorem for CW(A)-complexes.
• Use relative CW-complexes and strong deformation retracts to compare spaces obtained by different cell attachments and prove homotopy equivalence results.

Experimental results

Research questions

• RQ1 Under what conditions is the dimension of a CW(A)-complex well-defined, particularly when A is a finite-dimensional CW-complex?
• RQ2 How does the homotopy type of a CW(A)-complex change when the core space A is replaced by another space B via a map α:A→B?
• RQ3 When is a map f:X→Y between CW(A)-complexes a homotopy equivalence, and can this be detected via A-weak equivalence?
• RQ4 What is the connectivity of the pair (Xⁿ, Xⁿ⁻¹) in a CW(A)-complex when A is l-connected and of dimension k?
• RQ5 Can a space that is a generalized CW(A)-complex also be realized as a generalized CW(B)-complex when A is a CW(B)-complex?

Key findings

• The dimension of a CW(A)-complex is well-defined when the core space A is a finite-dimensional CW-complex, generalizing the classical invariance of dimension.
• If A is a CW(B)-complex of finite dimension, then any generalized CW(A)-complex X is also a generalized CW(B)-complex, and hence has the homotopy type of a standard CW-complex.
• When A is l-connected and of dimension k, the pair (Xⁿ, Xⁿ⁻¹) in a CW(A)-complex X is A-(n−k+l+1)-connected, generalizing classical connectivity bounds.
• The pair (Xⁿ, Xⁿ⁻¹) is (n+l+1)-connected when A is l-connected, showing that higher connectivity of A improves the connectivity of the skeleta filtration.
• A map f:X→Y between CW(A)-complexes is a homotopy equivalence if and only if it is an A-weak equivalence, establishing a generalized Whitehead Theorem.
• If βα = id_A, then X (a CW(A)-complex) is homotopy equivalent to a CW(B)-complex Y via maps φ:X→Y and ψ:Y→X with ψφ = id_X, showing that core changes via retractions preserve homotopy type.