Skeleta, CW subcomplexes, and relative CW complexes

Depends on

• CW complex with closure finiteness and weak topology

Used by

• The image of a compact space lies in a finite CW subcomplex Corollary
• Based cellular chains of a universal cover as finite free right group-ring modules Definition
• Connectivity of a CW pair Definition
• Elementary expansions and collapses of finite CW complexes Definition
• Incidence number of two CW cells Definition
• Skeletal mapping telescope of a cw pair Definition
• Unreduced homology theory on cw pairs Definition
• A handle decomposition gives a relative CW complex Lemma
• A low-dimensional disk can be pushed off a higher cell Lemma
• Any ordinary homology theory has a cellular chain complex on a cw pair Lemma
• Cellular attachments with finite boundary support form a CW complex Lemma
• Cellular mapping cylinders and relative cylinders are CW complexes Lemma
• Compact CW images have finite cell support without choice Lemma
• CW quotients and collapse of a contractible subcomplex Lemma
• Finite dimensional axiomatic homology has finite subcomplex support Lemma
• Finite relative homotopy lifting across a weak equivalence Lemma
• High relative cells do not change lower homotopy Lemma
• Homotopy excision for a single relative cell layer Lemma
• The Davis complex as a CW complex: disk cells and the Cayley skeleta Lemma
• Vanishing relative homotopy extends an inverse over cells Lemma
• A CW complex is the colimit of its skeleta in the weak topology Proposition
• A fibration has path lifting and homotopy lifting relative to a subspace Proposition
• Cellular maps induce cellular chain maps Proposition
• CW skeleta are closed and cells form a disjoint partition Proposition
• Euler characteristic is additive for finite CW pairs Proposition
• Relative CW inclusions are cofibrations Proposition
• Cellular approximation for maps of CW pairs Theorem
• Relative cellular homology computes relative singular homology Theorem
• Relative homology of consecutive CW skeleta Theorem
• The Davis complex is simply connected Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Peter May, A Concise Course in Algebraic Topology, Chapter 10 (standard reference, not scraped)