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Reduced suspension pinch map

Definition

Let $(Z, z_{0})$ be a based topological space. Define its reduced suspension as $Σ Z = (Z \times [0, 1]) / (Z \times {0, 1} \cup {z_{0}} \times [0, 1])$ , with the collapsed subset as basepoint. Let $ι_{1}, ι_{2} : Σ Z \to Σ Z \lor Σ Z$ be the wedge inclusions. The reduced suspension pinch map $p_{Z} : Σ Z \to Σ Z \lor Σ Z$ is induced by $p_{Z} ([z, t]) = {\begin{cases} ι_{1} ([z, 2 t]), & 0 \leq t \leq \frac{1}{2}, \\ ι_{2} ([z, 2 t - 1]), & \frac{1}{2} \leq t \leq 1. \end{cases}$ At $t = \frac{1}{2}$ both values are the wedge basepoint.

Definition references

Hatcher, Algebraic Topology, Appendix §4.L

May, A Concise Course in Algebraic Topology, Chapter 7

Facts & Assumptions

Given: The based space and reduced suspension quotient in the Definition.

[F1]

The cited supplier gives the reduced-suspension quotient formula under its well-pointed compactly generated weak Hausdorff hypotheses. For arbitrary based spaces in this item, use the quotient defined above with ordinary product and quotient topologies; the wedge is the quotient of two disjoint copies of $Σ Z$ identifying their basepoints. Its inclusions are continuous by the quotient construction. The verification below establishes the pinch map directly without the supplier's additional hypotheses. Reduced cone suspension and cofiber sequence

[F2]

A continuous map constant on the equivalence classes of a quotient descends uniquely to a continuous map on the quotient. For a quotient map $q : X \to Y$ , a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$ ; and a composite of quotient maps is a quotient map

Verification

1.1F1givenconstruct

On the closed subspaces $Z \times [0, \frac{1}{2}]$ and $Z \times [\frac{1}{2}, 1]$ , the two formulas in the Definition are continuous. They agree at $t = \frac{1}{2}$ , since each suspension endpoint is collapsed, and they send the basepoint track and both outer end slices to the wedge basepoint. For any closed subset $C$ of the wedge, the preimage under each restricted formula is closed in its closed half, hence closed in $Z \times [0, 1]$ ; their finite union is closed. Thus the pasted map is continuous and constant on every equivalence class of the reduced suspension quotient.

2.1F1F2step 1.1algebra∎

By [F2], the map descends uniquely and continuously to $p_{Z} : Σ Z \to Σ Z \lor Σ Z$ . On the first half it is the first wedge inclusion with linear rescaling of the suspension coordinate, and on the second half it is the second inclusion with linear rescaling; the midpoint maps to the common basepoint. Thus the displayed formula is well-defined at all endpoints and defines the reduced suspension pinch map.

Reduced suspension pinch map

Definition

Definition references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources