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The prism operator for a path homotopy

Example

Let $H : I \times I \to X$ be a path homotopy rel ${0, 1}$ from $α$ to $β$ . For the identity $1$ -simplex $ι_{1} : Δ^{1} \to I$ , the prism operator is the sum of the two oriented triangles cutting the square $I \times I$ along its main diagonal. If $c_{0}$ and $c_{1}$ are the constant singular $1$ -simplices at the two common endpoints, then its boundary in the unnormalized singular chain complex is $\partial P_{H} (ι_{1}) = β - α - c_{1} + c_{0} .$ The degenerate side terms vanish only after passing to normalized singular chains.

Facts & Assumptions

Given: A path homotopy $H : I \times I \to X$ rel ${0, 1}$ from $α$ to $β$ .

[L1]

A path homotopy rel ${0, 1}$ keeps the two endpoint tracks constant (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints).

[L2]

The prism operator is built by triangulating $Δ^{1} \times I$ into two oriented $2$ -simplices (The prism operator of a homotopy).

[L3]

The prism operator satisfies $g_{#} - f_{#} = \partial P_{H} + P_{H} \partial$ (The singular chain homotopy formula).

Verification

technique · direct

1.1

By [L2], $P_{H} (ι_{1})$ is the sum of the two oriented triangles obtained from the standard triangulation of the square $I \times I$ . Applying [L3] to the singular simplex $ι_{1}$ gives $\partial P_{H} (ι_{1}) = β - α - P_{H} (\partial ι_{1}) .$

L2L3given

2.1

The chain $\partial ι_{1}$ is the terminal vertex minus the initial vertex. By [L1], these two vertices trace the constant singular $1$ -simplices $c_{1}$ and $c_{0}$ , respectively, so $P_{H} (\partial ι_{1}) = c_{1} - c_{0} .$ Substitution into step 1.1 gives $\partial P_{H} (ι_{1}) = β - α - c_{1} + c_{0} .$ Constant singular $1$ -simplices are genuine generators here, so they cannot be discarded in the unnormalized complex.

L1step 1.1algebra∎

Depends on

Used by

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Sources

Allen Hatcher, Algebraic Topology (standard reference, not scraped)
Haynes Miller, Algebraic Topology I, Lecture 6 (standard reference, not scraped)