bibkey: hatcher2002algebraic
authors: Allen Hatcher
year: 2002
title: Algebraic Topology
doi: null
url: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
claim: "The alternating simplex boundary has square zero and defines the simplicial chain complex. The ordinary quotient join satisfies the classical sphere identity S^m * S^n homeomorphic to S^(m+n+1)."
strata_touched: []
license: citation-only
triage: anchor

Simplicial boundary convention

The author-hosted book, Section 2.1, pp. 105–106, defines the alternating deletion boundary and proves its square is zero in Lemma 2.1. These are PDF pages 114–115. The displayed complex there is unaugmented at degree zero. The consumer explicitly adds the empty-face line in degree minus one and sends each vertex to its unit; the edge boundary still has augmentation zero.

This supplies the chain convention in the five-face projection, Definition 22.1 and Theorem 22.2. Exterior degree is one more than the chain degree in that chosen basis dictionary. The citation supplies neither a chain retraction onto the sparse support nor a homology interpretation of a probability-observation kernel.

Ordinary joins and the sphere identity

Locator

Source: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf

Chapter 0, “Operations on Spaces,” printed p. 9 (PDF page 18), defines the ordinary join $X\ast Y$ as the quotient of $X\times Y\times I$ by $(x,y_1,0)\sim (x,y_2,0)$ and $(x_1,y,1)\sim (x_2,y,1)$. Thus the zero end retains $X$ and the one end retains $Y$. The same page describes the iterated join of $n$ copies of the two-point sphere $S^0$, for $n\ge 1$, using the two unit vectors on each coordinate axis, and gives a homeomorphism to $S^{n-1}$ by radial projection from the origin.

Chapter 0, Exercise 18, printed p. 19 (PDF page 28), asks to show $S^1\ast S^1=S^3$ and, more generally, $S^m\ast S^n=S^{m+n+1}$. This is the literature attribution for the classical homeomorphism identity, stated as an exercise rather than a supplied proof.

The Euclidean sphere join consumer uses this ordinary quotient convention and realizes the identity for every natural pair of dimensions, including zero, with actual Euclidean unit spheres and their subspace topologies. Its explicit intermediate maximum-norm normalization, coordinate concatenation, and selected first-coordinate endpoint formula are local refinements. These locators do not prescribe those formulas or certify an information-escape registration, and do not resolve Carnero Bravo Question 22.