fix: proofread Algebraic Topology II: Homology part - #334
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Fix typos, grammar, and small notational errors across the six homology
chapters (singular homology, long exact sequence, excision, cellular
homology, cohomology, cup product):
- singular.tex: "simplex in X", $H_1$ vs $H^1$, "we have $ab \neq ba$",
"to denote the trivial group".
- long-exact.tex: $\wt H_1$/$H_1$ subscripts, "is because", "gets sent to",
"does not find", "lets us obtain", $\oplus$ in Mayer-Vietoris read-off.
- excision.tex: "must contain", "everything is finite", "$1$-cycle",
"induce", "$\iota \colon$", base case "$n=0$".
- cellular.tex: "one can construct", "vanishes", "a $k$-cell",
"is the degree", "cells that form", "$(k-1)$-cell", "is actually",
"paths".
- cohomology.tex: "rather than", $A_n$ subscript, $f^\sharp$ dual map,
dropped duplicate $F_0'$, "There are so many", "groups are",
"every term", dimensions "$n=0,1,2$", "$1$st", $C_n(X)$, "we care".
- cup-product.tex: $\times \colon R \times R \to R$, "an anticommutative",
"$(m+n)$", $C_{m+n}$/$H_{m+n}$ subscripts, "This formalizes",
"we write", "better have", "give identical results", $\Cells_n(Y)$,
"are formed", "is clean", "should assign", "determine", "we declare",
closing $)^\vee$, "pairs of holes", "$\RP^3$", "computing it".
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01DokygnRwF3vavW8JDWermg
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Summary
Proofreads the Algebraic Topology II: Homology part (the first unchecked box in #315), covering the six chapters:
tex/homology/singular.textex/homology/long-exact.textex/homology/excision.textex/homology/cellular.textex/homology/cohomology.textex/homology/cup-product.texAll findings are minor: typos, grammar, and small notational slips. No mathematical arguments were rewritten.
Notable notational fixes
singular.tex: "we don't have$H^1(S^1)$→$H_1(S^1)$(homology subscript, not cohomology).long-exact.tex:$\wt H^1(S^1)$/$H^1(S^1)$→$\wt H_1$/$H_1$; in the punctured-$\RR^n$ solution,$\wt H_k(U) \cap \wt H_k(V)$→$\oplus$(Mayer–Vietoris gives a direct sum); "homotopy does not found" → "find".excision.tex: base case$\wt H_n(S^0)$ is $\ZZ$ for $m=0$→for $n=0$(variable mismatch); missing\colonin the excision inclusion.cohomology.tex:$\delta(A_n \to G) = A_{n+1} \to A \to G$→A_n(missing subscript); upward cochain map$f_\sharp$→$f^\sharp$; dropped a duplicated$F_0'$in a free resolution; torus cohomology "in dimensions$n=0,1,2$;$C_n(X;G)$→$C_n(X)$for the free abelian group on simplices.cup-product.tex:$\times \colon R \to R$→$R \times R \to R$; cross-product target$C_{m\times n}$/$H_{m\times n}$→$C_{m+n}$/$H_{m+n}$;$e^n \in \Cells_m(Y)$→$\Cells_n(Y)$; missing closing)^\veein the wedge-product definition;$RP^3$→$\RP^3$.The remaining changes are straightforward subject–verb agreement, article ("a"/"an"), and pluralization fixes.
Part of the proofreading campaign (#315).
🤖 Generated with Claude Code
https://claude.ai/code/session_01DokygnRwF3vavW8JDWermg
Generated by Claude Code