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Integral homology comparison gives finite-range homotopy comparison for simply connected CW complexes

Statement

Assume AC. Let N≥2 and let f:(X,x₀)→(Y,y₀) be a based continuous map between nonempty path-connected simply connected CW complexes. Suppose f_:H_i(X;Z)→H_i(Y;Z) is an isomorphism for 0≤i<N and a surjection for i=N. Then f_:π_i(X,x₀)→π_i(Y,y₀) is an isomorphism for 1≤i<N and a surjection for i=N. No conclusion about π_{N+1} or homotopy equivalence of the spaces is included.

Facts & Assumptions

Given: AC; an integer $N \geq 2$ ; nonempty path-connected simply connected CW complexes $X, Y$ with based continuous map $f : (X, x_{0}) \to (Y, y_{0})$ ; and $f_{*} : H_{i} (X; Z) \to H_{i} (Y; Z)$ an isomorphism for $0 \leq i < N$ and a surjection for $i = N$ .

[F1]

A based map of CW complexes is homotopic to a cellular map, and mapping cylinders of cellular maps of CW complexes are CW complexes with the source as a subcomplex and the target a deformation retract (Cellular approximation for maps of CW pairs, Cellular mapping cylinders and relative cylinders are CW complexes).

[F2]

The singular homology of a pair is related by a long exact sequence, and homotopic maps induce the same homology map (Long exact sequence of a pair, Homotopic maps induce the same map on singular homology).

[F3]

The relative homotopy groups of a pair fit in a long exact sequence, and relative Hurewicz identifies the first nonzero relative homotopy group of an $(m - 1)$ -connected pair with the first nonzero relative integral homology when the pair is simply connected in the appropriate sense (Long exact sequence of relative homotopy groups, Relative Hurewicz theorem in the simple-connectivity range); based homotopy groups transport along homotopy tracks (Higher homotopy basepoint transport and moving homotopies).

[F4]

AC underlies the CW approximation and cellular-approximation selections used to put the map in cellular form (The Axiom of Choice).

Proof

technique · direct

1.1givenF1F2

Use cellular approximation to replace $f$ by a cellular map $g$ through a homotopy. If the homotopy moves the basepoint, its track gives the canonical target basepoint-transport isomorphism, so both the hypotheses and conclusions transfer between $f$ and $g$ . Form the CW mapping cylinder $M_{g}$ , with source inclusion $j : X ↪ M_{g}$ and target deformation retraction $r : M_{g} \to Y$ , so $r j = g$ . The homology exact sequence contains $H_{i} (X) \overset{j_{*}}{\to} H_{i} (M_{g}) ⟶ H_{i} (M_{g}, X) ⟶ H_{i - 1} (X) \overset{j_{*}}{\to} H_{i - 1} (M_{g}) .$ For $1 \leq i \leq N$ , the first arrow is surjective and the last injective. Exactness therefore forces $H_{i} (M_{g}, X) = 0$ . The component condition gives $H_{0} (M_{g}, X) = 0$ as well.

2.1step 1.1F3

Both $X$ and $M_{g}$ are path connected and simply connected. The relative homotopy exact sequence consequently gives $π_{1} (M_{g}, X) = *$ : every relative path has its initial endpoint connected to the basepoint inside $X$ , and the resulting based loop is null in $M_{g}$ . Induct on $m = 2, \dots, N$ . If the relative groups below $m$ vanish, the CW pair $(M_{g}, X)$ is $(m - 1)$ -connected, and $X$ is nonempty simply connected. Relative Hurewicz identifies $π_{m} (M_{g}, X)$ with $H_{m} (M_{g}, X) = 0$ . This proves all relative groups through $N$ vanish.

3.1step 2.1F3F4

For $2 \leq i < N$ , the two adjacent relative groups in $π_{i + 1} (M_{g}, X) ⟶ π_{i} (X) \overset{j_{*}}{\to} π_{i} (M_{g}) ⟶ π_{i} (M_{g}, X)$ vanish, so $j_{*}$ is injective and surjective. At $i = N$ , vanishing of the last term gives surjectivity. At $i = 1$ both absolute groups are trivial. Composition with $r_{*}$ and the basepoint-transport isomorphism proves the claims for $f$ . Every Hurewicz invocation is within its stated simple-connectivity range.

4.1step 3.1∎

Endpoint justification. Homology isomorphisms through degree $L$ imply homotopy isomorphisms only through degree $L - 1$ by this argument: apply the lemma with $N = L$ . To deduce an isomorphism at $L$ , one also needs homology surjectivity at $L + 1$ . This loss of one degree is essential to this proof, since injectivity at $L$ uses $π_{L + 1} (M_{g}, X)$ .

Depends on

Used by

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Sources

Allen Hatcher, Algebraic Topology, Chapter 4 (standard reference, not scraped)