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Finite-range rational homology vanishing implies rational homotopy vanishing

Statement

Assume AC. If Y is simply connected and H_j(Y;Q)=0 for 0<j≤D, then π_j(Y)⊗Q=0 for 2≤j≤D.

Facts & Assumptions

Given: AC; a simply connected space $Y$ and $D \geq 2$ with $H_{j} (Y; Q) = 0$ for $0 < j \leq D$ .

[F1]

The rational first-Hurewicz-after-killing-lower-torsion-homotopy lemma identifies $π_{j} (Y) \otimes Q$ with $H_{j} (Y; Q)$ for $2 \leq j \leq D$ when the lower homotopy groups are torsion, and rationalization is exact (First rational Hurewicz after killing lower torsion homotopy, Rationalization is exact and commutes with singular homology).

[F2]

Torsion groups have vanishing rationalization by [F1]; AC is inherited from the rationalization and rational first-Hurewicz suppliers, including their Eilenberg–Mac Lane and representing-map choices. The weak CW approximation itself is choice-free (The Axiom of Choice).

Proof

technique · direct

1.1givenF1

Induct on j. For j=2, the rational first-Hurewicz lemma identifies rational homotopy with the zero rational homology. At the next j, all previous homotopy groups are torsion by the rationalization lemma, so the rational first-Hurewicz lemma again applies.

2.1step 1.1F1F2∎

Finite induction proves the assertion. This implication is valid for arbitrary simply connected spaces because the rational first-Hurewicz lemma includes their weak CW replacement.

Depends on

Used by

Rational sphere homotopy below the first unstable degree Lemma

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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