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Freudenthal identifies the eventual suspension system for spheres

Statement

For integers $k \geq 0$ and $n \geq 1$ , the sphere-system bonding map

$E : π_{n + k} (S^{n}) ⟶ π_{n + k + 1} (S^{n + 1})$

is an isomorphism when $n > k + 1$ and is a surjection when $n = k + 1$ .

Facts & Assumptions

[F1]

Every based map $S^{j} \to S^{n}$ with $0 \leq j < n$ is based nullhomotopic, including the path-connectedness assertion at $j = 0$ (Lower-dimensional sphere maps are based nullhomotopic). With the standard two-cell CW structure, $S^{n}$ is therefore $(n - 1)$ -connected for $n \geq 1$ .

[F2]

For $n \geq 1$ , Freudenthal says that suspension $π_{i} (X) \to π_{i + 1} (Σ X)$ is an isomorphism for $1 \leq i < 2 n - 1$ and a surjection for $i = 2 n - 1$ when $X$ is an $(n - 1)$ -connected based CW complex. In degree zero it instead gives a bijection of the two singleton pointed sets (Freudenthal suspension theorem).

[F3]

The stable-stem definition uses the reduced-suspension bonding map determined by $S^{1} \land S^{n} ≅ S^{n + 1}$ (Stable stems of the sphere).

[F4]

Collapsing a nonempty contractible CW subcomplex is a weak homotopy equivalence (CW quotients and collapse of a contractible subcomplex).

Proof

Given: Integers $k \geq 0$ and $n \geq 1$ .

1.1F1F2

Apply [F2] to $X = S^{n}$ using [F1] and set $i = n + k$ . The isomorphism inequality becomes [F1, F2] $n + k < 2 n - 1 ⟺ n > k + 1.$

1.2F2

The endpoint equation $n + k = 2 n - 1$ is exactly $n = k + 1$ , so [F2] gives the claimed surjection there and makes no injectivity claim.

2.1F3F4step 1.1step 1.2

The quotient from the two-cone suspension to the reduced suspension collapses precisely the basepoint track, a nonempty contractible CW subcomplex. By [F4] it induces an isomorphism on the displayed homotopy group, while [F3] identifies the resulting reduced-suspension map with the sphere-system bonding map. Thus the computed range applies to that system itself. $□$

Depends on

Dependency tree · 2 levels

Sources

J. P. May, A Concise Course in Algebraic Topology