LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Singular complexes preserve the sequential group convention

PDF p. 3: Filtered S1 spectra: stable enhancement prerequisites

Statement

Let $E$ be a sequential prespectrum of well-pointed based compactly generated weak Hausdorff spaces with structure maps $σ_{n} : S^{1} \land E_{n} \to E_{n + 1}$ , sphere first. Let $K = Δ^{1} / \partial Δ^{1}$ , with $∣ K ∣$ identified with $S^{1}$ in the chosen orientation, and put $X_{n} = Sing E_{n}$ with its constant basepoint simplex. Give $X$ the structure map $K \land X_{n} \to X_{n + 1}$ adjoint to $∣ K \land X_{n} ∣ ≅ S^{1} \land ∣ X_{n} ∣ \to S^{1} \land E_{n} \to E_{n + 1}$ , where the middle map uses evaluation. Then every $X_{n}$ is Kan and $G_{k} (X) ≅ π_{k} (E)$ naturally for every integer $k$ , with the published sphere-first bonding convention.

Facts & Assumptions

Given: $E$ , $K$ and the chosen oriented realization as in the Statement. The group $G_{k}$ is the sequential colimit of the tail-invariance supplier.

Goal: Construct the simplicial prespectrum and compare its stable groups to the published groups.

[F1]

The singular-realization input gives the natural evaluation weak equivalence and pointed realization–singular adjunction. External singular realization comparison

[F2]

The sequential definitions fix sphere-first structure maps and bonding maps. Sequential prespectra, spectra, and adjoint structure maps, Stable homotopy groups of a sequential prespectrum

[F3]

Tail invariance gives cutoff independence and passage of compatible level isomorphisms to colimits. Cofinal tails and level invariance of stable groups

[F4]

The direct sphere-comparison lemma identifies the simplicial groups with topological groups by evaluated sphere representatives and verifies the ordered suspension formula. Singular sphere classes and ordered suspension

Proof

1.1F1F2given

The adjunction defines the displayed structure map. A horn in $Sing E_{n}$ is a continuous map from a realized topological horn to $E_{n}$ ; the simplex retracts continuously onto that horn, so the map extends and $X_{n}$ is Kan. Naturality of evaluation and the defining adjunction make the square relating the realized structure map to $σ_{n}$ commute.

2.1F1F2F4step 1.1

For $d \geq 1$ , use [F4] to identify $π_{d}^{s i m p} (X_{n}) ≅ π_{d} (E_{n})$ , not just the homotopy groups of $∣ X_{n} ∣$ . For a sphere representative $a$ , the evaluated simplicial bonding map is $σ_{n} \circ (1_{S^{1}} \land (ϵ_{n} ∣ a ∣))$ by [F4]. This is exactly the bonding representative of [F2] with the same sphere-coordinate identification. Thus the group isomorphisms commute with the actual bonding maps, without permuting coordinates.

3.1F2F3step 2.1∎

For each integer $k$ , choose a tail with $n + k \geq 2$ . Step 2.1 gives compatible isomorphisms of its stage groups. Passing to colimits by [F3] gives the asserted natural isomorphism $G_{k} (X) ≅ π_{k} (E)$ for every integer $k$ .

Depends on

Used by · 1

Stable spectra and exact triangles

Dependency tree · 3 levels

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