Expository Telescope-Conjecture Fragment

Definitions and mathematical statement

The supplied record is an expository fragment, not a recoverable standalone problem. No definite conjecture or question can be stated from it.

Let $\mathrm{Sp}$ denote the category of spectra, and let $\Sigma$ denote suspension. For a spectrum $E$, the fragment denotes Bousfield localization with respect to $E$ by \[ L_E\colon \mathrm{Sp}\longrightarrow\mathrm{Sp}, \] and its image category by $L_E\mathrm{Sp}$. It specializes to a spectrum denoted $K(n)$; write $K(n)_*(X)$ for the associated graded homology of a spectrum $X$. The excerpt does not define $K(n)$, the spectra $V(n)$, or the elements $v_n$.

With $d_n=|v_n|$ denoting the unspecified suspension degree, the corrupted display appears to describe a cofiber sequence \[ \Sigma^{d_n}V(n-1)\xrightarrow{w_n}V(n-1)\longrightarrow V(n). \] This is a tentative reconstruction of the extracted notation. The fragment gives the parameter restrictions \[ 0\leq n\leq 3,\qquad p\geq 2n+1, \] but the placement of this qualification is disrupted. Its precise scope, the domain of $p$, and the meaning of $V(-1)$ at $n=0$ are not specified. In particular, a prime domain for $p$ must not be silently imposed.

The fragment asserts that the induced map \[ K(n)_*(w_n)\colon K(n)_*\left(\Sigma^{d_n}V(n-1)\right) \longrightarrow K(n)_*\left(V(n-1)\right) \] is an isomorphism, that all iterates of $w_n$ are essential, and that the telescope \[ \operatorname*{hocolim}\left( V(n-1) \xrightarrow{\Sigma^{-d_n}w_n}\Sigma^{-d_n}V(n-1) \xrightarrow{\Sigma^{-2d_n}w_n}\Sigma^{-2d_n}V(n-1) \longrightarrow\cdots \right) \] is noncontractible. Here essential means not nullhomotopic, and the telescope maps are interpreted as the indicated desuspensions of $w_n$. These are supplied expository assertions, not an independently verified theorem statement. The excerpt does not explicitly supply the nonvanishing hypothesis on $K(n)_*(V(n-1))$ needed to justify its stated inference from an isomorphism to essentiality and noncontractibility.

The excerpt also qualitatively contrasts computing $\pi_*L_{K(2)}V(1)$ with computing $\pi_*V(1)$; it supplies neither a computation nor a precise computability claim. Despite the heading concerning the telescope conjecture, no formulation of that conjecture or requested mathematical conclusion is present.

Short English statement

This record contains damaged background exposition about localization, a proposed cofiber sequence, and its telescope. It does not specify a standalone problem. Its parameter conventions and some hypotheses remain unresolved, so the displayed reconstruction requires human review.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $30006094$, title "Expository Telescope-Conjecture Fragment," source code catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2024.
• Supplied report attribution: Arbeitsgemeinschaft: Algebraic $K$-Theory and the Telescope Conjecture (2025), Oberwolfach Reports, pp. 2751--2804, DOI 10.4171/owr/2024/47. Supplied source link: https://doi.org/10.4171/owr/2024/47. The citation year 2025 and the registry's 2024 year designation are retained without reconciliation.
• Supplied literature source: "Oberwolfach Report 47," https://doi.org/10.4171/OWR/2024/47.
• Supplied literature source: "Hopkins and Smith, Periodicity theorem," https://doi.org/10.2307/120991.
• Literature assessment checked at 2026-08-22: this is supplied dated curation, not independently verified. The supplied status label is "open," while the assessment says that the incomplete cofiber-sequence fragment admits no determinate solved or disproved status.
• The supplied statement-verification note says that the official report identifies this extraction as background exposition and discusses the higher-height telescope conjecture as disproved. This is a supplied verification claim, not an independent source check here. The record is marked unrecoverable; no clean statement is supplied, and the original is retained for provenance. The record's update timestamp is 2026-09-16T12:06:53Z.