Secondary K-Theory Comparison Map

Definitions and mathematical statement

The supplied record concerns Lott's secondary K-group and uses the notation \[ G(M):=KR^0(M),\qquad D(M):=KR_d^0(M),\qquad T(M):=KR_{\mathbb{R}/\mathbb{Z}}^{-1}(M). \] These are labels for the groups appearing in the source record, not independent definitions of them. The record does not specify the coefficient object $R$, the class of admissible spaces $M$, or the constructions defining these groups. It describes generators of $G(M)$ by triples $(V,g_V,\eta)$, but does not supply their admissibility conditions or the defining relations. The operations $\operatorname{cycl}$ and $a$ below likewise retain their source notation; their domains and definitions are not supplied.

The extracted assertion is that there is a well-defined group homomorphism \[ \Phi_M:G(M)\longrightarrow D(M), \qquad D(M)\cong T(M), \] whose value on every admissible generator is \[ \Phi_M\left([(V,g_V,\eta)]\right) =\operatorname{cycl}(V,g_V)-a(\eta). \] Here brackets denote the class of the indicated generator in the secondary K-group. Well-definedness requires that the displayed expression belong to $D(M)$ and respect every defining relation of $G(M)$. The supplied record does not provide the hypotheses or relations needed to state those requirements explicitly, nor does it explain the asserted identification $D(M)\cong T(M)$.

The supplied statement-verification field says that this extracted assertion is a construction and result proved in the cited report, rather than an unresolved question; it marks the statement as unrecoverable and supplies no clean replacement. Separately, the supplied dated literature assessment describes an existence part of Lott's conjecture as settled and a transfer-compatibility part as unproved. No transfer maps, compatibility equation, or hypotheses for that remaining part are supplied. Consequently, the displayed assertion preserves the extracted comparison-map claim, but the supplied evidence does not determine a precise remaining open problem.

Short English statement

The extracted claim sends each generator $(V,g_V,\eta)$ of Lott's secondary K-group to $\operatorname{cycl}(V,g_V)-a(\eta)$ in the indicated differential algebraic K-theory group. The supplied verification calls this a proved result. Its separate dated assessment mentions unresolved transfer compatibility, but gives insufficient definitions and hypotheses to formulate that question precisely.

Sources

• Local UnsolvedMath registry, permanent ID $30002362$, title Secondary K-Theory Comparison Map, collection Oberwolfach Reports Open Problems, exact source problem code catalogue entry. The record attributes the conjecture and secondary K-group to Lott; no individual proposer is supplied. Its proposed-year field is $2013$.
• Supplied report citation: Algebraic K-theory and Motivic Cohomology (2014), Oberwolfach Reports, pp. 1861--1913, DOI 10.4171/owr/2013/32. https://doi.org/10.4171/owr/2013/32. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "Oberwolfach report on Lott's conjecture": https://ems.press/content/serial-article-files/46461?nt=1.
• Supplied literature source, labeled "Bunke and Gepner, differential K-theory transfer": https://arxiv.org/abs/1306.0247.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the differential algebraic K-theory theorem settles the existence part of Lott's conjecture, while transfer compatibility remains unproved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The supplied statement-verification field reports a check against the official Oberwolfach report and classifies the extracted sentence as a proved construction/result. It supplies no separate clean statement and retains the original record for provenance. This verification report is reproduced as supplied metadata, not as a check performed for this draft.