Weighted $q$-Boundaries for Presimplicial Modules

Definitions and mathematical statement

A presimplicial module consists of modules $C_n$, indexed by integers $n\geq 0$, over a coefficient ring $k$, together with $k$-linear face maps $d_i^{(n)}:C_n\to C_{n-1}$ for $n\geq 1$ and $0\leq i\leq n$, satisfying \[ d_i^{(n-1)}\circ d_j^{(n)}=d_{j-1}^{(n-1)}\circ d_i^{(n)} \qquad(n\geq 2,\ 0\leq i<j\leq n). \] Its alternating boundary in positive degrees is \[ \partial_n=\sum_{i=0}^{n}(-1)^i d_i^{(n)}. \] The source does not specify the coefficient ring, module-sidedness, or an augmentation in degree zero.

Let $q$ denote a formal indeterminate. The source proposes the expressions \[ C_n^q=C_n\otimes_k\mathbb Z[q],\qquad \partial_n^q=\sum_{i=0}^{n}q^i\left(d_i^{(n)}\otimes 1\right):C_n^q\longrightarrow C_{n-1}^q \quad(n\geq 1). \] These expressions are conditional on compatible coefficient structures making the tensor product, induced face maps, and multiplication by $q$ well-defined. Such structures are not specified in the record: in particular, $\mathbb Z[q]$ is not automatically a $k$-algebra for an arbitrary coefficient ring $k$. The displayed tensor product is retained exactly as supplied, without replacing it by a different scalar extension.

Exploratory problem. For a given presimplicial module, and subject to resolving the coefficient issue above, investigate whether the weighted operators $\partial_n^q$ yield a useful homological structure. The source explicitly allows the general failure of \[ \partial_{n-1}^q\circ\partial_n^q=0\qquad(n\geq 2). \] Thus ordinary chain-complex homology is not automatically defined by these operators. The original wording asks whether this construction is useful; the normalized wording asks what useful homological structure can be extracted. Neither specifies a proposed invariant, axioms, a class of modules, restrictions or specializations on $q$, or a criterion of usefulness. Consequently, the supplied problem is an exploratory research question rather than a fully specified proposition with a fixed conclusion.

Short English statement

Given a presimplicial module, investigate whether weighting its face maps by $1,q,q^2,\ldots$ produces a useful homological structure, even though the resulting operators generally do not square to zero. The coefficient extension $C_n\otimes_k\mathbb Z[q]$ requires unspecified compatibility data, and the intended meaning of a useful homological structure remains open to clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002598$, title Weighted $q$-Boundaries for Presimplicial Modules, collection Oberwolfach Reports Open Problems; exact source problem code catalogue entry. The record gives proposed year 2014 and identifies no individual proposer.
• Supplied report citation: Algebraic Structures in Low-Dimensional Topology (2015), Oberwolfach Reports, pp. 1403--1458, DOI 10.4171/owr/2014/26. https://doi.org/10.4171/owr/2014/26. The supplied citation's year 2015 differs from the report designation 26/2014 and proposed year 2014; this discrepancy is unresolved here.
• Supplied literature source: Oberwolfach Report 26/2014. https://publications.mfo.de/bitstream/handle/mfo/3416/OWR_2014_26.pdf?isAllowed=y&sequence=1.
• Supplied literature source: Przytycki, Progress in distributive homology. https://arxiv.org/abs/1406.6499. This reference does not identify Przytycki as the proposer of this particular problem.
• Supplied literature assessment, checked 2026-08-21: the registry labels the problem open, describes the report's question as exploratory, notes that the weighted map is not generically a chain differential, and reports that no authoritative source was found settling a useful general theory for the construction. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; the status and cited claims require independent verification.