Homology Growth in Residual Towers

Definitions and mathematical statement

For a prime number $p$ and a homological degree $k\in{\mathbb{Z}}_{\geq 0}$, define the mod-$p$ Betti number of a space $X$ by \[ b_k(X;{\mathbb{F}}_p)=\dim_{{\mathbb{F}}_p}H_k(X;{\mathbb{F}}_p), \qquad {\mathbb{F}}_p={\mathbb{Z}}/p{\mathbb{Z}}. \] For an abelian group $A$, its torsion subgroup is \[ \operatorname{tors}A=\{a\in A:\text{there exists }n\in{\mathbb{Z}}_{>0}\text{ with }na=0\}. \] When this subgroup is finite, write \[ t_k(X)=\left|\operatorname{tors}H_k(X;{\mathbb{Z}})\right|. \] The trivial torsion subgroup has order $1$. Finitely generated integral homology suffices for this torsion order to be finite; finite-dimensional mod-$p$ homology is needed for the Betti number to be a finite integer. The supplied record does not state hypotheses ensuring these finiteness properties.

A tower of finite covers consists of compatible finite-sheeted covering maps \[ \cdots\longrightarrow M_{i+1}\longrightarrow M_i\longrightarrow\cdots\longrightarrow M_0=M. \] In a connected, based setting where covering-space classification applies, a common meaning of residual is that the associated finite-index subgroups $\Gamma_i\leq\Gamma=\pi_1(M)$ satisfy \[ \Gamma_{i+1}\leq\Gamma_i, \qquad \bigcap_{i\geq 0}\Gamma_i=\{1\}. \] Some conventions additionally require each $\Gamma_i$ to be normal in $\Gamma$. The supplied record does not specify its convention, so normality is not imposed here as an established source hypothesis.

Extracted question, with unresolved scope. For a residual tower of finite covers $M_i\to M$, determine how the sequences \[ b_k(M_i;{\mathbb{F}}_p) \qquad\text{and}\qquad t_k(M_i) \] grow as $i\to\infty$, whenever these quantities are finite. These are the two measures of homology size named in the record; no comparison between them is asserted. The source does not specify the class of base spaces $M$, compactness or boundary conditions, restrictions on $k$ or $p$, or whether growth is to be measured directly, relative to covering degree, or through a logarithmic normalization. Consequently this is a research question with unresolved hypotheses and target asymptotics, rather than a fully specified conjectural formula.

Short English statement

The question asks how mod-$p$ Betti numbers and orders of integral homology torsion grow along a residual tower of finite covers. The permitted base spaces, precise tower convention, finiteness hypotheses, and intended growth normalization are unspecified. The supplied registry labels the question open; this statement-only draft requires human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record for permanent ID $30002765$, title Homology Growth in Residual Towers, exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2015; no individual proposer is identified.
• Supplied report citation: Geometric Topology (2015), Oberwolfach Reports, pp. 187--233, DOI 10.4171/owr/2015/3. https://doi.org/10.4171/owr/2015/3. The record describes this as workshop-report provenance; it supplies no exact page or passage for the extracted question.
• Supplied literature source: Sauer, Geometry and Topology. https://doi.org/10.2140/gt.2016.20.1035.
• Supplied literature source: Lueck, approximation survey. https://arxiv.org/abs/1501.07446.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open and reports important upper bounds and special-case results, but no general growth law resolving the extracted question. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. These are supplied editorial assertions, not independent verification of the report passage, mathematical completeness, or current literature status.