Averaged Morse Numbers and Lefschetz Theory

Definitions and mathematical statement

The supplied record asks three questions concerning Bagchi's lower-bound conjecture for the $g$-vector, the $g$-conjecture, and invariants denoted by $\mu_i$, described as averaged discrete Morse invariants. It does not supply the definitions of these invariants, their normalization or coefficient field, the admissible class of spaces or simplicial complexes, or the precise versions of the two conjectures. These omissions remain unresolved; consequently, the questions below cannot yet be expressed as a fully specified universally quantified conjecture.

For a finitely generated group $G$, define its minimum number of generators by \[ d(G)=\min\left\{|S|: S\subseteq G,\ S\text{ is finite},\ \langle S\rangle=G\right\}. \] For a path-connected space $X$ with basepoint $x_0$ and finitely generated fundamental group, $d(\pi_1(X,x_0))$ is independent of the choice of basepoint.

1. Determine the relationship between Bagchi's lower-bound conjecture for the $g$-vector and the $g$-conjecture. The source does not specify the conjectural inequalities, hypotheses, or intended meaning of "relationship," so no particular implication or equivalence is asserted here.
2. Determine the relationship between the invariants $\mu_i$ and discrete Morse theory. The intended indices, admissible discrete Morse functions, and type of comparison are not specified in the supplied record.
3. Determine whether $\mu_1$ is a lower bound for the minimum number of generators of the fundamental group. For an admissible path-connected object $X$ with finitely generated fundamental group, the proposed lower-bound direction is \[ \mu_1(X)\leq d\left(\pi_1(X,x_0)\right). \] If $\mathcal C$ denotes the intended class of such objects, this asks whether the displayed inequality holds for every $X\in\mathcal C$. Both $\mathcal C$ and the definition of $\mu_1$ remain unspecified. This conditional transcription does not claim that the inequality is intended for all path-connected spaces or all finite simplicial complexes. The treatment of disconnected objects, if any, also remains unspecified.

Short English statement

The problem asks how Bagchi's lower-bound conjecture relates to the $g$-conjecture, how the averaged invariants $\mu_i$ relate to discrete Morse theory, and whether $\mu_1$ is at most the minimum number of generators of the fundamental group. The supplied record leaves the invariant definitions, conjecture versions, and domain unresolved. Its open status is a supplied dated assessment, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30002781$, exact source problem code catalogue entry, title "Averaged Morse Numbers and Lefschetz Theory," in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year 2015 and names no proposer. Bagchi is associated with the conjecture mentioned in the question; this does not establish attribution of the question itself.
• Supplied report citation: Geometric and Algebraic Combinatorics (2015), Oberwolfach Reports, pp. 285--368, DOI 10.4171/owr/2015/5. https://doi.org/10.4171/owr/2015/5. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No precise page or passage within this report is supplied.
• Supplied literature reference, labelled "Bagchi, the mu vector and generalized lower bounds": https://doi.org/10.1016/j.ejc.2015.04.007. The record does not reproduce the relevant definitions or conjecture statement from this reference.
• Supplied literature reference, labelled "original OWR report": https://doi.org/10.4171/owr/2015/5.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that known Morse inequalities and low-dimensional results do not resolve these broad relationship and lower-bound questions in general. This is supplied dated curation derived from the OWR record, not independently verified here; time-sensitive status and cited mathematical claims require independent verification.
• The registry labels its statement corrected_verified and reports that the official Oberwolfach passage was checked directly and that the question, definitions, and notation were restored without strengthening the claim. This is a supplied verification assertion, not a source visit or mathematical verification performed for this draft.