Problem 1F — Is there any convenient topological characteristic of the function $f_{-\varepsilon}$ which allows to…

Definitions and mathematical statement

For a real smooth function on an $n$-manifold, the Morse index of a nondegenerate critical point is the number of negative eigenvalues of its Hessian. The supplied registry explains the otherwise unspecified phrase "these indices" as the integer Morse indices of two critical points created in a Morse birth surgery.

In that intended setting, let $f_t$ be the relevant one-parameter deformation, with no newborn pair at $t=-\varepsilon$ and the pair $p_+,q_+$ at $t=+\varepsilon$, for sufficiently small $\varepsilon>0$. At a generic birth their indices are adjacent integers $k,k+1$. Vassiliev's Problem 1F asks whether a convenient topological characteristic of the pre-birth function $f_{-\varepsilon}$ can predict these integers, beyond the parity information reported as already available.

An eventual answer would specify allowable topological data $T(f_{-\varepsilon})$ and a rule recovering the unordered pair \[ \{\operatorname{ind}_{p_+}(f_{+\varepsilon}), \operatorname{ind}_{q_+}(f_{+\varepsilon})\} \] for the admissible birth surgeries. The excerpt does not give the ambient dimension, function/deformation class, local domain and boundary data, the birth location or other information retained, or the meaning of topological equivalence of pre-birth functions. Accordingly no assertion is made that an unspecified global topological type alone determines the indices, nor is a particular characteristic substituted for the requested one.

Short English statement

Can topological data available before a Morse birth predict the exact integer indices of the two newborn critical points, rather than only their parity? The permitted data and family hypotheses are missing from the excerpt.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10800006$, exact source code catalogue entry, attributed to Victor Vassiliev.
• Vassiliev, A Few Problems on Monodromy and Discriminants, Problem 1F: https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/15-11/. Supplied access date: 2026-07-29; extraction: html-structure; eighteen labeled problem environments.
• Supplied background reference: Vassiliev, Applied Picard--Lefschetz Theory, AMS, 2002, for the parity prediction in real Morsification theory. Supplied computational-work identifiers: https://arxiv.org/abs/2109.12287 and https://arxiv.org/abs/2512.12738; titles and exact relevant results are not supplied.