Topological Morse Functions

• Topological Morse functions are continuous functions defined via homeomorphism-based local normal forms, eliminating the need for derivatives.
• They extend classical Morse theory by capturing topology changes through handle attachments and controlling homological shifts in nonsmooth contexts.
• Min-type constructions and distance functions offer practical methods to analyze topology in settings like moduli spaces and persistent homology.

Topological Morse functions are continuous functions whose local behavior is specified up to homeomorphism rather than up to smooth change of coordinates. In the classical formulation used by Morse and in later treatments, a point is regular when the function is locally a coordinate projection, and a critical point is non-degenerate of index $m$ when the function is locally topologically conjugate to a quadratic form with $m$ negative squares. The framework retains Morse-theoretic control of topology while allowing genuinely nonsmooth examples, including distance functions on Voronoi skeletons, Min-type functions built from convex pieces, and continuous energy functions associated with topological flows (Arnal, 2024, Irmer, 5 Aug 2025, Medvedev et al., 2019).

1. Definition and local normal form

Let $U \subset \mathbb{R}^n$ be open and $f:U\to\mathbb{R}$ continuous. A point $z\in U$ is topologically regular if there exist neighborhoods $V_1$ of $0$ in $\mathbb{R}^n$ and $V_2$ of $z$ in $m$0, together with a homeomorphism $m$1 satisfying $m$2 and

$m$3

A point is topologically critical if it is not topologically regular. A topological critical point is non-degenerate of index $m$4 if, in suitable neighborhoods and topological coordinates,

$m$5

A function is a topological Morse function if all of its topological critical points are non-degenerate in this sense, and the index is the number of negative squares (Arnal, 2024).

On an $m$6-dimensional topological manifold, an equivalent local model is written as

$m$7

with index $m$8. The defining feature is that the local normal form is imposed by a homeomorphism, not by a $m$9 coordinate change. Accordingly, no derivative, gradient, or Hessian is needed in the definition (Irmer, 5 Aug 2025).

This distinguishes the theory sharply from classical smooth Morse theory. In the smooth setting, one requires $U \subset \mathbb{R}^n$0, $U \subset \mathbb{R}^n$1, and non-degenerate Hessian $U \subset \mathbb{R}^n$2; the index is then the number of negative eigenvalues of $U \subset \mathbb{R}^n$3. In the topological setting, the index is defined purely by the homeomorphism normal form, which is essential for functions that are continuous or only piecewise smooth and fail to be differentiable on large singular sets (Arnal, 2024).

2. Morse-theoretic consequences in the topological category

Topological Morse functions inherit most of the familiar consequences of smooth Morse theory. In particular, they support handle or cell attachment descriptions at isolated critical levels, and topology changes across critical values are determined by the index of the critical point (Irmer, 5 Aug 2025).

For sublevel sets, the standard topological Morse picture persists: when a parameter crosses a non-degenerate topological critical value of index $U \subset \mathbb{R}^n$4, the corresponding sublevel set changes by attaching an $U \subset \mathbb{R}^n$5-cell. In the finite-set distance setting this is stated explicitly for

$U \subset \mathbb{R}^n$6

where crossing a critical value of index $U \subset \mathbb{R}^n$7 attaches an $U \subset \mathbb{R}^n$8-cell, index-$U \subset \mathbb{R}^n$9 points create connected components, and the resulting attachments control homology changes of offsets (Arnal, 2024).

Standard topological Morse theory also yields Morse inequalities relating the number of index-$f:U\to\mathbb{R}$0 critical points to homology ranks. In this sense, the theory is not merely a topological analogue of the local Morse lemma; it remains a computational and structural tool for global topology. What is not currently parallel to the smooth case is existence and deformability. Topological Morse functions are not known to be generic, and it is unknown whether every topological manifold admits one (Irmer, 5 Aug 2025).

3. Min-type constructions and local convexity

A major source of topological Morse functions is the Min-type construction. Let $f:U\to\mathbb{R}$1 be a family of convex functions and define

$f:U\to\mathbb{R}$2

On a Riemannian manifold, if the $f:U\to\mathbb{R}$3 are convex and locally finite in the sense that near each $f:U\to\mathbb{R}$4 only finitely many functions realize the minimum, then $f:U\to\mathbb{R}$5 is a topological Morse function. A chartwise version on topological manifolds requires only that, near each point, the active functions become convex functions on an open set of $f:U\to\mathbb{R}$6 in local coordinates (Irmer, 5 Aug 2025).

The local analysis is geometric rather than differential. At a point where several convex pieces realize the minimum, one studies the intersection of their sublevel sets, its tangent cone, and the set of directions in which the minimum increases to first order. In the Riemannian formulation, a critical point is detected by the absence of a direction in which all active functions increase to first order, and the index is identified via the dimension of the span of the relevant gradients together with a monotonicity condition. Regular points can occur even when the set of increase has empty interior; the local level set is then still homeomorphic to a hyperplane because it is the boundary of the tangent cone to a convex intersection (Irmer, 5 Aug 2025).

This viewpoint also yields a simple deformation mechanism. If

$f:U\to\mathbb{R}$7

where the positive scalars $f:U\to\mathbb{R}$8 remain close to $f:U\to\mathbb{R}$9, then convexity and the local finiteness or local chartwise convexity hypotheses persist. A continuous path $z\in U$0 therefore produces a continuous family of topological Morse functions. This provides a concrete source of deformability in a category where full Cerf theory is not available (Irmer, 5 Aug 2025).

4. Distance functions to finite sets

A canonical nonsmooth example is the distance to a finite set $z\in U$1,

$z\in U$2

For each $z\in U$3, let

$z\in U$4

be the set of nearest points. Although $z\in U$5 fails to be smooth on the Voronoi skeleton and at points equidistant to multiple sites, it is nevertheless a topological Morse function for every finite $z\in U$6, with no general position assumptions (Arnal, 2024).

The topological critical points admit a precise classification. If $z\in U$7, then $z\in U$8 is a topological critical point of index $z\in U$9. If $V_1$0 and $V_1$1, one considers

$V_1$2

If there exists a nonzero $V_1$3 such that

$V_1$4

then $V_1$5 is topologically regular. Otherwise $V_1$6 is a non-degenerate topological critical point with

$V_1$7

All other points, namely those with $V_1$8 and $V_1$9, are topologically regular. This makes topological criticality a strict refinement of differential criticality (Arnal, 2024).

The same work compares topological and differential criteria using the generalized gradient

$0$0

where $0$1 is the Euclidean projection of $0$2 onto $0$3. Differential critical points are exactly the points of $0$4 together with points $0$5 satisfying $0$6. Topological critical points form a strict subset of these differential critical points. The proof uses the Clarke subdifferential

$0$7

and Lebourg’s Mean Value Theorem to establish the local injectivity and topological linearization needed for the regular and critical normal forms (Arnal, 2024).

The index formula is robust under degeneracy. It depends on $0$8, not on $0$9. Thus multiple collinear nearest points still give index $\mathbb{R}^n$0, while many cocircular nearest points in $\mathbb{R}^n$1 can give index $\mathbb{R}^n$2. This robustness is precisely what removes general-position hypotheses and makes the resulting Morse theory applicable to unions of balls, offsets, and persistent-homology event structures (Arnal, 2024).

5. Energy functions, Reeb-type invariants, and global classification

Topological Morse functions also arise from dynamics. For a topological flow on a closed $\mathbb{R}^n$3-manifold with finite hyperbolic chain recurrent set, every chain recurrent point is a fixed point, each such fixed point is locally topologically conjugate to

$\mathbb{R}^n$4

and there exists a continuous Morse energy function whose critical set is exactly the chain recurrent set and whose Morse index at each critical point equals the dynamical index $\mathbb{R}^n$5 (Medvedev et al., 2019).

At the global level, Reeb-type quotients and their refinements encode the organization of level sets. For an invariant decomposition $\mathbb{R}^n$6 on a topological space, the abstract weak element space $\mathbb{R}^n$7 and the abstract element space $\mathbb{R}^n$8 provide quotient objects adapted to recurrence and quasi-recurrence. The associated Morse hyper-graph exists for any topological space and any invariant decomposition, and the abstract weak element space generalizes the Reeb graph of a Morse function: if $\mathbb{R}^n$9 is the decomposition into connected components of level sets of a Morse function, then the Reeb graph is precisely $V_2$0 (Yokoyama, 2021).

On specific manifolds, this combinatorial viewpoint can be complete. For simple Morse functions on $V_2$1, the oriented Reeb graph is a complete topological invariant under fiber equivalence. Such a graph is necessarily a tree, it has exactly one vertex of degree $V_2$2, all other vertices have degree $V_2$3 or $V_2$4, and degree-$V_2$5 vertices are exactly the extrema (Bilun et al., 2023). In higher-dimensional settings, a related classification appears for simple sphere–torus–fibered Morse functions on closed orientable $V_2$6-manifolds that are connected sums of $V_2$7 and lens spaces: the oriented labeled Reeb graph records the sphere or torus fiber type along edges and controls the connected-sum structure at a coarse level (Kitazawa, 2024).

6. Scope, applications, and open problems

Topological Morse functions have become useful in settings where smooth Morse theory is either unavailable or too rigid. In moduli theory, the systole function on moduli spaces is topologically Morse, and a family of smooth approximants

$V_2$8

is used to transfer this topological Morse structure into a $V_2$9-Morse framework compatible with the Deligne–Mumford stratification. Critical points are described by eutacticity, the Morse index is recovered from the span of gradients of shortest geodesics, and the resulting handle decomposition provides a natural cell decomposition of $z$0 in theory for all $z$1 (Chen, 2023).

In applied topology, the finite-set distance theorem gives a degeneration-robust description of the topology changes of offsets

$z$2

with critical radii acting as event times and indices determining the type of cell attachment. This is directly relevant to unions of balls, alpha-complex intuition, and persistent homology. At the same time, the finite-set hypothesis is essential: distance functions to arbitrary compact sets need not be topological Morse, and differential critical points need not control homotopy changes of sublevel sets (Arnal, 2024).

Two foundational issues remain unresolved. First, it is unknown whether every topological manifold admits a topological Morse function. Second, topological Morse functions are not known to be generic, and full deformability results analogous to smooth Cerf theory are not established. Existing constructions provide continuous families in important Min-type regimes, but a general parametric theory of births, deaths, and handle slides in the topological category is still absent (Irmer, 5 Aug 2025). These limitations delimit the current scope of the subject, but they also define its central research program: extending Morse-theoretic topology beyond the differentiable setting while retaining enough local structure to control global topology.