Homotopic PMD: Parametrized Homotopic Distance

• Homotopic PMD is a framework that measures minimal homotopy-induced complexity between fibrewise maps, extending classical distance concepts.
• It refines invariants like Lusternik–Schnirelmann category and topological complexity via open covers and fibrewise homotopies.
• The approach applies to motion planning, topological data analysis, and policy optimization, offering practical computational methods and enhanced invariants.

Homotopic PMD (HPMD) appears in several theoretical and applied mathematical contexts, variously referring to generalized notions of homotopic distance and distance-like functionals that incorporate homotopy-theoretic constraints into their definitions. The unifying principle is the measurement of minimal “complexity” or “distance” between objects (typically maps, filtrations, policies, or modules) up to homotopy, often yielding a pseudometric or complexity invariant with rich algebraic or topological structure. The most developed, precise notion closely matching "Homotopic PMD" is the parametrized homotopic distance $D_B(f,g)$ in fibrewise topology, as described by Daundkar and García-Calcines (Daundkar et al., 27 Jan 2025), along with related (non-fibrewise) homotopic pseudometrics (Vergili et al., 2020) and emerging connections to policy optimization and multiparameter persistence.

1. Parametrized Homotopic Distance: Definition and Fundamental Properties

Let $B$ be a base space. For fibrewise spaces $X \to B$ and $Y \to B$ (i.e., spaces equipped with projection maps $p_X, p_Y$ over $B$), a map $f : X \to Y$ is fibrewise if $p_Y \circ f = p_X$. The parametrized homotopic distance $D_B(f,g)$ between two such fibrewise maps is defined as the minimum integer $n \ge 0$ for which there exists an open cover $X = U_0 \cup \cdots \cup U_n$ with $f|_{U_i}$ fibrewise homotopic to $g|_{U_i}$ for each $i$—that is, there exists a fibrewise homotopy $H: X \times_B I \to Y$ connecting $f$ and $g$ over each $U_i$ (Daundkar et al., 27 Jan 2025).

This invariant generalizes the classical (non-fibrewise) homotopic distance $D(f,g)$ (Vergili et al., 2020), and by varying the projections and specializations, recovers the Lusternik–Schnirelmann category and the (parametrized) topological complexity as particular cases:

$$TC_B(X) = D_B(\operatorname{pr}_1, \operatorname{pr}_2).$$

Here, $TC_B(X)$ is the fibrewise or parametrized topological complexity, and $$\operatorname{pr}_1, \operatorname{pr}_2 : X \times_B X \to X$$ are the fibrewise projections.

$D_B(f,g)$ is shown to coincide with $\operatorname{secat}_B(\Pi_Y^*)$, where $\Pi_Y^*$ is the pullback of the fibrewise path fibration

$$\Pi_Y: P_B(Y) \to Y \times_B Y,\quad \alpha \mapsto (\alpha(0), \alpha(1)).$$

Key properties:

• Cohomological lower bounds: If there exist $z_1,\dots,z_k\in H^*(Y\times_B Y;R)$ with $\Delta_Y^*(z_i)=0$ and $(f,g)^*(z_1 \cdots z_k) \neq 0$, then $D_B(f,g)\ge k$.
• Connectivity (dimension) upper bounds: If $Y$ is path-connected, $X$ has homotopy type of a CW-complex of dimension $\operatorname{hdim}(X)$, and $p_Y$ is an $r$-equivalence, then $D_B(f,g)\le \operatorname{hdim}(X)/r$.
• Triangle inequality: $D_B(f,g) \le D_B(f,h) + D_B(h,g)$ when $X$ is normal.
• Functoriality: Pre- and post-composition with fibrewise maps does not increase $D_B$.

2. Product, Composition, and Pointed/Unpointed Versions

The parametrized homotopic distance is sensitive to operations on fibrewise maps:

• Products: For $f,g : X \to Y$ and $f',g' : X' \to Y'$, if $X\times_B X'$ is normal then

$$D_B(f\times_B f',\,g\times_B g') \le D_B(f,g) + D_B(f',g').$$

This supports the use of HPMD as a complexity measure in parametrized settings, especially for group actions and equivariant structures.

• Fibrewise $H$-spaces and division maps: If $X$ is a fibrewise pointed $H$-space with a division map, the LS-category and topological complexity coincide with $D_B$:

$TC_B(X) = \cat_B^*(X).$

(Here, $\cat_B^*(X)$ is the fibrewise unpointed LS-category.)

• Pointed version: For pointed fibrewise maps (i.e., maps fixing a section $s_X$), a pointed parametrized homotopic distance $D_B^B(f,g)$ is defined, with

$D_B(f,g) \le D_B^B(f,g) \le D_B(f,g)+1,$

and under certain dimension-connectivity hypotheses $D_B^B(f,g)=D_B(f,g)$.

3. Interaction with Fibrewise Fibrations and Motion Planning

Estimates for the parametrized homotopic distance in the context of fibrewise fibrations exhibit a recursive structure:

• For a fibrewise fibration $\pi: E \to X$ with fibre $F$, and fibre-preserving maps $f,g: E \to E'$, the following holds:

$D_B(f,g)+1 \le (D_B(f_0,g_0)+1)\,(\cat_B^*(X)+1),$

where $f_0, g_0$ are the induced maps on the fibre.

• Analogous estimates for topological complexity and LS-category propagate through the fibration structure.

A worked example appears with Cayley $S^7$-sphere bundles, where $TC_B(X)$ can be sharply estimated in terms of the dimension of $B$ and the bundle’s nontriviality (Daundkar et al., 27 Jan 2025).

4. Generalized Homotopic Distances and Connections to Other Contexts

Homotopic pseudometric distances $D(f,g)$ on mapping spaces $\mathrm{Map}(X,Y)$ induce extended pseudometric topologies and are related to LS-category and other minimal-cover invariants (Vergili et al., 2020). Notably:

• $D(f,g)$ evaluates to $0$ if $f$ and $g$ are homotopic, and to the LS-category or topological complexity in fundamental cases.
• The induced topology on $\mathrm{Map}(X,Y)$ is non-Hausdorff unless quotiented by homotopy, and exhibits connectedness/disconnectedness properties according to the minimal covering number.

A plausible implication is that parametrized homotopic distance generalizes these non-parametric pseudometrics, integrating additional structure from the base space and supporting refined invariants in settings where symmetry, fibration, and equivariance play central roles.

5. Related Notions: Homotopy Classes in Complex Analysis and Policy Optimization

In several branches of mathematics, “homotopy” in domains with extra structure yields complexity invariants and classification results:

• Proper holomorphic mappings: Homotopy equivalence for rational proper holomorphic maps $f : B^n \to B^N$ (unit balls) is studied through continuous and rational homotopies; the resulting homotopy classes are finite in fixed codimension and target-dimension, and the degree is not a homotopy invariant in positive codimension (D'Angelo et al., 2014).
• Policy Mirror Descent: Homotopic Policy Mirror Descent (HPMD) in infinite-horizon Markov decision processes introduces a mirror descent update with a vanishing homotopy-based regularization term. This results in global linear convergence, local superlinear convergence, and last-iterate convergence to the maximum-entropy optimum, with a variety of Bregman divergences supported. Stochastic generalizations of HPMD achieve improved sample complexity bounds compared to prior policy gradient algorithms (Li et al., 2022).

This suggests that, while the specific term “Homotopic PMD” is used variously across domains, the unifying property remains the quantification of minimal “homotopy-induced cost” under a relevant covering, interpolation, or deformation analytic framework.

6. Computational Aspects and Software

Parametrized and path-based homotopic distances admit explicit algorithmic treatment in topological data analysis:

• For multiparameter persistent modules, “path-based” distances generalize matching distance by taking a supremum over monotone path projections in parameter space, leading to greater sensitivity to complex topological differences (Sun et al., 31 Jul 2025).
• The computation proceeds via restriction to one-parameter submodules along monotone paths, followed by bottleneck or Wasserstein distance computation for persistence diagrams, using standard algorithms (e.g., PHAT, Hera). The core routines are implemented in C++ and Python, with user-accessible APIs, example code, and integration with common TDA libraries.

In the homotopic policy optimization context, HPMD iterations are explicit and rely on sequence parameter tuning, with analytic convergence guarantees (Li et al., 2022). No explicit connection is made between HPMD in RL and homotopic distances in topology, though both leverage homotopy-induced or regularized structures in their definitions.

7. Applications and Broader Significance

Parametrized homotopic distance and related invariants have broad applicability:

• Parameter-dependent motion planning: $TC_B(X)$ realizes the minimal number of local rules required for continuous, fibrewise-compatible path selection in parameterized configuration spaces.
• Complexity in fibration structures: Estimates for $D_B(f,g)$ underpin sharp lower and upper bounds for motion planning and navigation problems with symmetry, e.g., for sphere bundles, group actions, and bundles with division structures.
• Topological data analysis: Path-based, supremum-over-path distances in multiparameter persistence modules capture finer topological variations between data sets than classical matching distances, supporting more nuanced data comparison tasks.
• Policy optimization: Implicit homotopy-regularization in policy-gradient updates (HPMD) yields improvements in convergence speed, sample complexity, and optimal policy characterization.

The parametrized homotopic distance thus serves as a core invariant connecting fibrewise homotopy theory, dynamical systems, data science, and optimization, ultimately mediating the interplay between local deformations, global structure, and computational tractability in a wide variety of mathematical and applied contexts (Daundkar et al., 27 Jan 2025, Vergili et al., 2020, D'Angelo et al., 2014, Li et al., 2022, Sun et al., 31 Jul 2025).