Persistent Simple-homotopy invariants via discrete Morse theory
演讲者
Divya Ahuja
时间
2026年06月25日 15:00 至 16:00
地点
A3-4-301
线上
Zoom 928 682 9093
(BIMSA)
摘要
Persistent homology is a central tool in topological data analysis, capturing multiscale homological features of filtered spaces. However, it detects only additive invariants such as Betti numbers and thus misses finer distinctions, including differences between homotopy equivalence and simple-homotopy equivalence. In classical topology, simple-homotopy theory refines homotopy equivalence by distinguishing maps realizable through elementary collapses and expansions, with the obstruction measured by Whitehead torsion. In this talk, we will introduce refinements of persistent homology that capture such phenomena beyond homology.
Given a filtered simplicial complex $\{K_i\}_{i \ge 0}$, we define the Morse complexity profile, recording the minimal number of critical simplices at each filtration level. This invariant captures combinatorial information beyond homology and highlights Morse spikes, where homology remains unchanged but the minimal Morse complexity increases. We will show that this profile is invariant under levelwise simple-homotopy equivalence and illustrate these phenomena through explicit examples.
We will also introduce a persistent version of Whitehead torsion, assigning to each inclusion $K_s \hookrightarrow K_t$ an obstruction that vanishes precisely when the inclusion is a simple-homotopy
equivalence. This provides a strictly finer invariant detecting phenomena invisible to persistent homology, and we will show that it is invariant under levelwise simple-homotopy equivalence.
This talk is based on a joint work with Prof. Jaya N. Iyer.
Given a filtered simplicial complex $\{K_i\}_{i \ge 0}$, we define the Morse complexity profile, recording the minimal number of critical simplices at each filtration level. This invariant captures combinatorial information beyond homology and highlights Morse spikes, where homology remains unchanged but the minimal Morse complexity increases. We will show that this profile is invariant under levelwise simple-homotopy equivalence and illustrate these phenomena through explicit examples.
We will also introduce a persistent version of Whitehead torsion, assigning to each inclusion $K_s \hookrightarrow K_t$ an obstruction that vanishes precisely when the inclusion is a simple-homotopy
equivalence. This provides a strictly finer invariant detecting phenomena invisible to persistent homology, and we will show that it is invariant under levelwise simple-homotopy equivalence.
This talk is based on a joint work with Prof. Jaya N. Iyer.