Persistence for Circle Valued Maps

We study circle valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real valued maps, circle valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circle valued map on an input simplicial complex.

💡 Research Summary

The paper “Persistence for Circle Valued Maps” develops a comprehensive persistence theory for maps (f\colon X\to S^{1}) that parallels the well‑established theory for real‑valued functions but captures phenomena unique to the circular codomain. The authors begin by reviewing sublevel and level persistence for real‑valued maps, emphasizing that barcodes completely describe the birth and death of homology classes. They then point out that for circle‑valued maps sublevel sets are not defined, and level persistence alone cannot record the effect of traversing the circle multiple times.

To overcome this, the authors introduce the infinite cyclic covering (\tilde f\colon \tilde X\to\mathbb R) associated to (f). The covering lifts each angle (\theta) to a real parameter (t) with (p(t)=\theta) (where (p) is the universal covering (\mathbb R\to S^{1})). While (\tilde f) behaves like a real‑valued map and admits ordinary level persistence, the periodic identification of angles forces additional structure to be recorded.

The key technical contribution is the use of representation theory of the cyclic quiver. The linear maps induced by inclusions of level sets are assembled into a block matrix that encodes a quiver representation (\rho_{r}) for each homological dimension (r). Decomposing this representation yields two types of invariants:

1. Barcodes – intervals (