Periodic Points and Topological Restriction Homology

@article{Malkiewich2018PeriodicPA,
  title={Periodic Points and Topological Restriction Homology},
  author={Cary Malkiewich and Kate Ponto},
  journal={arXiv: Algebraic Topology},
  year={2018}
}
We answer in the affirmative two conjectures made by Klein and Williams. First, in a range of dimensions, the equivariant Reidemeister trace defines a complete obstruction to removing $n$-periodic points from a self-map $f$. Second, this obstruction defines a class in topological restriction homology. We prove these results using duality and trace for bicategories. This allows for immediate generalizations, including a corresponding theorem for the fiberwise Reidemeister trace. 

Unwinding the relative Tate diagonal

We show that a spectral sequence developed by Lipshitz and Treumann, for application to Heegaard Floer theory, converges to a localized form of topological Hochschild homology with coefficients. This

Isovariant homotopy theory and fixed point invariants

An isovariant map is an equivariant map between G-spaces which strictly preserves isotropy groups. We consider an isovariant analogue of Klein–Williams equivariant intersection theory for a finite

Periodic Points on Tori: Vanishing and Realizability

OF DISSERTATION Periodic Points on Tori: Vanishing and Realizability Let X be a finite simplicial complex and f : X Ñ X be a continuous map. A point x P X is a fixed point if fpxq “ x. Classically

Applications of topological cyclic homology to algebraic K-theory

A bstract. Algebraic K-theory has applications in a broad range of mathematical subjects, from number theory to functional analysis. It is also fiendishly hard to calculate. Presently there are two

Coherence for indexed symmetric monoidal categories

Indexed symmetric monoidal categories are an important refinement of bicategories -- this structure underlies several familiar bicategories, including the homotopy bicategory of parametrized spectra,

F eb 2 02 0 Unwinding the relative Tate diagonal

We show that a spectral sequence developed by Lipshitz and Treumann, for application to Heegaard Floer theory, converges to a localized form of topological Hochschild homology with coefficients. This

$K$-theory of endomorphisms, the $\mathit{TR}$-trace, and zeta functions

We show that the characteristic polynomial and the Lefschetz zeta function are manifestations of the trace map from the $K$-theory of endomorphisms to topological restriction homology (TR). Along the

References

SHOWING 1-10 OF 44 REFERENCES

Topological Hochschild homology and higher characteristics

We show that an important classical fixed point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the

The multiplicativity of fixed point invariants

We prove two general factorization theorems for fixed-point invariants of fibrations: one for the Lefschetz number and one for the Reidemeister trace. These theorems imply the familiar

Homotopical intersection theory, II: equivariance

This paper is a sequel to (Klein and Williams in Geom Topol 11:939–977, 2007). We develop here an intersection theory for manifolds equipped with an action of a finite group. As in Klein and Williams

Cancelling periodic points

Abstract. We prove that a self map $f : M \to M$ of a PL-manifold of dimension $\ge 4 $ is homotopic to a map with no periodic points of period n iff the Nielsen numbers $N(f^k)$ (k divides n)

Coincidence invariants and higher Reidemeister traces

The Lefschetz number and fixed point index can be thought of as two different descriptions of the same invariant. The Lefschetz number is algebraic and defined using homology. The index is defined

Fixed Point Theory and Trace for Bicategories

The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in

The cyclotomic trace and algebraic K-theory of spaces

The cyclotomic trace is a map from algebraic K-theory of a group ring to a certain topological refinement of cyclic homology. The target is naturally mapped to topological Hochschild homology, and

Shadows and traces in bicategories

Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some

Comparing cyclotomic structures on different models for topological Hochschild homology

The topological Hochschild homology THH(A) of an orthogonal ring spectrum A can be defined by evaluating the cyclic bar construction on A or by applying Bökstedt's original definition of THH to A .

Parametrized spectra, a low-tech approach.

We give an alternative treatment of the foundations of parametrized spectra, with an eye toward applications in fixed-point theory. This treatment includes most of the central results from the book