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. 
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7 Citations
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7 Citations

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References

SHOWING 1-10 OF 51 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)…

Cyclotomic structure in the topological Hochschild homology of $DX$

Let $X$ be a finite CW complex, and let $DX$ be its dual in the category of spectra. We demonstrate that the Poincar\'e/Koszul duality between $THH(DX)$ and the free loop space $\Sigma^\infty_+ LX$…

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…

Homotopical intersection theory I

We give a new approach to intersection theory. Our "cycles" are closed manifolds mapping into compact manifolds and our "intersections" are elements of a homotopy group of a certain Thom space. The…

The universal functorial Lefschetz invariant

We introduce the universal functorial Lefschetz invariant for endomorphisms of finite CW -complexes in terms of Grothendieck groups of endomorphisms of finitely generated free modules. It encompasses…

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…
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