The linearity of fixed point invariants

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 multiplicativity results for the Lefschetz and Nielsen numbers of a fibration. Moreover, the proofs of these theorems are essentially formal, taking place in the abstract context of bicategorical traces. This makes generalizations to other contexts straightforward.

Equivariant Nielsen invariants for discrete groups

For discrete groups G, we introduce equivariant Nielsen invariants. They are equivariant analogs of the Nielsen number and give lower bounds for the number of fixed point orbits in the G-homotopy class of an equivariant endomorphism f:X->X. Under mild hypotheses, these lower bounds are sharp. We use the equivariant Nielsen invariants to show that a G-equivariant endomorphism f is G-homotopic to a fixed point free G-map if the generalized equivariant Lefschetz invariant of f is zero. Finally, we prove a converse of the equivariant Lefschetz fixed point theorem.

迈向 $p=3$ Kervaire 不变量问题：计算 $π_*({E_6}^{hC_9})$ 的同伦不动点谱序列的 $E_2$-页

Towards the $p=3$ Kervaire Invariant Problem: The $E_2$-page for the homotopy fixed points spectral sequence computing $π_*({E_6}^{hC_9})$

针对 $p=3$ 的 Kervaire 不变量问题，本文证明了相关检测定理，并利用 $E_6$ 上 $C_9$ 作用的猜想形式，计算了 $π_* E_6^{hC_9}$ 同伦不动点谱序列在非零同调度处的 $E_2$ 页。

Axiomatic Description of Lefschetz Type Equivariant Homotopy Invariants

We describe generators of universal Lefschetz groups consisting of self-maps of equivariant 1-spheres. This allows to formulate a normalization axiom which, together with the usual axioms, determines an equivariant Lefschetz number uniquely. We apply this technique to identify the fixed orbit detecting equivariant Lefschetz number of Chorny with the fixed orbit index of Dzedzej.