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Equivariant $v_{1,\vec{0}}$-self maps

William Balderrama, Yueshi Hou, Shangjie Zhang

Abstract

Let $G$ be a cyclic $p$-group or generalized quaternion group, $X\in π_0 S_G$ be a virtual $G$-set, and $V$ be a fixed point free complex $G$-representation. Under conditions depending on the sizes of $G$, $X$, and $V$, we construct a self map $v\colonΣ^V C(X)_{(p)}\rightarrow C(X)_{(p)}$ on the cofiber of $X$ which induces an equivalence in $G$-equivariant $K$-theory. These are transchromatic $v_{1,\vec{0}}$-self maps, in the sense that they are lifts of classical $v_1$-self maps for which the telescope $C(X)_{(p)}[v^{-1}]$ can have nonzero rational geometric fixed points.

Equivariant $v_{1,\vec{0}}$-self maps

Abstract

Let be a cyclic -group or generalized quaternion group, be a virtual -set, and be a fixed point free complex -representation. Under conditions depending on the sizes of , , and , we construct a self map on the cofiber of which induces an equivalence in -equivariant -theory. These are transchromatic -self maps, in the sense that they are lifts of classical -self maps for which the telescope can have nonzero rational geometric fixed points.

Paper Structure

This paper contains 16 sections, 18 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Constructing equivariant v_1-self maps
  3. Technical lemmas and propositions
  4. Representations of cyclic and dicyclic groups
  5. Cyclic groups
  6. Dicyclic groups
  7. Equivariant J-theory
  8. Symmetric Toda brackets
  9. v_1-self maps and the Bott class
  10. The proof of Theorem 1.0.1
  11. Step 1
  12. Step 2
  13. Step 3
  14. Examples
  15. Cyclic groups
  16. ...and 1 more sections

Key Result

Theorem 1.0.1

Let $G$ be a cyclic $p$-group or generalized quaternion group, of order $p^n$. Let $X \in \pi_0 S_G$ be a virtual $G$-set of virtual cardinality $p^tc$ with $p\nmid c$. Let $V$ be a fixed point free complex $G$-representation of complex dimension $p^kc(p-1)$, or $2^{k-1}c$ when $p=2$, with $p\nmid c

Theorems & Definitions (42)

  • Theorem 1.0.1
  • Example 1.0.2
  • Example 1.0.3
  • Remark 1.0.4
  • Lemma 2.0.1
  • proof
  • Example 2.0.2
  • Theorem 3.1.1
  • Lemma 3.1.2
  • proof
  • ...and 32 more