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Cobordism of G-manifolds

Jack Carlisle

Abstract

We survey some results in the field of equivariant cobordism. In particular, we use methods from equivariant stable homotopy theory to calculate the unoriented $C_2$-equivariant bordism ring $Ω^{C_2}_*$, which was originally calculated by Alexander using other methods. Our proof method generalizes well to other settings, such as equivariant complex cobordism, and affords a formal group theoretic interpretation of Alexander's calculation.

Cobordism of G-manifolds

Abstract

We survey some results in the field of equivariant cobordism. In particular, we use methods from equivariant stable homotopy theory to calculate the unoriented -equivariant bordism ring , which was originally calculated by Alexander using other methods. Our proof method generalizes well to other settings, such as equivariant complex cobordism, and affords a formal group theoretic interpretation of Alexander's calculation.
Paper Structure (4 sections, 9 theorems, 61 equations)

This paper contains 4 sections, 9 theorems, 61 equations.

Table of Contents

  1. Representing cobordism of $G$-manifolds
  2. The Tate square for $\Omega^{C_2}_*$
  3. Generators and relations for $\Omega^{C_2}_*$
  4. The extended cobordism ring $\Omega^{C_2}_\diamond$

Key Result

Theorem 1.1

(Schwede2, Theorem 6.2.33) For any finite group $G$, the assignment as defined above, determines a ring isomorphism \begin{tikzcd} \Omega^G_* \ar[r,"\cong"] & \pi^G_*(\Omega_G). \end{tikzcd}In fact, this holds true at the level of $H$-fixed points for any $H \leq G$, so the construction above determines an isomorphism of $G$-Mackey functors \begin{tikzcd} \underlin

Theorems & Definitions (16)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 6 more