# The equivariant Spivak normal bundle and equivariant surgery for compact Lie groups

## TL;DR

This paper extends equivariant surgery theory to compact Lie groups by developing equivariant Poincare complexes and establishing foundational properties like the existence of equivariant Spivak normal fibrations and the Pi-Pi Theorem.

## Contribution

It introduces a generalized framework for equivariant surgery for compact Lie groups using new equivariant homology and cohomology theories.

## Key findings

- Every compact G-manifold is an equivariant Poincare complex
- Finite equivariant Poincare complexes have equivariant spherical Spivak normal fibrations
- The Pi-Pi Theorem holds for equivariant Poincare pairs under certain conditions

## Abstract

We generalize the results of a previous paper of ours to compact Lie groups. Using a recently developed ordinary equivariant homology and cohomology, we define equivariant Poincare complexes with the properties that (1) every compact G-manifold is an equivariant Poincare complex, (2) every finite equivariant Poincare complex (with some mild additional hypotheses) has an equivariant spherical Spivak normal fibration, and (3) the Pi-Pi Theorem holds for equivariant Poincare pairs under suitable gap hypotheses. The nice behavior of the ordinary equivariant homology and cohomology theories allows us to follow Wall's original line of argument closely.