Equivariant surgery theory for homology equivalences under the gap condition

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Abstract

In the present paper, we discuss an obstruction theory to modify equivariant framed maps on even-dimensional compact smooth manifolds to homology equivalences by equivariant surgery. In 1974, Cappell-Shaneson already developed such obstruction theory in the nonequivariant setting. Our definition of the surgery-obstruction group presents a new aspect of Cappell-Shaneson's group in the nonequivariant setting and enables us to define directly the surgery obstructions of certain framed maps that are not necessarily connected up to the middle dimension. Using our framework defining the equivariant surgery obstruction, we prove a basic conjecture related to geometric connected sums and algebraic sums of surgery obstructions.

Original languageEnglish
Pages (from-to)481-506
Number of pages26
JournalPublications of the Research Institute for Mathematical Sciences
Volume42
Issue number2
DOIs
Publication statusPublished - Jun 1 2006

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Keywords

  • Equivariant surgery
  • Gap condition
  • Homology equivalence
  • Surgery obstruction

ASJC Scopus subject areas

  • Mathematics(all)

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