TL;DR
This paper explores the relationships between different real K-theories, revealing new connections and interpretations, including a link between KR-theory and equivariant K-theory, with applications to computing Witt groups of real curves.
Contribution
It demonstrates how a real projective bundle theorem connects Atiyah's KR-theory with equivariant K-theory and interprets KR-theory as a special case of twisted K-theory.
Findings
Established a relation between KR-theory and equivariant K-theory.
Applied the relation to compute the Witt group of real curves.
Interpreted Atiyah's KR-theory within the framework of twisted K-theory.
Abstract
The purpose of this short paper is to investigate relations between various real K-theories. In particular, we show how a real projective bundle theorem implies an unexpected relation between Atiyah's KR-theory and the usual equivariant K-theory of real vector bundles. This relation has been used recently in a new computation of the Witt group of real curves by Schlichting, Weibel and the author. We also interpret Atiyah's theory as a special case of twisted K-theory.
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