Rho-invariant
1 Introduction
The -invariant is an invariant of odd-dimensional closed manifolds closely related to the equivariant signature. Its definition is motivated by the equivariant signature defect of even-dimensional manifolds with boundary. Namely, for manifolds with boundary the classical index formula for equivariant signature must be corrected by an additional term. It is this term which gives rise to the -invariant. There is also another definition using bordism theory. Finally there is also an analytic defintion as a relative -invariant.
2 Definition
Let be a compact Lie group acting smoothly on a smooth manifold . The middle intersection form becomes a non-degenerate -symmetric bilinear form on which acts. As explained earlier, such a form yields an element in the representation ring which we denote by . The discussion in section \ref{sec:ses} also tells us that we have which in terms of characters means that we obtain a real (purely imaginary) character, which will be denoted as . The (cohomological version of the) Atiyah-Singer -index theorem [Atiyah-Singer-III(1968), Theorem (6.12)] tells us that if is closed then for all
where is an expression obtained by evaluating certain cohomological classes on the fundamental classes of the -fixed point submanifolds of . In particular if the action is free then if . This means that is a multiple of the regular representation. This theorem was generalized by Wall to topological semifree actions on topological manifolds, which is the case we will need in this paper [Wall(1999), chapter 14B]. The assumption that is closed is essential here, and motivates the definition of the -invariant. In fact, Atiyah and Singer provide two definitions. For the first one one also needs the result of Conner and Floyd [Conner-Floyd(1964)] that for an odd-dimensional manifold with a finite fundamental group there always exists a and a manifold with boundary such that and .
Definition 2.1 (Atiyah-Singer).
Let be a closed manifold with a finite group. Define
for some and such that and . The symbol denotes the ideal generated by the regular representation.
See [Atiyah-Singer-III(1968), Remark after Corollary 7.5]
By the Atiyah-Singer -index theorem [Atiyah-Singer-III(1968), Theorem (6.12)] is well defined.
Definition 2.2 ??.
Let be a compact Lie group acting freely on a manifold . Suppose in addition that there is a manifold with boundary on which acts (not necessarily freely) and such that . Define
In this definition we think about the -invariant as about a function . When both definitions apply (that means when is a finite group), then they coincide, that means .
3 References
- [Atiyah-Singer-III(1968)] Template:Atiyah-Singer-III(1968)
- [Conner-Floyd(1964)] Template:Conner-Floyd(1964)
- [Wall(1999)] Template:Wall(1999)
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