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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.
2.1 G-index theorem
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. The positive and negative definite subspaces are -invariant and hence such a form yields an element in the representation ring denoted by . In fact which in terms of characters means that we obtain a real (case even) / purely imaginary (case odd) character, which will be denoted as
The (cohomological version of the) Atiyah-Singer -index theorem [Atiyah&Singer1968b, 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 here [Wall1999, chapter 14B].
The assumption that is closed is essential, and motivates the definition of the -invariant.
2.2 Cobordism theory
Also the result from cobordism theory is needed which says that for an odd-dimensional manifold with a finite fundamental group there always exists a and a manifold with boundary such that and . In other words
3.1 G finite
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&Singer1968b, Remark after Corollary 7.5] for more details. Note that the manifold in the definition always exists by the above mentioned result in cobodism theory. Furthermore the invariant is well-defined thanks to the cohomological version of the -index theorem.
More generally, the invariant can be defined for all closed manifolds together with a homomorphism to the finite group . One uses such that and factors through , and one considers the representation on the middle homology of the induced -cover of .
3.2 G compact Lie group
Definition 3.2 (Atiyah-Singer). 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
See [Atiyah&Singer1968b, Theorem 7.4] for more details.
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 .
Note that the invariant is natural under restriction from a group to a subgroup: if , and acts freely on , then .
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