Rho-invariant

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== Introduction ==
== Introduction ==
<wikitex>;
<wikitex>;
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<wikitex>;
<wikitex>;
Let $G$ be a compact Lie group acting smoothly on a smooth manifold $Y^{2d}$. The middle intersection form becomes a non-degenerate $(-1)^d$-symmetric bilinear form on which
Let $G$ be a compact Lie group acting smoothly on a smooth manifold $Y^{2d}$. The middle intersection form becomes a non-degenerate $(-1)^d$-symmetric bilinear form on which
$G$ acts. The positivie and negative definite subspaces are $G$-invariant and hence such a form yields an element in the representation ring $R(G)$ denoted by $\mathrm{G-sign} (Y)$.
+
$G$ acts. The positive and negative definite subspaces are $G$-invariant and hence such a form yields an element in the representation ring $R(G)$ denoted by $\mathrm{sign_G} (Y)$.
In fact $\mathrm{G-sign} (Y) \in R^{(-1)^d} (G)$ which in terms of characters means that we obtain a real (purely imaginary) character, which will be denoted as
+
In fact $\mathrm{sign_G} (Y) \in R^{(-1)^d} (G)$ which in terms of characters means that we obtain a real (case $d$ even) / purely imaginary (case $d$ odd) character, which will be denoted as
$$
$$
\mathrm{G-sign} (-,Y) \colon g \in G \mapsto \mathrm{G-sign} (g,Y) \in \Cc.
+
\mathrm{sign_G} (-,Y) \colon g \in G \mapsto \mathrm{sign_G} (g,Y) \in \Cc.
$$
$$
The (cohomological version of the) Atiyah-Singer $G$-index theorem {{cite|Atiyah&Singer1968b|Theorem (6.12)}} tells us that if $Y$ is closed then for all $g \in G$
The (cohomological version of the) Atiyah-Singer $G$-index theorem {{cite|Atiyah&Singer1968b|Theorem (6.12)}} tells us that if $Y$ is closed then for all $g \in G$
$$
$$
\mathrm{G-sign} (g,Y) = L(g,Y) \in \Cc,
+
\mathrm{sign_G} (g,Y) = L(g,Y) \in \Cc,
$$
$$
where $L(g,Y)$ is an expression obtained by evaluating certain cohomological classes on the fundamental classes of the $g$-fixed point submanifolds $Y^g$ of $Y$. In particular if the action is free then $\mathrm{G-sign} (g,Y) = 0$ if $g \neq 1$. This means that $\mathrm{G-sign} (Y)$
+
where $L(g,Y)$ is an expression obtained by evaluating certain cohomological classes on the fundamental classes of the $g$-fixed point submanifolds $Y^g$ of $Y$. In particular if the action is free then $\mathrm{sign_G} (g,Y) = 0$ if $g \neq 1$. This means that $\mathrm{sign_G} (Y)$
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
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
{{cite|Wall1999|chapter 14B}}.
{{cite|Wall1999|chapter 14B}}.
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This is due to {{cite|Conner&Floyd1964}}, {{cite|Williamson1966}} and {{cite|Madsen&Milgram1979}}.
This is due to {{cite|Conner&Floyd1964}}, {{cite|Williamson1966}} and {{cite|Madsen&Milgram1979}}.
</wikitex>
</wikitex>
== Definition ==
== Definition ==
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Let $X^{2d-1}$ be a closed manifold with $\pi_1 (X) \cong G$ a finite group. Define
Let $X^{2d-1}$ be a closed manifold with $\pi_1 (X) \cong G$ a finite group. Define
$$
$$
\rho (X) = \frac{1}{k} \cdot \mathrm{G-sign} (\widetilde Y) \in \Qq R^{(-1)^d}
+
\rho (X) = \frac{1}{k} \cdot \mathrm{sign_G} (\widetilde Y) \in \Qq R^{(-1)^d}
(G)/ \langle \mathrm{reg} \rangle
(G)/ \langle \mathrm{reg} \rangle
$$
$$
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See {{cite|Atiyah&Singer1968b|Remark after Corollary 7.5}} for more details. Note that the manifold $Y$ 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 $G$-index theorem.
See {{cite|Atiyah&Singer1968b|Remark after Corollary 7.5}} for more details. Note that the manifold $Y$ 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 $G$-index theorem.
+
+
More generally, the invariant can be defined for all closed manifolds $X^{2d-1}$ together with a homomorphism $f:\pi_1 (X)\to G$ to the finite group $G$. One uses $Y$ such that $\partial Y = k \cdot X$ and $f$ factors through $\pi_1 (Y)$, and one considers the representation on the middle homology of the induced
+
$G$-cover of $Y$.
</wikitex>
</wikitex>
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Let $G$ be a compact Lie group acting freely on a manifold $\widetilde{X}^{2d-1}$. Suppose in addition that there is a manifold with boundary $(Y,\partial Y)$ on which $G$ acts (not necessarily freely) and such that $\partial Y = \widetilde X$. Define
Let $G$ be a compact Lie group acting freely on a manifold $\widetilde{X}^{2d-1}$. Suppose in addition that there is a manifold with boundary $(Y,\partial Y)$ on which $G$ acts (not necessarily freely) and such that $\partial Y = \widetilde X$. Define
$$
$$
\rho_G (\widetilde X) \co g \in G \mapsto \mathrm{G-sign} (g,Y) - L(g,Y) \in \Cc.
+
\rho_G (\widetilde X) \co G \ni g \mapsto \mathrm{sign_G} (g,Y) - L(g,Y) \in \Cc.
$$
$$
{{endthm}}
{{endthm}}
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When both definitions apply (that means when $G$ is a finite group), then they coincide, that means $\rho (X) = \rho_G (\widetilde X)$.
When both definitions apply (that means when $G$ is a finite group), then they coincide, that means $\rho (X) = \rho_G (\widetilde X)$.
</wikitex>
== G finite cyclic ==
+
Note that the invariant is natural under restriction from a group $G$ to a subgroup: if $H \le G$, and $G$ acts freely on $\widetilde X$, then $\rho_G (\widetilde X)|_{H\smallsetminus\{1\}}=\rho_H (\widetilde X)$.
<wikitex>;
+
For finite $G < S^1$ we will use special notation following
+
\cite[Proof of Proposition 14E.6 on page 222]{Wall(1999)}. By
+
$\widehat G$ is denoted the Pontrjagin dual of $G$, the group
+
$\Hom_\Zz (G,S^1)$. Recall that for a finite cyclic $G$ the
+
representation ring $R(G)$ can be canonically identified with the
+
group ring $\Zz \widehat G$. Then we also have $\Qq R(G) = \Qq
+
\otimes R(G) = \Qq \widehat G$. Dividing out the regular
+
representation corresponds to dividing out the norm element, denoted
+
by $Z$, hence $R(G)/\langle \textup{reg} \rangle = R_{\widehat G} =
+
\Zz \widehat G/\langle Z \rangle$ and $\Qq R(G)/\langle \textup{reg}
+
\rangle = \Qq R_{\widehat G} = \Qq \widehat G/\langle Z \rangle$.
+
Choosing a generator $\widehat G = \langle \chi \rangle$ gives the
+
identifications $\Qq R_{\widehat G} = \Qq [\chi]/\langle 1+\chi +\cdots +
+
\chi^{N-1} \rangle$ where $N$ is the order of $G$. In order to save
+
space we also use the following notation $I \langle K \rangle =
+
\langle 1 + \chi + \cdots + \chi^{N-1} \rangle$.
+
</wikitex>
</wikitex>
== References == {{#RefList:}}
== References == {{#RefList:}}
+
[[Category:Definitions]]
[[Category:Theory]]
+

Latest revision as of 10:51, 13 June 2013

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

The \rho-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 \rho-invariant. There is also another definition using bordism theory. Finally there is also an analytic defintion as a relative \eta-invariant.

2 Background

2.1 G-index theorem

Let G be a compact Lie group acting smoothly on a smooth manifold Y^{2d}. The middle intersection form becomes a non-degenerate (-1)^d-symmetric bilinear form on which G acts. The positive and negative definite subspaces are G-invariant and hence such a form yields an element in the representation ring R(G) denoted by \mathrm{sign_G} (Y). In fact \mathrm{sign_G} (Y) \in R^{(-1)^d} (G) which in terms of characters means that we obtain a real (case d even) / purely imaginary (case d odd) character, which will be denoted as

\displaystyle  \mathrm{sign_G} (-,Y) \colon g \in G \mapsto \mathrm{sign_G} (g,Y) \in \Cc.

The (cohomological version of the) Atiyah-Singer G-index theorem [Atiyah&Singer1968b, Theorem (6.12)] tells us that if Y is closed then for all g \in G

\displaystyle   \mathrm{sign_G} (g,Y)  = L(g,Y) \in \Cc,

where L(g,Y) is an expression obtained by evaluating certain cohomological classes on the fundamental classes of the g-fixed point submanifolds Y^g of Y. In particular if the action is free then \mathrm{sign_G} (g,Y) = 0 if g \neq 1. This means that \mathrm{sign_G} (Y) 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 Y is closed is essential, and motivates the definition of the \rho-invariant.

2.2 Cobordism theory

Also the result from cobordism theory is needed which says that for an odd-dimensional manifold X with a finite fundamental group G there always exists a k \in \Nn and a manifold with boundary (Y,\partial Y) such that \pi_1 (Y) \cong \pi_1 (X) \cong G and \partial Y = k \cdot X. In other words

\displaystyle  \Omega^{\textup{STOP}}_{2d-1} (BG) \otimes \Qq = 0

This is due to [Conner&Floyd1964], [Williamson1966] and [Madsen&Milgram1979].

3 Definition

3.1 G finite

Definition 3.1.

Let X^{2d-1} be a closed manifold with \pi_1 (X) \cong G a finite group. Define

\displaystyle  \rho (X) = \frac{1}{k} \cdot \mathrm{sign_G} (\widetilde Y) \in \Qq R^{(-1)^d} (G)/ \langle \mathrm{reg} \rangle

for some k \in \Nn and (Y,\partial Y) such that \pi_1 (Y) \cong \pi_1 (X) and \partial Y = k \cdot X. The symbol \langle \textup{reg} \rangle denotes the ideal generated by the regular representation.

See [Atiyah&Singer1968b, Remark after Corollary 7.5] for more details. Note that the manifold Y 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 G-index theorem.

More generally, the invariant can be defined for all closed manifolds X^{2d-1} together with a homomorphism f:\pi_1 (X)\to G to the finite group G. One uses Y such that \partial Y = k \cdot X and f factors through \pi_1 (Y), and one considers the representation on the middle homology of the induced G-cover of Y.

3.2 G compact Lie group

Definition 3.2 (Atiyah-Singer). Let G be a compact Lie group acting freely on a manifold \widetilde{X}^{2d-1}. Suppose in addition that there is a manifold with boundary (Y,\partial Y) on which G acts (not necessarily freely) and such that \partial Y = \widetilde X. Define

\displaystyle  \rho_G (\widetilde X) \co G \ni g \mapsto \mathrm{sign_G} (g,Y) - L(g,Y) \in \Cc.

See [Atiyah&Singer1968b, Theorem 7.4] for more details.

In this definition we think about the \rho-invariant as about a function G \smallsetminus \{1\} \rightarrow \Cc.

When both definitions apply (that means when G is a finite group), then they coincide, that means \rho (X) = \rho_G (\widetilde X).

Note that the invariant is natural under restriction from a group G to a subgroup: if H \le G, and G acts freely on \widetilde X, then \rho_G (\widetilde X)|_{H\smallsetminus\{1\}}=\rho_H (\widetilde X).

4 References

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