# Rho-invariant

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

## 1 Introduction

The $\rho$$ == 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. == Definition == ; 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. As explained earlier, such a form yields an element in the representation ring R(G) which we denote by \mathrm{G-sign} (Y). The discussion in section \ref{sec:ses} also tells us that we have \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 \mathrm{G-sign} (-,Y) \colon g \in G \mapsto \mathrm{G-sign} (g,Y) \in \Cc. The (cohomological version of the) Atiyah-Singer G-index theorem \cite[Theorem (6.12)]{Atiyah-Singer-III(1968)} tells us that if Y is closed then for all g \in G \mathrm{G-sign} (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) 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 \cite[chapter 14B]{Wall(1999)}. The assumption that Y is closed is essential here, and motivates the definition of the \rho-invariant. In fact, Atiyah and Singer provide two definitions. For the first one one also needs the result of Conner and Floyd \cite{Conner-Floyd(1964)} that for an odd-dimensional manifold X with a finite fundamental group there always exists a k \in \Nn and a manifold with boundary (Y,\partial Y) such that \pi_1 (Y) \cong \pi_1 (X) and \partial Y = k \cdot X. {{beginthm|Definition|(Atiyah-Singer)}} \label{defn-rho-1} 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} (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. {{endthm}} See \cite[Remark after Corollary 7.5]{Atiyah-Singer-III(1968)} By the Atiyah-Singer G-index theorem \cite[Theorem (6.12)]{Atiyah-Singer-III(1968)} is \rho well defined. {{beginthm|Definition|??}} \label{defn-rho-2} 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. {{endthm}} 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). == References == {{#RefList:}} [[Category:Theory]] {{Stub}}\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$$\rho$-invariant. There is also another definition using bordism theory. Finally there is also an analytic defintion as a relative $\eta$$\eta$-invariant.

## 2 Definition

Let $G$$G$ be a compact Lie group acting smoothly on a smooth manifold $Y^{2d}$$Y^{2d}$. The middle intersection form becomes a non-degenerate $(-1)^d$$(-1)^d$-symmetric bilinear form on which $G$$G$ acts. As explained earlier, such a form yields an element in the representation ring $R(G)$$R(G)$ which we denote by $\mathrm{G-sign} (Y)$$\mathrm{G-sign} (Y)$. The discussion in section \ref{sec:ses} also tells us that we have $\mathrm{G-sign} (Y) \in R^{(-1)^d} (G)$$\mathrm{G-sign} (Y) \in R^{(-1)^d} (G)$ which in terms of characters means that we obtain a real (purely imaginary) character, which will be denoted as $\mathrm{G-sign} (-,Y) \colon g \in G \mapsto \mathrm{G-sign} (g,Y) \in \Cc$$\mathrm{G-sign} (-,Y) \colon g \in G \mapsto \mathrm{G-sign} (g,Y) \in \Cc$. The (cohomological version of the) Atiyah-Singer $G$$G$-index theorem [Atiyah-Singer-III(1968), Theorem (6.12)] tells us that if $Y$$Y$ is closed then for all $g \in G$$g \in G$

$\displaystyle \mathrm{G-sign} (g,Y) = L(g,Y) \in \Cc,$

where $L(g,Y)$$L(g,Y)$ is an expression obtained by evaluating certain cohomological classes on the fundamental classes of the $g$$g$-fixed point submanifolds $Y^g$$Y^g$ of $Y$$Y$. In particular if the action is free then $\mathrm{G-sign} (g,Y) = 0$$\mathrm{G-sign} (g,Y) = 0$ if $g \neq 1$$g \neq 1$. This means that $\mathrm{G-sign} (Y)$$\mathrm{G-sign} (Y)$ is a multiple of the regular representation. This theorem was generalized by Wall to topological semifree actions on topological manifolds, which is the case we will need in this paper [Wall(1999), chapter 14B]. The assumption that $Y$$Y$ is closed is essential here, and motivates the definition of the $\rho$$\rho$-invariant. In fact, Atiyah and Singer provide two definitions. For the first one one also needs the result of Conner and Floyd [Conner-Floyd(1964)] that for an odd-dimensional manifold $X$$X$ with a finite fundamental group there always exists a $k \in \Nn$$k \in \Nn$ and a manifold with boundary $(Y,\partial Y)$$(Y,\partial Y)$ such that $\pi_1 (Y) \cong \pi_1 (X)$$\pi_1 (Y) \cong \pi_1 (X)$ and $\partial Y = k \cdot X$$\partial Y = k \cdot X$.

Definition 2.1 (Atiyah-Singer).

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

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

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

See [Atiyah-Singer-III(1968), Remark after Corollary 7.5]

By the Atiyah-Singer $G$$G$-index theorem [Atiyah-Singer-III(1968), Theorem (6.12)] is $\rho$$\rho$ well defined.

Definition 2.2 ??.

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

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

In this definition we think about the $\rho$$\rho$-invariant as about a function $G \smallsetminus \{1\} \rightarrow \Cc$$G \smallsetminus \{1\} \rightarrow \Cc$. When both definitions apply (that means when $G$$G$ is a finite group), then they coincide, that means $\rho (X) = \rho_G (\widetilde X)$$\rho (X) = \rho_G (\widetilde X)$.