# Rho-invariant

## 1 Introduction


## 2 Background

### 2.1 G-index theorem

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. The positive and negative definite subspaces are $G$$G$-invariant and hence such a form yields an element in the representation ring $R(G)$$R(G)$ denoted by $\mathrm{sign_G} (Y)$$\mathrm{sign_G} (Y)$. In fact $\mathrm{sign_G} (Y) \in R^{(-1)^d} (G)$$\mathrm{sign_G} (Y) \in R^{(-1)^d} (G)$ which in terms of characters means that we obtain a real (case $d$$d$ even) / purely imaginary (case $d$$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$$G$-index theorem [Atiyah&Singer1968b, Theorem (6.12)] tells us that if $Y$$Y$ is closed then for all $g \in G$$g \in G$

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

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{sign_G} (g,Y) = 0$$\mathrm{sign_G} (g,Y) = 0$ if $g \neq 1$$g \neq 1$. This means that $\mathrm{sign_G} (Y)$$\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$$Y$ is closed is essential, and motivates the definition of the $\rho$$\rho$-invariant.

### 2.2 Cobordism theory

Also the result from cobordism theory is needed which says that for an odd-dimensional manifold $X$$X$ with a finite fundamental group $G$$G$ 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) \cong G$$\pi_1 (Y) \cong \pi_1 (X) \cong G$ and $\partial Y = k \cdot X$$\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}$$X^{2d-1}$ be a closed manifold with $\pi_1 (X) \cong G$$\pi_1 (X) \cong G$ a finite group. Define

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

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&Singer1968b, Remark after Corollary 7.5] for more details. Note that the manifold $Y$$Y$ in the definition always exists by the above mentioned result in cobodism theory. Furthermore the invariant is well-defined thanks to the cohomological version of the $G$$G$-index theorem.

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

### 3.2 G compact Lie group

Definition 3.2 (Atiyah-Singer). 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

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

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

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

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