# GKM manifold

## 1 Definition


1. the fixed point set $M^T$$M^T$ is finite,
2. the set $E$$E$ of the complex one-dimensional orbits is finite,
3. the action is equivariantly formal, that is, the Leray-Serre cohomology spectral sequence for the Borel construction
$\displaystyle M \hookrightarrow ET\times_T M \to BT$
collapses at $E_2$$E_2$-term.

The letters G', K' and `M' correspond to the first initials of the authors of [Goresky&Kottwitz&MacPherson1998] where GKM manifolds were first introduced.

Let $m=\dim_\C(M)$$m=\dim_\C(M)$. Then it is shown in [Goresky&Kottwitz&MacPherson1998] that $(M^T,E)$$(M^T,E)$ forms an $m$$m$-valent simple graph in the sense that the following hold:

1. The closure of each orbit $e\in E$$e\in E$ is an embedded $\C P^1$$\C P^1$ containing exactly two fixed points at the north and the south pole.
2. At any $p\in M^T$$p\in M^T$, the closures of $m$$m$ orbits meet.

For any edge connecting $p,q\in M^T$$p,q\in M^T$, let $\alpha_{pq} \in H^2(BT)$$\alpha_{pq} \in H^2(BT)$ be the weight of the isotropic representation of $T$$T$ on $T_p( \C P^1 )$$T_p( \C P^1 )$, where $\C P^1$$\C P^1$ is the corresponding closure of the one-dimensional $T$$T$-orbit. The graph $(M^T,E)$$(M^T,E)$ together with this data often referred to as the GKM graph associated to $M$$M$. The main theorem in [Goresky&Kottwitz&MacPherson1998] states that the rational $T$$T$-equivariant Borel cohomology of $M$$M$, $H^*_T(M;\Q)$$H^*_T(M;\Q)$, is described in terms of the GKM graph:

Theorem 1.1 [Goresky&Kottwitz&MacPherson1998].

$\displaystyle H^*_T(M;\Q) \simeq \{ (f_p) \in \bigoplus_{p \in M^T} H^*(BT;\Q) \mid f_p - f_q \in (\alpha_{pq}), \text{ for any edge connecting p and q } \},$
where $(\alpha_{pq})$$(\alpha_{pq})$ is the ideal generated by $\alpha_{pq} \in H^2(BT; \Q)$$\alpha_{pq} \in H^2(BT; \Q)$.

## 2 Examples

Projective toric manifolds and flag manifolds are examples of GKM manifolds and the above theorem corresponds to the piecewise polynomial description ([Brion&Vergne1997]) and Arabia's description ([Arabia1989]) of the $T$$T$-equivariant cohomology of those manifolds.

## 3 Further discussion

Several generalisation of the theory are known; for example, [Braden&MacPherson2001] for equivariant intersection homology, [Rosu2003] for equivariant $K$$K$-theory, [Harada&Henriques&Holm2005] for complex oriented cohomology theories, [Guillemin&Holm2004] for action with non-isolated fixed points, and [Goertsches&Mare2013] for non-abelian group action. For introductory surveys of GKM manifolds, see [Tymoczko2005][Kuroki2009]. The theory of GKM manifolds has applications also in combinatorics: see, for example, [Guillemin&Zara2001] or [Bolker&Guillemin&Holm2002].