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Allen Knutson
Allen Knutson
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Here's are two natural things to ask about any compact group action on a compact manifold: (1) what are the (finitely many) conjugacy classes of stabilizer groups? Assume our group is a torus, so we can omit "conjugacy classes of". (2) For each such stabilizer, what are the (finitely many) components of its fixed point set?

So, we have a poset of such submanifolds, and a torus subgroup for each. Now comes the additional question in the Hamiltonian situation: (3) what's the moment polytope for each of these submanifolds?

In actual examples, (1-3) are easy to figure out, and it's silly to answer (3) only for the whole manifold (or one might say, only for the fixed points).

You ask if $H^*_T(X;\mathbb Q)$ can be computed from this sort of data alone. In the case that the minimal strata in (2) are all $S^2$s, $X$ is called a GKM space (for Goresky-Kottwitz-MacPherson), and indeed it can; it's easy to find references with that keyword.

Here's are two natural things to ask about any compact group action on a compact manifold: (1) what are the (finitely many) conjugacy classes of stabilizer groups? Assume our group is a torus, so we can omit "conjugacy classes of". (2) For each such stabilizer, what are the (finitely many) components of its fixed point set?

So, we have a poset of such submanifolds, and a torus subgroup for each. Now comes the additional question in the Hamiltonian situation: (3) what's the moment polytope for each of these submanifolds?

In actual examples, (1-3) are easy to figure out, and it's silly to answer (3) only for the whole manifold (or one might say, only for the fixed points).

You ask if $H^*_T(X;\mathbb Q)$ can be computed from this sort of data alone. In the case that the minimal strata in (2) are all $S^2$s, $X$ is called a GKM space (for Goresky-Kottwitz-MacPherson), and indeed it can; it's easy to find references with that keyword.

Here's two natural things to ask about any compact group action on a compact manifold: (1) what are the (finitely many) conjugacy classes of stabilizer groups? Assume our group is a torus, so we can omit "conjugacy classes of". (2) For each such stabilizer, what are the (finitely many) components of its fixed point set?

So, we have a poset of such submanifolds, and a torus subgroup for each. Now comes the additional question in the Hamiltonian situation: (3) what's the moment polytope for each of these submanifolds?

In actual examples, (1-3) are easy to figure out, and it's silly to answer (3) only for the whole manifold (or one might say, only for the fixed points).

You ask if $H^*_T(X;\mathbb Q)$ can be computed from this sort of data alone. In the case that the minimal strata in (2) are all $S^2$s, $X$ is called a GKM space (for Goresky-Kottwitz-MacPherson), and indeed it can; it's easy to find references with that keyword.

Source Link
Allen Knutson
Allen Knutson
  • 27.8k
  • 4
  • 54
  • 152

Here's are two natural things to ask about any compact group action on a compact manifold: (1) what are the (finitely many) conjugacy classes of stabilizer groups? Assume our group is a torus, so we can omit "conjugacy classes of". (2) For each such stabilizer, what are the (finitely many) components of its fixed point set?

So, we have a poset of such submanifolds, and a torus subgroup for each. Now comes the additional question in the Hamiltonian situation: (3) what's the moment polytope for each of these submanifolds?

In actual examples, (1-3) are easy to figure out, and it's silly to answer (3) only for the whole manifold (or one might say, only for the fixed points).

You ask if $H^*_T(X;\mathbb Q)$ can be computed from this sort of data alone. In the case that the minimal strata in (2) are all $S^2$s, $X$ is called a GKM space (for Goresky-Kottwitz-MacPherson), and indeed it can; it's easy to find references with that keyword.