Subject: Topology of 7-manifolds [for toplist]
Date: Sun, 8 Apr 2001 08:39:33 -0700
From: Greg Kuperberg <greg@math.ucdavis.edu>
To: dmd1@lehigh.edu

Dear toplist,

For a period last year I was interested in G_2 manifolds, which are
Riemannian 7-manifolds with exceptional holonomy.  There is a theorem that
a closed G_2 manifold has to have finite pi_1.  So to get some context,
how do you classify smooth, simply connected, closed 7-manifolds?
Such a 7-manifold M has a cohomology ring.  Poincare duality and the
coefficient theorem place some restrictions on this ring.  The group of
smooth structures of S^7, which is Z/28, acts transitively on the smooth
structures on M by connected sum.  So

1) Are there further restrictions on H^*(M)?
2) How do you compute the number of smooth structures?
3) What other structure might M have?

--
  /\  Greg Kuperberg (UC Davis)
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