Finiteness in the minimal models of Sullivan

Abstract

Let X be a 1-connected topological space such that the vector spaces I7I*(X) 0 Q and H*(X; Q) are finite dimensional. Then H*(X; Q) satisfies Poincare duality. Set Xr, = E(I)Pdim rlp(X) 0 Q and X, = X(I)Pdim HP(X; Q). Then Xri 0. Moreover the conditions: (1) Xrn = 0, (2) X, > 0, H*(X; Q) evenly graded, are equivalent. In this case H*(X; Q) is a polynomial algebra truncated by a Borel ideal. Finally, if X is a finite 1-connected C.W. complex, and an r-torus acts continuously on X with only finite isotropy, then Xn < r.