Finiteness in the minimal models of Sullivan

Let X be a 1-connected topological space such that the vector spaces ${\Pi _ \ast }(X) \otimes {\mathbf{Q}}$ and ${H^\ast}(X;{\mathbf{Q}})$ are finite dimensional. Then ${H^\ast}(X;{\mathbf{Q}})$ satisfies Poincaré duality. Set ${\chi _\Pi } = \sum {( - 1)^p}\dim {\Pi _p}(X) \otimes {\mathbf{Q}}$ and ${\chi _c} =$ $\sum {( - 1)^p}\dim {H^p}(X;{\mathbf{Q}})$. Then ${\chi _\Pi } \leqslant 0$ and ${\chi _c} \geqslant 0$. Moreover the conditions: (1) ${\chi _\Pi } = 0$, (2) ${\chi _c} > 0,{H^\ast}(X;{\mathbf{Q}})$ evenly graded, are equivalent. In this case ${H^\ast}(X;{\mathbf{Q}})$ 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 ${\chi _\Pi } \leqslant - r$.