Finiteness in the Minimal Models of Sullivan

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