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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Finiteness in the minimal models of Sullivan

Author: Stephen Halperin
Journal: Trans. Amer. Math. Soc. 230 (1977), 173-199
MSC: Primary 55H05; Secondary 55D15
MathSciNet review: 0461508
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Abstract: 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$.

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Keywords: Minimal models, homotopy Euler characteristic, Koszul complex, torus action, finite isotropy
Article copyright: © Copyright 1977 American Mathematical Society