Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 ${\chi _\Pi } \leqslant - r$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55H05, 55D15

Retrieve articles in all journals with MSC: 55H05, 55D15

Additional Information

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