Exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1-connected complex

• 成果类型: Article
• 署名作者: Felix, Yves; Halperin, Steve; Thomas, Jean-Claude
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2009.170.443
• 发表日期: 2009
• 页码: 443-464
• 关键词: lie-algebras

摘要

Let X be an n-dimensional, finite, simply connected CW complex and set alpha(X) = lim sup(i) (log rank pi(i) (X))/i. We prove that either rank pi(i) (X) = 0, i >= 2n or else that 0 < alpha(X) < infinity and that for any epsilon > 0 there is a K = K(epsilon) such that e((alpha X - epsilon)k) <= Sigma(k+n)(i=k+2) rank pi(i) (X) <= e((alpha X + epsilon)k), for all k >= K. In particular, this sum grows exponentially in k.