An asymptotic formula for the ranks of homotopy groups
- Felix, Yves
- Halperin, Steve
- Thomas, Jean-Claude
DOIAbstract
AbstractLet X be a finite simply connected CW complex of dimension n. The loop space homology H∗(ΩX;Q) is the universal enveloping algebra of a graded Lie algebra LX isomorphic with π∗−1(X)⊗Q. Let QX⊂LX be a minimal generating subspace, and set α=lim supilogrkπi(X)i.Theorem: If dimLX=∞ and lim sup(dim(QX)k)1/k<lim supdim(LX)k1/k, then∑i=1n−1rkπk+i(X)=e(α+εk)k,where εk→0 as k→∞. In particular ∑i=1n−1rkπk+i(X) grows exponentially in k
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