Abstract
We show that S^n vee S^m is {mathbb {Z}}/p^r-hyperbolic for all primes p and all r in {mathbb {Z}}^+, provided n,m ge 2, and consequently that various spaces containing S^n vee S^m as a p-local retract are {mathbb {Z}}/p^r-hyperbolic. We then give a K-theory criterion for a suspension Sigma X to be p-hyperbolic, and use it to deduce that the suspension of a complex Grassmannian Sigma Gr_{k,n} is p-hyperbolic for all odd primes p when n ge 3 and 0<k<n. We obtain similar results for some related spaces.
Highlights
A space X is called rationally elliptic if π∗(X ) ⊗ Q is finite dimensional, and rationally hyperbolic if the dimension of i≤m πi (X ) ⊗ Q grows exponentially in m
We say that X is p-hyperbolic if the number of p-torsion summands in π∗(X ) grows exponentially, in the sense that lim inf ln(Tm) > 0, m m where Tm is the number of p-torsion summands in i≤m πi (X )
One advantage of ordinary cohomology is that it is connected to the homotopy groups integrally, via the universal coefficient theorem and Hurewicz map
Summary
A space X is called rationally elliptic if π∗(X ) ⊗ Q is finite dimensional, and rationally hyperbolic if the dimension of i≤m πi (X ) ⊗ Q grows exponentially in m. By a result of Henn [17], any co-H space, and in particular any suspension, decomposes rationally as a wedge of spheres It follows from the Hilton-Milnor theorem [19] and the computation of the rational homotopy groups of spheres [31] that such a suspension is rationally hyperbolic precisely when there are at least two spheres (of dimension ≥ 2) in this decomposition. One advantage of ordinary cohomology is that it is connected to the homotopy groups integrally, via the universal coefficient theorem and Hurewicz map By the universal coefficient theorem relating ordinary homology and cohomology, μ1 ∨ μ2 induces a surjection on integral cohomology, so by Corollary 2, X is p-hyperbolic for all odd primes p. An overview of the proof strategy can be found at the start of Sect. 7
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callSimilar Papers
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
