AbstractWe show that $$S^n \vee S^m$$
S
n
∨
S
m
is $${\mathbb {Z}}/p^r$$
Z
/
p
r
-hyperbolic for all primes p and all $$r \in {\mathbb {Z}}^+$$
r
∈
Z
+
, provided $$n,m \ge 2$$
n
,
m
≥
2
, and consequently that various spaces containing $$S^n \vee S^m$$
S
n
∨
S
m
as a p-local retract are $${\mathbb {Z}}/p^r$$
Z
/
p
r
-hyperbolic. We then give a K-theory criterion for a suspension $$\Sigma X$$
Σ
X
to be p-hyperbolic, and use it to deduce that the suspension of a complex Grassmannian $$\Sigma Gr_{k,n}$$
Σ
G
r
k
,
n
is p-hyperbolic for all odd primes p when $$n \ge 3$$
n
≥
3
and $$0<k<n$$
0
<
k
<
n
. We obtain similar results for some related spaces.
The homotopy groups of the algebraic K–theory
of the sphere spectrum
