In this paper we study self-maps of
Ω
k
S
m
+
1
{\Omega ^k}{S^{m + 1}}
and show that, except for the cases
m
=
1
,
3
,
7
m = 1,\,3,\,7
, or
p
p
and
m
m
, if
f
f
induces an isomorphism on
H
m
+
1
−
k
(
Ω
k
S
m
+
1
;
Z
/
p
Z
)
{H_{m + 1 - k}}({\Omega ^k}{S^{m + 1}};\,Z/pZ)
with
k
>
m
k > m
, then
f
(
p
)
{f_{(p)}}
is a homotopy equivalence.