AbstractLet $${\mathbb {K}}={\mathbb {R}},\,{\mathbb {C}}$$
K
=
R
,
C
, the field of reals or complex numbers and $${\mathbb {H}}$$
H
, the skew $${\mathbb {R}}$$
R
-algebra of quaternions. We study the homotopy nilpotency of the loop spaces $$\Omega (G_{n,m}({\mathbb {K}}))$$
Ω
(
G
n
,
m
(
K
)
)
, $$\Omega (F_{n;n_1,\ldots ,n_k}({\mathbb {K}}))$$
Ω
(
F
n
;
n
1
,
…
,
n
k
(
K
)
)
, and $$\Omega (V_{n,m}({\mathbb {K}}))$$
Ω
(
V
n
,
m
(
K
)
)
of Grassmann $$G_{n,m}({\mathbb {K}})$$
G
n
,
m
(
K
)
, flag $$F_{n;n_1,\ldots ,n_k}({\mathbb {K}})$$
F
n
;
n
1
,
…
,
n
k
(
K
)
and Stiefel $$V_{n,m}({\mathbb {K}})$$
V
n
,
m
(
K
)
manifolds. Additionally, homotopy nilpotency classes of p-localized $$\Omega (G^+_{n,m}({\mathbb {K}})_{(p)})$$
Ω
(
G
n
,
m
+
(
K
)
(
p
)
)
and $$\Omega (V_{n,m}({\mathbb {K}})_{(p)})$$
Ω
(
V
n
,
m
(
K
)
(
p
)
)
for certain primes p are estimated, where $$G^+_{n,m}({\mathbb {K}})_{(p)}$$
G
n
,
m
+
(
K
)
(
p
)
is the oriented Grassmann manifolds. Further, the homotopy nilpotency classes of loop spaces of localized homogeneous spaces given as quotients of exceptional Lie groups are investigated as well.
Homotopy nilpotency of some homogeneous spaces
