One of the oldest variations of
homotopy groups
is the mod
p
homotopy group introduced by Peterson
[
Pet56
]
. It is deﬁned by replacing spheres by mod
p
Moore space
. The properties are studied by Neisendorfer
[
Nei80
]
in detail. And the results are applied to the famous
work on the
exponent problem of homotopy groups of spheres
by Cohen, Moore, and Neisendorfer
[
CMN79b
,
CMN79a
]
.
We may also replace spheres by other spaces. For
example, the Hawaiian earring group is deﬁned
by using a countable wedge sum of spheres (Hawaiian
earring) in place of spheres. It is deﬁned and
used in
[
KR06
,
KR10
]
.
Homotopy groups (homotopy sets) are deﬁned by
quotienting out the set of continuous maps under an
equivalence relation. Thus it has a topology induced
from the compactopen topology.
An important variation related to the mod
p
homotopy group is the
v
_{
1
}
periodic homotopy group.
From the viewpoint of
chromatic homotopy theory
, the
v
_{
1
}
periodic homotopy group comes next to the
rational homotopy group
.
Besides the chromatic decomposition of homotopy
groups, there are some other ways to ﬁlter
homotopy groups. For example, Guth
[
Gutb
]
deﬁned a ﬁltration by using
k
dilations.
According to Guth
[
Guta
]
, it was Gromov
[
Gro78
]
who ﬁrst studied the relation between
1dilations and homotopy classes of maps. Gromov also
addressed problems on the relation between
k
dilations and homotopy in
[
Gro96
]
.
Baues and Muro
[
BM08a
]
introduced secondary homotopy groups in order to
study secondary operations. They also deﬁned
the secondary Whitehead product in
[
BM08b
]
. See also
[
BM08c
]
.
The homotopy groups of
simplicial sets
can be deﬁned by taking geometric
realizations. Or we may directly deﬁne homotopy
groups from the simplicial structure.

the (simplicial) homotopy groups of simplicial sets

the homotopy groups of simplicial Abelian groups
Chigogidze
[
Chi02
]
introduced [
L
]homotopy groups for a ﬁnite
CW complex
L
. The deﬁnition can be also found in
[
CK03
]
. There is an attempt of calculation in
[
Kar01
]
.
Chigogidze and Karasev
[
CK03
]
showed that the category of topological spaces can be
made into a
model category
in which weak equivalences are maps that induce
isomorphisms of [
L
]homotopy groups.
For compact metrizable spaces, we may deﬁne
the Steenrod homotopy groups. Melikhov
[
Mel09
]
describes Steenrod homotopy theory.
A variation for manifolds with singularities is the
intersection homotopy group.
There is another attempt due to Woolf
[
Woo10
]
. He deﬁnes transversal homotopy monoids for
Whitney stratiﬁed spaces.

transversal homotopy monoid
Fro the viewpoint of functional analysis, DeJarnette,
Hajlasz, Lukyanenko, and Tyson
[
DHLT
]
introduced Lipschitz homotopy groups and horizontal
homotopy groups.

Lipschitz homotopy groups

horizontal homotopy groups
They are deﬁned in order to study the
Sobolev spaces of maps between smooth manifolds
Wenger and Young
[
WY
]
study Lipschitz homotopy groups.
[BM08a]
HansJoachim Baues and Fernando Muro. Secondary
homotopy groups.
Forum Math.
, 20(4):631–677, 2008,
arXiv:math/0604029
.
[BM08b]
HansJoachim Baues and Fernando Muro. Smash products
for secondary homotopy groups.
Appl. Categ. Structures
, 16(5):551–616, 2008,
arXiv:math/0604031
.
[BM08c]
HansJoachim Baues and Fernando Muro. The symmetric
action on secondary homotopy groups.
Bull. Belg. Math. Soc. Simon Stevin
, 15(4):733–768, 2008,
arXiv:math/0604030
.
[Chi02]
Alex Chigogidze. Inﬁnite dimensional topology
and shape theory. In
Handbook of geometric topology
, pages 307–371. NorthHolland, Amsterdam,
2002.
[CK03]
Alex Chigogidze and Alexandr Karasev. Topological
model categories generated by ﬁnite
complexes.
Monatsh. Math.
, 139(2):129–150, 2003,
arXiv:math/0205014
.
[CMN79a]
F. R. Cohen, J. C. Moore, and
J. A. Neisendorfer. The double suspension and
exponents of the homotopy groups of spheres.
Ann.
of Math. (2)
, 110(3):549–565, 1979,
http://dx.doi.org/10.2307/1971238
.
[CMN79b]
F. R. Cohen, J. C. Moore, and
J. A. Neisendorfer. Torsion in homotopy
groups.
Ann. of Math. (2)
, 109(1):121–168, 1979,
http://dx.doi.org/10.2307/1971269
.
[DHLT]
Noel DeJarnette, Piotr Hajlasz, Anton Lukyanenko,
and Jeremy Tyson. On the lack of density of
Lipschitz mappings in Sobolev spaces with Heisenberg
target,
arXiv:1109.4641
.
[Gro78]
Mikhael Gromov. Homotopical effects of dilatation.
J. Differential Geom.
, 13(3):303–310, 1978,
http://projecteuclid.org/getRecord?id=euclid.jdg/1214434601
.
[Gro96]
Mikhael Gromov. CarnotCarathéodory spaces
seen from within. In
SubRiemannian geometry
, volume 144 of
Progr. Math.
, pages 79–323. Birkhäuser, Basel, 1996.
[Guta]
Larry Guth. Contraction of areas vs. topology of
mappings,
arXiv:1211.1057
.
[Gutb]
Larry Guth. Homotopically nontrivial maps with
small
k
dilation,
arXiv:0709.1241
.
[Kar01]
A. Karasev. On [
L
]homotopy groups.
JP J. Geom. Topol.
, 1(3):301–310, 2001,
arXiv:math/0001019
.
[KR06]
U. Kh. Karimov and D. Repovsh.
Hawaiian groups of topological spaces.
Uspekhi Mat. Nauk
, 61(5(371)):185–186, 2006,
http://dx.doi.org/10.1070/RM2006v061n05ABEH004363
.
[KR10]
Umed H. Karimov and Dušan Repovš.
On noncontractible compacta with trivial homology
and homotopy groups.
Proc. Amer.
Math. Soc.
, 138(4):1525–1531, 2010,
arXiv:0910.0531
.
[Mel09]
S. A. Melikhov. Steenrod homotopy.
Uspekhi Mat. Nauk
, 64(3(387)):73–166, 2009,
arXiv:0812.1407
.
[Nei80]
Joseph Neisendorfer. Primary homotopy theory.
Mem. Amer.
Math. Soc.
, 25(232):iv 67, 1980.
[Pet56]
Franklin P. Peterson. Generalized cohomotopy
groups.
Amer.
J. math.
, 78:259–281, 1956.
[Woo10]
Jonathan Woolf. Transversal homotopy theory.
Theory Appl.
Categ.
, 24:No. 7, 148–178, 2010,
arXiv:0910.3322
.
[WY]
Stefan Wenger and Robert Young. Lipschitz homotopy
groups of the Heisenberg groups,
arXiv:1210.6943
.