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# Variations of Homotopy Groups

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 ] .

• Hawaiian earring group

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 compact-open 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.

• Guth’s ﬁltration

According to Guth [ Guta ] , it was Gromov [ Gro78 ] who ﬁrst studied the relation between 1-dilations 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 ] .

• [ L ]-homotopy groups

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.

• Steenrod homotopy groups

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.