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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.

References

[BM08a]     Hans-Joachim Baues and Fernando Muro. Secondary homotopy groups. Forum Math. , 20(4):631–677, 2008, arXiv:math/0604029 .

[BM08b]     Hans-Joachim Baues and Fernando Muro. Smash products for secondary homotopy groups. Appl. Categ. Structures , 16(5):551–616, 2008, arXiv:math/0604031 .

[BM08c]     Hans-Joachim 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. North-Holland, 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. Carnot-Carathéodory spaces seen from within. In Sub-Riemannian 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 non-trivial 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 .