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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 55Q: Homotopy groups

Introduction

History

Applications and related fields

The "homotopy continuation methods" in numerical analysis and control are essentially unrelated to homotopy theory (but rather are more akin to analytic continuation in complex analysis.) One is, at best, using a linear homotopy between two constant maps into M_n(R).

Subfields

• 55Q05: Homotopy groups, general; sets of homotopy classes
• 55Q07: Shape groups
• 55Q10: Stable homotopy groups
• 55Q15: Whitehead products and generalizations
• 55Q20: Homotopy groups of wedges, joins, and simple spaces
• 55Q25: Hopf invariants
• 55Q35: Operations in homotopy groups
• 55Q40: Homotopy groups of spheres
• 55Q45: Stable homotopy of spheres
• 55Q50: J-morphism, See also 19L20
• 55Q51: v_n-periodicity [new in 2000]
• 55Q52: Homotopy groups of special spaces
• 55Q55: Cohomotopy groups
• 55Q70: Homotopy groups of special types, See also 55N05, 55N07
• 55Q91: Equivariant homotopy groups, See also 19L47
• 55Q99: None of the above but in this section

Parent field: 55: Algebraic Topology

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Software and tables

Other web sites with this focus

Selected topics at this site

• What is the fundamental group and what does it have to do with knot theory?
• Survey article on homotopy groups of spheres [John Baez]
• What is Bott periodicity (homotopy groups of SO(n) and related topics).
• Homotopy groups of SO(3) (special orthogonal group).
• Generator of homotopy group pi_7(O) of orthogonal group, spheres
• Fundamental group of the space of all unlabeled orthogonal frames in R^3.
• To what extent do homotopy groups (say) determine a topological space?
• What is the fundamental group of the Hawaiian earring?
• Calculating the fundamental groups of (compact, connected, orientable) surfaces.
• Do subsets of R^2 and R^3 have torsion-free fundamental groups?
• Will X and its quotient space X/A have the same fundmental group under nice circumstances?
• Applications of homotopy groups to defects in matter
• What is the Hopf map between two spheres (of different dimensions)? [Chris Hillman]
• Homotopy classes of maps between spheres

Some topics on the question of explicit computability of higher homotopy groups (not a trivial issue!)

• General discussion regarding explicit homotopy computation.
• Francis Sergeraert on his approach to computable homotopy.