# homotopy groups

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## homotopy groups

[hō′mäd·ə·pē ‚grüps]
(mathematics)
Associated to a topological space X, the groups appearing for each positive integer n, which reflect the number of different ways (up to homotopy) than an n-dimensional sphere may be mapped to X.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Abu-Saleem, "On chaotic homotopy group," Advanced Studies in Contemporary Mathematics, vol.
of the stable homotopy group of the homotopy fixed-point set of f/p.
Homotopy is an equivalence relation on [[OMEGA].sup.n] (G, [x.sub.0]) The homotopy group of the based graph (g, [x.sub.0]) is defined as [[product].sub.n](G, [x.sub.0]) = [[product].sub.n] (G, [x.sub.0])/ ~.
Then [[pi].sub.1] (M([A.sub.n,k]) [congruent to] [A.sup.n-k+1.sub.1](C([A.sub.n])), where [A.sup.q.sub.1] is a discrete homotopy group, to be defined below.
It had been shown previously by Babson  and independently by Bjorner that that the discrete fundamental group of the permutahedron is isomorphic to the classical homotopy group of the real complement of the k-equal arrangement, [M.sub.n,k].
In this section we explain briefly how Kenzo can compute the homotopy group [[pi].sub.4] ([SIGMA]K([A.sub.4],1)).
To a pointed topological topological space (X, [x.sub.0]) one can associate the n-th homotopy group [[pi].sub.n](X).
Thus the set [T.sub.0](X) is more sensitive than the number [r.sub.0](X) about degrees of the rational homotopy group of X.
Especially, there is a sequence of subgroups in the m-th homotopy group [[pi].sub.m](Z):
They cover an overview of geometry and physics, spin systems for mathematicians, the Arf-Brown topological quantum field theory of pin(su)- surfaces, a guide for computing stable homotopy groups, flagged higher categories, how to derive Feynman diagrams for finite-dimensional integrals directly from the Batalin-Vilkovisky formalism, homotopy RG flow and the non-linear s-model, and the holomorphic bosonic string.
Since MacPherson's work, some progress on this question has been made, most notably by Anderson [And99], who obtained results on homotopy groups of the matroid Grassmannian, and by Anderson and Davis [AD02], who constructed maps between the real Grassmannian and the matroid Grassmannian--showing that philosophically, there is a splitting of the map from topology to combinatorics--and thereby gained some understanding of the mod 2 cohomology of the matroid Grassmannian.

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