# 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 [2] 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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