cohomotopy groups


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, particularly
algebraic topology 250px, A torus, one of the most frequently studied objects in algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathemat ...
, cohomotopy sets are particular contravariant functors from the
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of pointed topological spaces and basepoint-preserving
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maps to the category of sets and
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. They are
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, but less studied.


The ''p''-th cohomotopy set of a pointed topological space ''X'' is defined by :\pi^p(X) = [X,S^p] the set of pointed homotopy classes of continuous mappings from X to the ''p''-hypersphere, sphere S^p. For ''p'' = 1 this set has an abelian group structure, and, provided X is a CW-complex, is group isomorphism, isomorphic to the first cohomology group H^1(X), since the circle S^1 is an Eilenberg–MacLane space of type K(\mathbb,1). In fact, it is a theorem of Heinz Hopf that if X is a CW-complex of dimension at most ''p'', then [X,S^p] is in bijection with the ''p''-th cohomology group H^p(X). The set [X,S^p] also has a natural group (mathematics), group structure if X is a suspension (topology), suspension \Sigma Y, such as a sphere S^q for q \ge 1. If ''X'' is not homotopy equivalent to a CW-complex, then H^1(X) might not be isomorphic to [X,S^1]. A counterexample is given by the Warsaw circle, whose first cohomology group vanishes, but admits a map to S^1 which is not homotopic to a constant map.Polish Circle
Retrieved July 17, 2014.


Some basic facts about cohomotopy sets, some more obvious than others: * \pi^p(S^q) = \pi_q(S^p) for all ''p'' and ''q''. * For q= p + 1 and p > 2, the group \pi^p(S^q) is equal to \mathbb_2. (To prove this result, Lev Pontryagin developed the concept of framed cobordism.) * If f,g\colon X \to S^p has \, f(x) - g(x)\, < 2 for all ''x'', then [f] = [g], and the homotopy is smooth if ''f'' and ''g'' are. * For X a compact space, compact smooth manifold, \pi^p(X) is isomorphic to the set of homotopy classes of smooth function, smooth maps X \to S^p; in this case, every continuous map can be uniformly approximated by a smooth map and any homotopic smooth maps will be smoothly homotopic. * If X is an m-manifold, then \pi^p(X)=0 for p > m. * If X is an m-manifold#Manifold with boundary, manifold with boundary, the set \pi^p(X,\partial X) is natural isomorphism, canonically in bijection with the set of cobordism classes of codimension-''p'' framed submanifolds of the Interior (topology), interior X \setminus \partial X. * The stable cohomotopy group of X is the colimit :\pi^p_s(X) = \varinjlim_k :which is an abelian group.


Homotopy theory {{topology-stub