cohomotopy groups

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, 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
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *C ...
of pointed topological spaces and basepoint-preserving
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
maps to the category of sets and
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
. They are
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
to the
homotopy groups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, but less studied.

# Overview

The ''p''-th cohomotopy set of a pointed topological space ''X'' is defined by :$\pi^p\left(X\right) = \left[X,S^p\right]$ 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 $H^1\left(X\right)$, since the circle $S^1$ is an Eilenberg–MacLane space of type $K\left(\mathbb,1\right)$. In fact, it is a theorem of Heinz Hopf that if $X$ is a CW-complex of dimension at most ''p'', then $\left[X,S^p\right]$ is in bijection with the ''p''-th cohomology group $H^p\left(X\right)$. The set $\left[X,S^p\right]$ 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 $H^1\left(X\right)$ might not be isomorphic to $\left[X,S^1\right]$. 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.

# Properties

Some basic facts about cohomotopy sets, some more obvious than others: * $\pi^p\left(S^q\right) = \pi_q\left(S^p\right)$ for all ''p'' and ''q''. * For $q= p + 1$ and $p > 2$, the group $\pi^p\left(S^q\right)$ 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 $\, f\left(x\right) - g\left(x\right)\, < 2$ for all ''x'', then $\left[f\right] = \left[g\right]$, and the homotopy is smooth if ''f'' and ''g'' are. * For $X$ a compact space, compact smooth manifold, $\pi^p\left(X\right)$ 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 $\pi^p\left(X\right)=0$ for $p > m$. * If $X$ is an $m$-manifold#Manifold with boundary, manifold with boundary, the set $\pi^p\left(X,\partial X\right)$ 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 :$\pi^p_s\left(X\right) = \varinjlim_k$ :which is an abelian group.

# References

Homotopy theory {{topology-stub