In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and point-preserving continuous maps to the category of sets and functions. They are dual to the homotopy groups, but less studied.
== Overview ==
The p-th cohomotopy set of a pointed topological space X is defined by
π p(X) = [X,S p]
the set of pointed homotopy classes of continuous mappings from X to the p-sphere S p. For p=1 this set has an abelian group structure, and, provided X is a CW-complex, is isomorphic to the first cohomology group H1...
Let $X,Y$ be compact connected manifolds and $\varphi\colon\pi_1(X)\to\pi_1(Y)$ be a homomorphism between their fundamental groups. Under what conditions on $X$, $Y$ and $\varphi$ is it true that $\varphi$ is the homomorphism induced by an appropriate continuous map $f\colon X\to Y$?
