One of the main advantages of working with infra-nilmanifolds, is the fact that every continuous map lies in the same homotopy class
as a map induced by such an affine map with similar properties.
It is usually formulated as finding navigation paths under homotopy class
Continuous modifications of the soliton field do not change Q and the homotopy class
of the configuration.
It is easy to show that the homotopy class
[bar.[gamma]] of [gamma] depends only upon the homotopy classes
In the context of closed orientable triangulable manifolds, we define a "pseudo-index" for BU-coincidence classes, then a Nielsen-Borsuk-Ulam number in such situation, demonstrating that said number is a lower bound for the number of pairs of coincidences between f and f [??] [tau] in the homotopy class
of f and that it can be realized (Wecken type theorem) when the dimension of the manifolds are greater than 2 (as usual in Nielsen theory).
Schirmer generalized the concept of CIPD and gave necessary and sufficient conditions for a nonempty closed subset A to be the fixed point set of a map g in the homotopy class
of a given selfmap f.
Those maps having the least number of fixed points possible among all maps in a given homotopy class
. The partition classes are single fixed points.
In our paper  we associated to a homotopy class
of a string link a certain group diagram and showed that two string links have the same closure, up to link-homotopy, if and only if there is a certain type of isomorphism between their group diagrams.
The naive notion of a homotopy action of a group G on a topological space X can be described as the choice of a homotopy class
of a map BG [right arrow] B haut(X), where haut(X) is the monoid of self-homotopy equivalences (see [section]1.1).
Thus the homotopy class
of c is uniquely determined by the homotopy class
Ten chapters discuss the skew field of quaternions; elements of the geometry of S3, Hopf bundles, and spin representations; internal variables of singularity free vector fields in a Euclidean space; isomorphism classes, Chern classes, and homotopy classes
of singularity free vector fields in 3-space; Heisenberg algebras, Heisenberg groups, Minkowski metrics, Jordan algebras, and special linear groups; the Heisenbreg group and natural C*-algebras of a vector field in 3-space; the Schrodinger representation and the metaplectic representation; the Heisenberg group as a basic geometric background of signal analysis and geometric optics; quantization of quadratic polynomials; and field theoretic Weyl quantization of a vector field in 3-space.
Let Ph (X, Y) denote the set of pointed homotopy classes
of phantom maps from X to Y.