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 constraints.

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 [[bar.[gamma]].sub.i].

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 [1] 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 of [bar.c].

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.