Parametric Analysis and Topological Studies on Differential Manifolds and Smooth Maps

Every manifold can be built by a successive attachment of handles of increasing dimension. To such a structure there is associated a chain complex yielding the homology of the manifold. The chains are linear combinations of handles and the boundary operator is given by a matrix of intersection numbers. Of course, the same is true for a triangulation or a cellular decomposition, but the relation between the handle presentation and the homology structure of the manifold is very transparent geometrically. The minimal number of handles necessary to build an n-dimensional sphere is two: two n-discs glued along boundaries, if we succeed in proving that a homotopy sphere admits a presentation with the minimal number of handles determined by its homology and then it must admit a presentation with two handles. The homology of the manifold is given by chain groups and homomorphisms described by matrices of intersection numbers. It is wellknown how this structure can be reduced through a sequence of algebraic operations to the most economical form, for instance, with all matrices diagonal, etc.

The main reason for introducing smooth structures was to enable us to define smooth functions on manifolds and smooth maps between manifolds. The study begins by defining smooth real-valued and vector-valued functions and then generalizes this to smooth maps between manifolds. We then focus our attention for a while on the special case of diffeomorphisms, which are bijective smooth maps with smooth inverses. If there is a diffeomorphism between two smooth manifolds, we say that they are diffeomorphic. The main objects of study in smooth manifold theory are properties that are invariant under diffeomorphisms.