Hello Liangping

You may try to look at the following paper by Misha Gromov:

Singularities, expanders and topology of map. Part 2: From combinatorics to topology via algebraic isoperimetry. GAFA, Geom. func. anal.,20 (2010), 416-526.

available at:

http://www.ihes.fr/~gromov/PDF/morse2_gafa.pdf

It deals with topological/geometric measures of complexity of maps from spaces (typically CW complexes) of higher dimension to those of lower dimension.

If you have a map s: V -> W  and the map is Fredholm, i.e. the derivative is Fredholm in every point of the zero's of s, then the zero cycle carries a fundamental class in locally finite homology of dimension the Fredholm index. The best (although not optimal) description of this construction  I know of is chapter 3 of my paper  The Canonical Class and the C^\infty properties of algebraic surfaces. 

http://www.emis.ams.org/journals/NYJM/j/1996/2-7p.pdf

See also Bernd Siebert's Paper  Gromov-Witten invariants for general symplectic manifolds http://arxiv.org/pdf/dg-ga/9608005.pdf

The term degree refers exclusively to maps between manifolds of the same dimension. When dealing with manifolds of different dimensions there is the so called Pontryagin-Thom construction that  is associated to a smooth map  f:M-->N, where M, an N are compact oriented manifolds, dim M  dim  N.   The degree will be the framed cobodism class of a  generic fiber of a map. When dim M = dim M, the framed cobordism class can be identified with an integer, the degree. For details I refer to Chapter IX of Kosinski's book   Differentiable Manifolds.  (By the way, the crew running REsearch Gate has not yet figured out how to  activate MathJax on this site rendering useless for mathematical conversations.)

Thanks very much for your help. 

This question arises from applications of coincidence degree theory to differential equations of the type Lx=Nx, where L is the differential operator, also a Fredholm operator of index zero. It happens that the operator N can not be well-defined on the whole function space (say Y). It is the case that the image N(Y)  is not contained in Y (some point of N(Y) is not in Y). So my idea is to restrict the domain of N to a subspace of Y, such as

dom N={ x in Y; N(x) is in Y},

and consider a degree for mappings between spaces of different dimensions (from a space of lower dimension to a space of higer dimension). I do not know whethe it is possible to do in this way.

I'm happy to receive so many valuble advices from all of you.

Best wishes,

Liangping Qi