I've read some beautiful mathematics about analogous definitions of degrees and their implications for degree theory. This is basically a question asking someone to dumb down Javier Álvarez's explanation. I don't know any algebraic topology.
The intuitive description via Wikipedia is that it "represents the number of times that the domain manifold wraps around the range manifold under the mapping".
Next for the formal definitions, I have
Let $M$, $N$, be orientable manifolds of equal dimension, $M$ compact, $N$ connected and $F : M \rightarrow N$ a mapping. The degree of $F$, is defined as $$\deg(F) := \sum_{p \in F^{-1}(\{q\})} \operatorname{sgn}( \det (d F_p)),$$ for a regular value $q \in N$ of $F$.
I do not understand why this is independent of regular value selected?
Wikipedia also has a definition of degree as $n = r - s$ where $ r $ is the number of components of a point which have their orientation preserved and $ s $ is the number of components of a point which have their orientation reversed.
The functions I would like to consider to aid my intuition is the antipodal identification of a sphere and a permutation of the indices of a point on the sphere. To be formal : $ f : S ^ { n } \rightarrow S ^ { n } $
- $ f ( x ) = - x $ . Now $ \det (d F_p) $ is -1 on the diagonal for $ n+1 $ points, so, I have $ \deg(F) = (-1)^{n+1} $.
The only thing I can picture is $ S ^ 2$, do I always swap the orientation of one component here? I can't picture this.
- $ f \left( x _ { 0 } , \dots , x _ { n } \right) = \left( x _ { \sigma ( 0 ) } , \dots , x _ { \sigma ( n ) } \right) $
for some permutation $ \sigma $.
I'm considering this because I'm hoping to understand the second definition through this. I don't know where to start though, or how to imagine this as some winding, or how to compare it to the differential definition. If anyone could spell this out for me I would be eternally grateful.
