 Topological degree

For a continuous map $f:M\to N$ between connected closed oriented manifolds of the same positive dimension $n$, the topological degree is the integer determined byThe chosen fundamental classes fix the signs. This homology definition makes sense without differentiability and is invariant under homotopy. For spheres, it is the integer by which the induced map multiplies their top homology group.^[ref]

If $f$ is smooth and $y$ is a regular value, the finite preimage giveswhere the sign compares the chosen orientations in source and target. Thus opposite local signs can cancel: topological degree is a signed count, rather than the number of preimages.

For the circle map $z\mapsto z^k$, with $|z|=1$ and $k$ an integer, the topological degree is $k$. An orientation-reversing reflection has topological degree $-1$, while a constant map has topological degree zero. A nonzero topological degree forces surjectivity; a homotopy between continuous maps preserves their topological degree.