Ramification - in Algebraic Topology

In Algebraic Topology

In a covering map the Euler-Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The z zn mapping shows this as a local pattern: if we exclude 0, looking at 0 < |z| < 1 say, we have (from the homotopy point of view) the circle mapped to itself by the n-th power map (Euler-Poincaré characteristic 0), but with the whole disk the Euler-Poincaré characteristic is 1, n – 1 being the 'lost' points as the n sheets come together at z = 0.

In geometric terms, ramification is something that happens in codimension two (like knot theory, and monodromy); since real codimension two is complex codimension one, the local complex example sets the pattern for higher-dimensional complex manifolds. In complex analysis, sheets can't simply fold over along a line (one variable), or codimension one subspace in the general case. The ramification set (branch locus on the base, double point set above) will be two real dimensions lower than the ambient manifold, and so will not separate it into two 'sides', locally―there will be paths that trace round the branch locus, just as in the example. In algebraic geometry over any field, by analogy, it also happens in algebraic codimension one.