Editing Ramification (mathematics) (section)

==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'' &rarr;&nbsp;''z''<sup>''n''</sup> 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 (mathematics)|disk]] the Euler–Poincaré characteristic is 1, ''n''&nbsp;−&nbsp;1 being the 'lost' points as the ''n'' sheets come together at&nbsp;''z''&nbsp;=&nbsp;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 manifold]]s. 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 (mathematics)|field]], by analogy, it also happens in algebraic codimension one.