##  [Branch Point](/branch-point-2) 

 Definition

A point in a covering map or projection where two or more local sheets meet and the local degree of the map drops; locally the map fails to be a homeomorphism (or is not locally bijective) because distinct preimages coalesce.

 

 

 

 

 

 





## Principle

Principle

A branch point is characterized by a change in local sheet structure of a covering: analytic continuation or monodromy around the image point permutes sheets nontrivially, and the local mapping degree is less than the generic degree.

 

 

 

 

 





## Demonstration

Demonstration

For the map z ↦ z^2 from the complex plane to itself, the origin is a branch point: nearby points have two distinct square roots (two sheets), but at 0 the two sheets meet and the local mapping degree drops from two to one.

 

 

 

 

## Misapplication

Misapplication

Calling any singular point of the domain a branch point is incorrect; a cusp or self-intersection of the domain that does not arise from the coalescence of covering sheets is not a branch point of the projection.

 

 

 

 

 





## Consequence

Consequence

Correct identification of branch points yields nontrivial monodromy representation of the fundamental group of the base, forces branch cuts or Riemann surface gluing in analytic models, and locates the discriminant locus of the covering.

 

 

 

 

## Reversal

Reversal

An unbranched (regular) point is one where the covering map is locally a homeomorphism and each point of the fiber is isolated with constant local degree equal to the generic degree.

 

 

 

 

 





## Boundary

Boundary

The term applies to coverings, projections, and finite analytic maps; it excludes arbitrary singularities that do not arise from sheet coalescence and does not by itself distinguish tame versus wild arithmetic ramification behaviors.

 

 

 

 

 





## Semantic Tension

Semantic Tension

In topology/complex analysis 'branch point' often overlaps with algebraic 'ramification point', but the two emphasize different features: branch point highlights sheet-gluing and monodromy, while ramification in algebraic/arithmetic contexts emphasizes algebraic inertia and valuation-theoretic indices.

 

 

 

 

 





## Synthesis

Synthesis

A branch point is a base-point of a covering or projection where the local multivalued structure degenerates: sheets meet, local degree drops, and analytic continuation around the point permutes the sheets nontrivially.