#
nLab

winding number

### Context

#### Homotopy theory

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

# Contents

## Idea

A continuous function between two topological spaces that are each weakly homotopy equivalent to the circle is, up to homotopy, defined by how much it “winds” one circle around the other. This is its *winding number*, an integer. Equivalently this is the element in the fundamental group of the codomain which is represented by the map.

A special case of the *degree of a continuous function*.

Last revised on July 1, 2017 at 11:49:54.
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