A cycle has several distinct meanings in mathematics. Some describe a return to a starting point, while others are defined using a boundary operator or classes associated with algebraic varieties.
In graph theory, a graph cycle follows edges back to its initial graph vertex, with no other repeated vertices. A cycle graph is a graph consisting of a single graph cycle.
A permutation cycle describes distinct symbols permuted in a cyclic order. For example, 
sends 1 to 2, 2 to 3, and 3 to 1. A map
cycle is a periodic trajectory under repeated application of a map.
A group cycle records successive powers of an element
of a finite group, returning to the identity
element.
In homology, a homology cycle is an element 
of a chain complex whose boundary is zero, 
. Cycles modulo homology
boundaries define homology groups.
In algebraic geometry, an algebraic cycle is a finite formal linear combination
of irreducible closed subvarieties, usually with integer coefficients. A Hodge
cycle is instead a rational cohomology class of
type 
on a smooth projective complex algebraic variety. The Hodge
conjecture relates Hodge cycles to the cohomology
classes of algebraic cycles.
