Circle

As a subset of the Euclidean plane
A circle with center $$O$$ and radius $$r > 0$$ is defined as the set of all points $$P$$ in the Euclidean plane at a distance of $$r$$ from $$O$$.

The unit circle is the circle whose center is at the origin and radius is $$1$$, it is defined as the following subset of the Euclidean plane:

$$\{ (x,y) \mid x^2 + y^2 = 1 \}$$

Under the identification of the Euclidean plane with the complex numbers, this can also be described as the set of complex numbers whose modulus is $$1$$.

Note that all circles are equivalent up to similarity transformations of the Euclidean plane.

As a quotient
The circle is defined as a quotient in either of the following equivalent ways:


 * 1) It is the quotient $$\R/\mathbb{Z}$$ in the sense of topological spaces. In other words it is the quotient of real numbers by the equivalence relation of differing by an integer.
 * 2) It is the quotient of the closed unit interval $$[0,1]$$ by the identification of the two points $$0$$ and $$1$$. In symbols, this is $$[0,1]/\{0,1\}$$.

As a topological space
As a topological space, a circle is defined as the topological space obtained by using the subspace topology from the Euclidean plane on any circle described above.

Homology groups
With coefficients in $$\mathbb{Z}$$, we have $$H_0(S^1) \cong H_1(S^1) \cong \mathbb{Z}$$. All the higher homology groups are zero.

For reduced homology groups, we have $$\tilde{H}_1(S^1) \cong \mathbb{Z}$$ and all the other reduced homology groups are zero.

More generally, for any module $$M$$ over any commutative unital ring $$R$$, we have $$H_0(S^1;M) \cong H_1(S^1;M) \cong M$$ and all higher homology groups are zero. For reduced homology groups, $$\tilde{H}_1(S^1;M) \cong M$$ and all the other reduced homology groups are zero.

Cohomology groups
With coefficients in $$\mathbb{Z}$$, we have $$H^0(S^1) \cong H^1(S^1) \cong \mathbb{Z}$$, and all the higher homology groups are zero. The cohomology ring is isomorphic to $$\mathbb{Z}[x]/(x^2)$$ where $$x$$ is an additive generator of the first cohomology group.

More generally, with coefficients in any commutative unital ring $$R$$, we have $$H^0(S^1;R) \cong H^1(S^1;R) \cong R$$ and the cohomology ring is isomorphic to $$R[x]/(x^2)$$ where $$x$$ is an additive generator of the first cohomology group.

Homotopy groups
We have $$\pi_0(S^1)$$ is the one-point set (or trivial group, if we use the H-space interpretation of $$S^1$$). The fundamental group $$\pi_1(S^1)$$ is isomorphic to the group of integers $$\mathbb{Z}$$, and all the higher homotopy groups are trivial.

In particular, the circle is a realization of the classifying space for the group of integers $$\mathbb{Z}$$. Its universal cover, the Euclidean line, is a contractible space.

Algebraic structure
The circle has the natural structure of an abelian group, which can be realized in many ways:


 * View it as the set of complex numbers with modulus 1, and perform multiplication of complex numbers.
 * View it as $$\R/\Z$$, i.e., the quotient of the additive group of real numbers by the discrete subgroup of integers.

In particular, the circle has the structure of a topological group. Further, the structure is that of a Lie group, and hence a compact connected Lie group.

Also, on account of being a topological group, the circle gets a H-space structure. Thus, its fundamental group is an abelian group and its zeroth homotopy set (which is a one-point set) naturally acquires a group structure (giving the trivial group).

Coalgebraic structure
Fix a base point $$p \in S^1$$. We can construct a comultiplication:

$$(S^1,p) \to (S^1,p) \vee (S^1,p)$$

where $$\vee$$ denotes the wedge sum and the map is a continuous based map, i.e., a continuous map preserving basepoint. This map is cocommutative and coassociative up to homotopy, and it is used to give a group structure to the set of homotopy classes from the based circle to any based topological space. This group is termed the fundamental group.