Circle

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Definition

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 $\mathbb{R}/\mathbb{Z}$ 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.

Equivalent spaces

Space	How strongly is it equivalent to the circle (in the geometric sense)?
Ellipse in ${\mathbb{R}}^2$	Equivalent up to an affine transformation
Simple closed convex curve of ${\mathbb{R}}^2$	Equivalent up to a self-homeomorphism of ${\mathbb{R}}^2$ arising from a straight line homotopy
Simple closed curve in ${\mathbb{R}}^2$	Equivalent up to a self-homeomorphism of ${\mathbb{R}}^2$
Compact differential 1-manifold	Diffeomorphic
Compact 1-manifold	Homeomorphic
Boundary of 2-simplex (i.e., triangle)	Equivalent up to a self-homeomorphism of ${\mathbb{R}}^2$ arising from a straight line homotopy
Boundary of a 2-cube (i.e., square)	Equivalent up to a self-homeomorphism of ${\mathbb{R}}^2$ arising from a straight line homotopy

Algebraic topology

Homology groups

Further information: homology computation for spheres

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

Further information: homotopy computation for spheres

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 and coalgebraic structure

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 $\mathbb{R}/\mathbb{Z}$, 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

Further information: comultiplication of spheres

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.