Torus

The torus is the product of two circles and can be formed by identifying opposite sides of a square.

For the surface of revolution of the circle with center $(a,R)$ and radius $r<R$ in the upper half-plane, use $X(u,v)=(a+r\cos u,(R+r\sin u)\cos v,(R+r\sin u)\sin v)$ and unit normal $n=(\cos u,\sin u\cos v,\sin u\sin v)$. Exactly four surface normals pass through the displayed $P$. The equation $P-X=tn$ forces $(t+r)\cos u=0$. The case $t=-r$ would put $P$ at radial distance $R$, so $\cos u=0$; its nonzero radial coordinate then forces $v=0$ or $\pi$. All four combinations work. By contrast, infinitely many normals pass through $Q$: both complete circles with $u=\pi /2$ or $3\pi /2$ have radial normals through that point. This is a concrete illustration of degenerate distance-critical sets versus isolated critical points.

Take $u,v$ modulo $\pi$; shifting either coordinate by $\pi$ leaves $h$ unchanged. Its partial derivatives are $h_u=\sin v\sin (2u-v)$ and $h_v=\sin u\sin (u-2v)$. The only critical points are $(0,0)$, $(\pi /3,2\pi /3)$ and $(2\pi /3,\pi /3)$. The latter are a nondegenerate minimum and maximum, with values $\mp 3\sqrt{3}/8$. At the origin the leading cubic is $uv(u-v)$, linearly equivalent to a monkey saddle; its critical point is isolated but degenerate. Thus three smooth critical points coexist with the Morse lower bound of four for the torus.

Collapsing an essential meridian circle of a torus to one point gives the pinched torus. In the torus's CW structure, the two-cell attaches to two circles by the commutator. Collapsing one circle leaves one one-cell and a two-cell attached by a null-homotopic word. Thus the quotient is homotopy equivalent to $S^1\vee S^2$. Its cellular chain groups are $\mathbb{Z}$ in degrees zero, one and two, with zero differentials, so its integral homology groups are $H_0=H_1=H_2=\mathbb{Z}$ and vanish in all higher degrees. This illustrates that collapsing a circle can preserve a top-dimensional homology class even while reducing the first homology rank.

The product of three circles is a compact smooth three-manifold. Quotient coordinates allow constant commuting vector fields. A rank-two plane field can have dense immersed leaves, as in a dense immersed cylinder in a three-dimensional torus.

Since $T^r$ is a product of circles, the Künneth theorem identifies its cohomology ring with the exterior algebra on the coordinate degree-one classes. Inversion negates each integral degree-one class, so it acts as $(-1{)}^q$ in degree $q$. Even in characteristic two, each degree-one class has square zero.

On the torus ${\mathbb{R}}^d/{\mathbb{Z}}^d$, an integer matrix $A$ defines an endomorphism. The coordinate circles give a basis of the first homology group, and the induced map on that group is $A$. The Künneth theorem identifies higher integral homology with the exterior powers of ${\mathbb{Z}}^d$, so the degree-$q$ map is ${\bigwedge }^{q}A$. In top degree it is multiplication by $\det A$. The map is a homeomorphism precisely when $A$ is a unimodular matrix; a merely nonsingular integer matrix need not have an integer inverse.

A solid torus is the product of a circle with a closed disk. Its boundary is the torus $S^1\times S^1$. It deformation retracts onto its central circle, so its degree-one de Rham cohomology is one dimensional and injects into that of its boundary.

A compact orientable irreducible three-manifold with connected torus boundary which is compressible is a solid torus. Compressing along a compression disk gives an embedded sphere. The compressed component has one sphere boundary and irreducibility makes it a ball; undoing compression attaches one one-handle, giving $D^2\times S^1$.

A meridian of a solid torus is a primitive curve on its boundary that bounds an embedded disk in the solid torus.

Suppose a compact $m$-manifold $N$ is embedded in the half-space $x_{m+1}>0$ of ${\mathbb{R}}^{m+1}$. Spinning it around the boundary hyperplane gives the embeddingStarting with a circle and iterating proves that $T^n=(S^1{)}^n$ embeds in ${\mathbb{R}}^{n+1}$.

For major radius $R$ and minor radius $r$ with $R>r>0$, the standard embedded torus of revolution in ${\mathbb{R}}^3$ isIt is parametrized bywhich identifies the opposite sides of the unit square.

For the standard longitude $a$ and meridian $b$,

For the universal coveringlift a based loop from the origin. Its endpoint lies in the fibre ${\mathbb{Z}}^2$, depends only on its based homotopy class, and turns concatenation into addition. This gives a natural isomorphism ${\pi }_1(T^2)\simeq {\mathbb{Z}}^2$.

Every $A\in G{L}_2(\mathbb{Z})$ induces a based homeomorphismIts inverse is $f_{A^{-1}}$, and under the lift-endpoint description of the fundamental group of the torus, its induced map on ${\pi }_1(T^2)$ is multiplication by $A$.

Any two connected degree-two coverings of the torus fit into a commuting square with homeomorphisms of their total and base spaces. They correspond to index-two subgroups of ${\mathbb{Z}}^2$, which are kernels of the three nonzero maps ${\mathbb{Z}}^2\to \mathbb{Z}/2\mathbb{Z}$. Reduction modulo two shows that $G{L}_2(\mathbb{Z})$ acts transitively on these subgroups; the classification of connected covering spaces and the lifting criterion for a covering space then supply the homeomorphism of total spaces.