## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

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Topology studies properties of spaces that are invariant under any continuous deformation. It is sometimes called "rubber-sheet geometry" because the objects can be stretched and contracted like rubber, but cannot be broken. For example, a square can be deformed into a circle without breaking it, but a figure 8 cannot. Hence a square is topologically equivalent to a circle, but different from a figure 8.

Here are some examples of typical questions in topology: How many holes are there in an object? How can you define the holes in a torus or sphere? What is the boundary of an object? Is a space connected? Does every continuous function from the space to itself have a fixed point?

Topology is a relatively new branch of mathematics; most of the research in topology has been done since 1900. The following are some of the subfields of topology.

General topology normally considers local properties of spaces, and is closely related to analysis. It generalizes the concept of continuity to define topological spaces, in which limits of sequences can be considered. Sometimes distances can be defined in these spaces, in which case they are called metric spaces; sometimes no concept of distance makes sense.**General Topology or Point Set Topology.**Combinatorial topology considers the global properties of spaces, built up from a network of vertices, edges, and faces. This is the oldest branch of topology, and dates back to Euler. It has been shown that topologically equivalent spaces have the same numerical invariant, which we now call the Euler characteristic. This is the number (V - E + F), where V, E, and F are the number of vertices, edges, and faces of an object. For example, a tetrahedron and a cube are topologically equivalent to a sphere, and any “triangulation” of a sphere will have an Euler characteristic of 2.**Combinatorial Topology.**Algebraic topology also considers the global properties of spaces, and uses algebraic objects such as groups and rings to answer topological questions. Algebraic topology converts a topological problem into an algebraic problem that is hopefully easier to solve. For example, a group called a homology group can be associated to each space, and the torus and the Klein bottle can be distinguished from each other because they have different homology groups.**Algebraic Topology.**

Torus |
Klein Bottle |

Algebraic topology sometimes uses the combinatorial structure of a space to calculate the various groups associated to that space.

Differential topology considers spaces with some kind of smoothness associated to each point. In this case, the square and the circle would not be smoothly (or differentiably) equivalent to each other. Differential topology is useful for studying properties of vector fields, such as a magnetic or electric fields.**Differential Topology.**

Topology is used in many branches of mathematics, such as differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex analysis. It is also used in string theory in physics, and for describing the space-time structure of universe.

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land promised to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Indigenous Initiatives Office.