Skip to main content

Mathematics Illuminated

Topology’s Twists and Turns

Can you imagine the shape of the universe? That's where Topology comes in: a branch of mathematics concerned with the study of spatial relationships that don't depend on measurement, and is more concerned with concepts like 'between' or 'inside,' and how things are connected.

View Transcript
Hyperbolic Purse

Hyperbolic Purse

Topology, known as “rubber sheet math,” is a field of mathematics that concerns those properties of an object that remain the same even when the object is stretched and squashed. In this unit we investigate topology’s seminal relationship to network theory, the study of connectedness, and its critical function in understanding the shape of the universe in which we live.

Unit Goals

  • Topology is the study of fundamental shape.
  • Objects are topologically equivalent if they can be continuously deformed into one another.  Properties that are preserved during this process are called topological invariants.
  • Intrinsic topology is the study of a surface or manifold from the perspective of being on or in it.
  • Extrinsic topology is concerned with properties of a surface or manifold seen from an external viewpoint.  This requires some kind of embedding.
  • The Euler characteristic is a topological invariant.
  • Orientability is a topological invariant.
  • A configuration space is a topological object that can be used to study the allowable states of a given system.
  • The question of the shape of our universe is a question of intrinsic topology.





Abrams, A. and R. Ghrist. “Finding Topology in a Factory: Configuration Spaces,” The American Mathematical Monthly, 109, (February 2002).

Alexander, J.C. “On the Connected Sum of Projective Planes, Tori, and Klein Bottles,” The American Mathematical Monthly, vol. 78, no. 2, (February 1971).

Arnold, B.H. Intuitive Concepts in Elementary Topology. Englewood Cliffs, NJ: Prentice-Hall, 1962.

Ban, Yih-En Andrew, Herber Edelsbrunner, and Johannes Rudolph. “Interface Surfaces for Protein-Protein Complexes,” RECOMB’04, San Diego, CA, (March 27–31, 2004).

Berlinghoff, William P. and Kerry E. Grant. A Mathematics Sampler: Topics for the Liberal Arts, 3rd ed. New York: Ardsley House Publishers, Inc., 1992.

Borges, Carlos R. Elementary Topology and Applications. (World Scientific). Singapore: World Scientific Press, 2000.

Boyer, Carl B. (revised by Uta C. Merzbach). A History of Mathematics, 2nd ed. New York: John Wiley and Sons, 1991.

Casti, John L. Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics. New York: John Wiley and Sons, 2000.

Devlin, Keith J. The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time. New York: Basic Books, 2002.

Kuijpers, B., Paredaens, J., and J. Van den Bussche. “Lossless Representation of Topological Spatial Data,” Advances in Spatial Databases (M.J. Egenhofer, J.R. Herring, editors), Lecture Notes in Computer Science, vol. 951, Springer-Verlanger, 1995.

Kurant, Maciej and Patrick Thiran. “Trainspotting: Extraction and Analysis of Traffic and Topologies of Transportation.” Networks. (Dated: May 23, 2006)

Luminet, Jean-Pierre. “The Topology of the Universe: Is the Universe Crumpled?” Laboratory Universe and Theories (LUTH). (accessed December 13, 2006).

Mackenzie, Dana. “Breakthrough of the Year: The Poincaré Conjecture—Proved,” Science, vol. 314, no. 5807 (2006).

Milnor, John. “Towards the Poincaré Conjecture and the Classification of 3-Manifolds,” Notices of the American Mathematical Society, vol. 50, no. 10 (November 2003).

Monastyrsky, Michael. [Translated by James King and Victoria King. Edited by R.O. Wells Junior] Riemann, Topology and Physics. Boston, MA: Birkhauser, 1979.

Montgomery, Richard “A New Solution to the Three-Body Problem,” Notice of the AMS, vol. 48, no. 5 (May 2001).

Newman, James R. Volume 1 of the World of Mathematics: A Small Library of the Literature of Mathematics from A’h-mose the Scribe to Albert Einstein, Presented with Commentaries and Notes. New York: Simon and Schuster, 1956.

Pickover, Clifford A. The Möbius Strip: Dr. August Möbius’s Marvelous Band in Mathematics, Game, Literature, Art, Technology, and Cosmology. New York: Thunder’s Mouth Press, 2006.

Poincaré, Henri (edited and introduced by Daniel L. Goroff) New Methods of Celestial Mechanics, vol. 1; Los Angeles, CA: American Institute of Physics: 1993.

Ray, Nicolas, Xavier Cavin, Jean-Claude Paul, and Bernard Maigret. “Dynamic Interface Between Proteins,” Journal of Molecular Graphics and Modelling, vol. 23, no. 4, (January 2005).

Rockmore, Dan. Stalking the Riemann Hypothesis: The Quest To Find the Hidden Law of Prime Numbers. New York: Vintage Books (division of Randomhouse), 2005.

Stewart, Ian. From Here to Infinity: A Guide to Today’s Mathematics. New York: Oxford University Press, 1996.

Sumners, De Witt. “Lifting the Curtain: Using Topology To Probe the Hidden Action of Enzymes,” Notices of the AMS, vol. 42, no. 5 (May 1995).

Tannenbaum, Peter. Excursions in Modern Mathematics, 5th ed. Upper Saddle River, NJ: Pearson Education, Inc., 2004.

Weeks, Jeffrey R. The Shape of Space, 2nd ed. (Pure and Applied Mathematics). New York: Marcel Dekker Inc., 2002.

Weeks, Jeffrey. “The Poincaré Dodecahedral Space and the Mystery of the Missing Fluctuations,” Notices of the AMS, vol. 51, no. 6 (June/July 2004).

Weisstein, Eric W. “Möbius, August Ferdinand (1790-1868)” Wolfram Research. (accessed 2007).


Hitchin, Nigel: “Lecture notes for course b3 2004: Geometry of Surfaces: Chapter 1, Topology.” Mathematical Institute, University of Oxford. 2007).

McMullen, Curtis. “The Geometry of 3-Manifolds.” Lecture presented as part of Harvard University’s Research Lecture for Non-Specialists, Cambridge, Massachusetts, October 11, 2006.