To classify shapes, or topological spaces, mathematicians distinguish between differences that matter and those that don\u2019t. Homotopy theory is a perspective from which to make those distinctions. It considers a ball and an egg to be fundamentally the same topological space, because you can bend and stretch one into the other without ripping either. In the same way, homotopy theory considers a ball and an inner tube to be fundamentally different because you have to tear a hole in the ball to deform it into the inner tube.<\/p>\n

Homotopy is useful for classifying topological spaces \u2014 creating a chart of all the kinds of shapes that are possible. It\u2019s also important for understanding something else mathematicians care about: maps between spaces. If you have two topological spaces, one way to probe their properties is to look for functions that convert, or map, points on one to points on the other \u2014 input a point on space A, get a point on space B as your output, and do that for all the points on A.<\/p>\n

To see how these maps work, and why they illuminate properties of the spaces involved, start with a circle. Now map it onto the two-dimensional sphere, which is the surface of a ball. There are infinitely many ways of doing this. If you imagine the sphere as Earth\u2019s surface, you could put your circle at any line of latitude, for example. From the perspective of homotopy theory, they\u2019re all equivalent, or homotopic, because they can all shrink down to a point at the north or south pole.<\/p>\n

Next, map the circle onto the two-dimensional surface of an inner tube (a one-holed torus). Again, there are infinitely many ways of doing this, and most are homotopic. But not all of them. You could place a circle horizontally or vertically around the torus, and neither can be smoothly deformed into the other. These are two (of many) ways of mapping a circle onto the torus, while there is just one way to map it onto a sphere, reflecting a fundamental difference between the two spaces: The torus has one hole while the sphere has none.<\/p>\n

It\u2019s easy to count the ways we can map from the circle to the two-dimensional sphere or torus. They\u2019re familiar spaces that are easy to visualize. But counting maps is much harder when higher-dimensional spaces are involved.<\/p>\n