A genus one surface, or torus, built from one single square by identifying opposite sides.

A trivial loop on a torus, with the loop retracting on the square once the torus has been cut open.

A genus one surface, or torus, built from one single square by identifying opposite sides.

A trivial loop on a torus, with the loop retracting on the square once the torus has been cut open.

A genus 2 surface, built from one single octogonal cell by identifying pieces of the boundary.

This one shows a trivial loop on a genus 2 surface, with the loop retracting on the octogonal cell. It's the surface constructed in the previous video.

A finite volume, non-compact manifold, and a quasi-isometry to a wedge of lines.

A non-orientable two-dimensional manifold, commonly called Klein bottle.

A Dehn twist on a torus. This is a homeomorphism which is the identity far from a simple closed curve: here the one along which the torus is cut. Away from it, the map is the identity. So, cut along the curve, give it a full turn and glue back. Any curve intersecting the curve along which we twist will be modified.

A genus 2 surface, obtained by identifying sides of the cross on a Swiss flag. All the curvature is concentrated in that middle point, elsewhere it's flat.

The universal cover of a torus glued on a genus 2 surface along a simple closed curve. The universal cover is a tree-like structure of Euclidean planes and hyperbolic planes, glued together along geodesics. This illustrates relative hyperbolicity, and was displayed at ICERM.

The universal cover of a torus, with a copy of a circle glued at one point. It is a tree of lines and Euclidean planes.

The universal cover of a non-compact hyperbolic 3-manifold is truncated hyperbolic 3-space, which is hyperbolic 3space with a collection of horoballs removed. Here are more explanations. This is an enhanced version of the next animation, showing the same thing.

The universal cover of a non-compact hyperbolic 3-manifold is truncated hyperbolic 3-space, which is hyperbolic 3-space with a collection of horoballs removed. This work has been featured in the AMS book Illustrating mathematics, originally as a gif called Neutered space.

A crown is a disk with a hole inside, and it can continuously be deformed into the circle. Hence both spaces have the same fundamental group, which is given by homotopy classes of loops based at a point. A few elements of the fundamental groups will be described on the next pictures, pictured on the crown, as on the circle they are more difficult to see.

The neutral element in the fundamental group is given by homotopically trivial loops: those are loops that can be continuously retracted to the base point.

Here is a non-trivial loop: with continuous deformations one cannot go through the hole.

The inverse of a loop consists in going backwards on the loop. Here is a non-trivial loop, followed by its inverse and then retracted to to trivial loop, showing that indeed composing a loop with its inverse gives the identity in the fundamental group.

A more complicated loop now wraps around the hole twice clockwise. Its inverse has to wrap around the hole twice counter-clockwise. By pulling a loop tight one can start to understand that the fundamental group of the circle is Z, the integers, counting the number of times the loop wraps around the hole clockwise or counter-clockwise.

Kind of unrelated to fundamental groups, here is a random walk on the Petersen graph, a graph with good expansion properties. Read my paper with Austin Lawson on that topic here

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