Gluing two discs to make a sphere

Loops on the sphere

Loops on the torus

A homotopy of the trefoil

Geometry of the Möbius strip

The Möbius strip and the circle

The punctured plane

The punctured sphere

The punctured torus

Wrapping an annulus

The exponential map

The exponential map is a covering

Path lifting

Homotopy lifting

Subdivision of a prism $[0,1]\times\Delta_2$ 
Boundary of a tetrahedron

Barycentric subdivision of a tetrahedron

The map $\mu\colonK'\toK$
