College Mathematics geometric topology

FLAT MANIFOLDS

A disk is a square, along with its interior.

A sphere is two disks with corresponding edges glued together.

A cylinder is a square with the left and right edges glued together in the same direction. a a

A M ö bius band is a square with a half-twist before gluing the left and right edges together. a a

b A torus is a square with the left and right edges glued together in the same direction and the top and bottom edges glued together in the same direction. a a

b b

A Klein bottle is a square with a half-twist before gluing the left and right edges together and the top and bottom edges glued together in the same direction. a a

b

A projective plane is a square with a half-twist before gluing the left and right edges together and with a half-twist before gluing the top and bottom edges together. b

a a

b MANIFOLDS

(0-dimensional) point

(1-dimensional) curve without endpoints: line, circle (1-dimensional) curve with endpoints: line segment

(2-dimensional) surface without edges: sphere, torus, Klein bottle, projective plane (2-dimensional) surface with edges: disk, cylinder, Möbius band

(3-dimensional) solid without walls: 3-d space (3-dimensional) solid with walls: cube

SUMS To add two surfaces, remove a small disk from each surface and glue the boundaries of the two holes together. The resulting surface is called the connected sum of two surfaces, written as A # B.

PRODUCTS To multiply two manifolds, attach a copy of one manifold to each point of the other manifold, with each copy perpendicular to the original. This is called the product of manifolds, written as A x B.

KNOTS & LINKS In topology, the two ends of a knot are always joined. Otherwise, every knot would be topologically equivalent to a line segment. The simplest knot is a circle, called the un-knot. It's not really knotted! A knot is truly knotted if it cannot be transformed into the un-knot without tearing.

Can you find diagrams in this table corresponding to each of the following: an overhand knot? a figure-8 knot? a square knot?

Is it possible to add two knots together?

Is there a knot that can be added to any other knot that leaves it unchanged?

Are any of the knots described equal to the connected sum of two knots in the table?