Images and animations

General facts on surfaces

The morphisms between fundamental groups induced by homotopic maps.

The Moebius strip has the 1-sphere as a strong deformation retract.

Both the cylinder and the Moebius strip have the same strong deformation retract.

Two isotopies in the disc. One relative to the boundary

The meridian of a torus and its core commute

The sphere as a quotient of the torus

Homeomorphism between the cylinder and the plane minus a point

The sphere as a quotient of the disk.

The torus as a quotient of the plane.

The Moebius strip as a quotient of the square.

Clasification of surfaces

The torus obtained by identifying the sides of a square

The torus with a hole is homeomorphic to two cylinders glued along a square

Another way to represent the torus with a hole.

Two moebius strips, glued along their boundary, form a Klein bottle.

The genus 2 surface obtained by gluing the sides of an octogon.

The klein bottle minus a disk.

The connected sum of the torus and a moebius strip coincides with the sum of a klein bottle and a moebius strip.

Cutting a Moebius srtip along a circle that does not separate, we get an orientable surface (a cylinder)

Surgeries in a torus

A cycle in a chain of connected sum of tori, is itself a torus.

Relating embeddings

An isotopy between two apparently different embeddings of the genus 2 surface in the space.

And also in the complement of the trivial knot

Knots and links

The trefoil knot as a torus knot.

The figure eight knot is its own mirror image.

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