NoteNextra-origin/content/Math4202/Math4202_L1.md

Math4202 Topology II (Lecture 1)

Topology of manifolds

Fundamental groups

Use fundamental group as invariant for topological spaces up to homeomorphism (exists bijective and continuous map with continuous inverse) / homotopy equivalence.

Classifying two dimensional surfaces.

• Sphere
• Torus
• \mathbb{R}P^2

Quotient spaces

Let X be a topological space and f:X\to Y is a

1. continuous
2. surjective map.
3. With the property that U\subset Y is open if and only if f^{-1}(U) is open in X.

Then we say f is a quotient map and Y is a quotient space.