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 continuous, surjective map. WIth the property that $U\subset Y$ is open if and only if $f^{-1}(U)$ is open in $X$, we say $f$ is a quotient map and $Y$ is a quotient space.