Subsection5.5.1SVK Theorem statement and first examples
Theorem5.5.1.SVK = "Seifert-Van Kampen Theorem".
Let X be a topological space with basepoint x_0 and PC subspaces A_\alpha (for indices \alpha in an index set J) such that each A_\alpha is an open set in X containing x_0, every pairwise and triple intersection A_\alpha \cap A\beta and A_\alpha \cap A\beta \cap Aγ is also PC, and X = \cup \alpha \in J A_\alpha. For each \beta,γ \in J, let iA\beta Aγ: A\beta \cap Aγ \to A\beta be the inclusion map. Then \pi_1(X) \cong ∗\alpha \in J \pi_1(A_\alpha )/N, where N is the normal subgroup generated by all elements of the form iA\beta Aγ∗([w]) iAγ A\beta ∗([w])-1 with [w] \in \pi_1(A\beta \cap Aγ) and \beta,γ \in J.
Examples.
Theorem5.5.2.
\pi_1(Sn) = 1 for all n \geq 2.
Definition5.5.3.
Let X be a disjoint union of singleton spaces V\alpha =\{v\alpha \} (for \alpha \in J) and spaces E\beta (for \beta \in K) homeomorpic to I. For each \beta \in K, let s\beta,t\beta \in J. Let \sim be the equivalence relation on X generated by the relation 0\beta \sim vs\beta and 1\beta \sim vt\beta. The quotient space X/\sim is a graph. Each point [v\alpha ] is a vertex of the graph X/\sim, and the image of each E\beta is an edge of the graph.
Definition5.5.4.
A bouquet of n circles is a graph with one vertex and n edges.
Definition5.5.5.
A tree is a connected graph that does not have a nonempty edge path from a vertex back to itself that does not traverse any edge (in any direction) twice. A maximal tree in a graph X is a subgraph that is a tree an includes every vertex of X.
Theorem5.5.6.
If X is a graph and T is a maximal tree in X, then \pi_1(X) \cong F(S), where S is the set of edges in X that are not in T.
Proposition5.5.7.
\pi_1(T^2) \cong \Z 2.
\pi_1(K2) \cong \langle a,b | ba = ab-1 \rangle.
More results from the exercises\dots
Subsection5.5.2Classification of surfaces
Definition5.5.8.
An n-manifold is a T_2 space X with a countable basis such that each point p of X is contained in an open subset Up of X that (as a subspace of X) is homeomorphic to \R^n (with the Euclidean topology). A surface is a 2-manifold.
Definition5.5.9.
Let S^1 and S^2 be two surfaces. For 1 \leq i \leq 2, let Ci \sse Si be a closed set that, as a subspace, has a homeomorphism hi: Ci \to D2. Let Ui := hi-1(B((0,0),1)) and let Si\setminus Ui have the subspace topology from Si. Let X := (S^1\setminus U1) ∐ (S^2\setminus U2) have the disjoint union topology, and let \sim be the smallest equivalence relation on X for which h1-1(p) \sim h2-1(p) for all p \in S^1. The quotient space X/\sim is the connected sum of S^1 and S^2, and is denoted S^1 # S^2.
Theorem5.5.10.
If S^1 and S^2 are compact connected surfaces, then so is S^1 # S^2.
Proposition5.5.11.
If S^1 is a surface, then S^1 # S^2 \cong S^1.
Example5.5.12.
Examples
Proposition5.5.13.
Let n \geq 1.
Let Y := #n T^2. (a-i) The space Y is homeomorphic to a quotient of a Euclidean 4n-gon with gluings labeled (counterclockwise) by a1b1a1-1b1-1 a2b2a2-1b2-1 ··· anbnan-1bn-1. (a-ii) \pi_1(Y) \cong \langle a1,b1, \dots,an, bn | [a1,b1] ··· [an,bn] = 1 \rangle. (a-ii) \pi_1(Y)ab \cong \Z 2n.
Let Z := #n P^2. (b-i) The space Z is homeomorphic to a quotient of a Euclidean 2n-gon with gluings labeled (counterclockwise) by a1a1 a2a2 ··· anan. (b-ii) \pi_1(Z) \cong \langle a1, \dots,an | a12 a22 ··· an2 = 1 \rangle. (b-ii) \pi_1(Z)ab \cong \Z n-1 \times \Z /2.
T^2 # P^2 \cong #3 P^2
Theorem5.5.14.Classification of Surfaces Theorem.
Every compact connected surface is homeomorphic to exactly one of S^2, #n T^2, or #n P^2 for some natural number n.
Subsection5.5.3Proof and corollaries of the Seifert-Van Kampen Theorem
Outline:
Building the homomorphism with the HBT,
using the Lebesgue Number Lemma and the "seashell method" to prove onto, and
using LNL again to prove 1-1.
Definition5.5.15.
Let X_\alpha be a topological space for all \alpha in J such that X\beta \cap Xγ = \es for all \beta \ne γ \in J, and let p_\alpha \in X_\alpha for all \alpha. Let ∐\alpha X_\alpha be the union of the X_\alpha with the disjoint union topology, and let \sim be the equivalence relation on ∐\alpha X_\alpha generated by p_\beta \sim pγ for all \beta,γ \in J. The quotient space (∐\alpha X_\alpha )/\sim is the wedge product of the spaces ∐\alpha X_\alpha (with respect to the basepoints p_\alpha ); this space is also denoted ∨\alpha X_\alpha.
Theorem5.5.16.
For each \alpha let X_\alpha be a space with basepoint p_\alpha \in X_\alpha, and let p := [p_\alpha ] be the basepoint of the wedge product ∨\alpha X_\alpha. If for each \alpha the singleton space\{p_\alpha \} is a deformation retract of an open set U\alpha in X_\alpha in X containing p_\alpha, then \pi_1(∨\alpha X_\alpha,p) \cong ∗\alpha \pi_1(X_\alpha,p_\alpha ) is a free product.
