Let the topological space $X_n$ be obtained from $S^n$ by identifying three distinct points. Find the fundamental group of $X_n$.

Question

Let the topological space $X_n$ be obtained from $S^n$ by identifying three distinct points, i.e. $X_n = S^n/\{p, q, r\}.$ Find the fundamental group of $X_n$.

Seifert-van Kampen theorem should be useful. – A. Bellmunt Jul 02 '13 at 16:39 — A. Bellmunt, Jul 02 '13 at 16:39

score 2 · Answer 1 · answered Jul 02 '13 at 17:17

2

Your space is homotopically equivalent to $S^n \bigvee_2 S^1$ (i've used a little bit exotic notation, it's the wedge sum of $S^n$ with two copies of $S^1$), can you see how?

answered Jul 02 '13 at 17:17

Edoardo Lanari

3,564

Not really, could you help me a little? – dunkindonuts Jul 02 '13 at 17:35
I suppose that you know that if you have a cw-complex $X$ and a subcomplex $A$ which is contractible then the map $X \to X/A$ is an homotopy equivalence (chapter 0 of Hatcher). You can think of your space as a copy of $S^n$ whith three segment whith one extremity respectively in a point of {p,q,r} and the other in common, after having collapsed them to a point (example 0.8 is almost this case) – Edoardo Lanari Jul 02 '13 at 23:18

Let the topological space $X_n$ be obtained from $S^n$ by identifying three distinct points. Find the fundamental group of $X_n$.

1 Answers1