Simply connected topological space
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Contents
[hide]Definition
Let (X,J) be a topological space, we say X is simply connected if[1]:
- X is a path-connected topological space, and
- π1(X) is the trivial group[Note 1]
See next
- (X,J) is simply connected ⟺ ∀p,q∈C([0,1],X)[(p(0)=q(0)∧p(1)=q(1))⟹p≃q (rel {0,1}))] - note C([0,1],X) is the set of all paths into X
- If a topological space is simply connected then any retraction of that space is simply connected
Examples of simply connected spaces
- All convex subsets of Rn are simply connected topological subspaces
- As Rn is itself convex, we see as a corollary: Rn is a simply connected topological space
- Sn is simply connected for n≥2
- Rn−{0} is a simply connected topological subspace for n≥3
- Follows from Rn−{0} strongly deformation retracts to Sn−1, which means Rn−{0} and Sn−1 are homotopy equivalent topological spaces and then homotopy invariance of the fundamental group tells us π1(Rn−{0})≅π1(Sn−1) and we know Sn is simply connected for n≥2 from above.
- ˉBn−{0} is a simply connected topological subspace for n≥3
- Follows from ˉBn−{0} strongly deformation retracts to Sn−1, which means ˉBn−{0} and Sn−1 are homotopy equivalent topological spaces and then homotopy invariance of the fundamental group tells us π1(ˉBn−{0})≅π1(Sn−1) and we know Sn is simply connected for n≥2 from above.
Notes
- Jump up ↑ Notice we do not specify the basepoint of the fundamental group here, that is we write π1(X) not π1(X,x0) for some x0∈X, that is because for a path-connected topological space all the fundamental groups are isomorphic
- That is: ∀p,q∈X[π1(X,p)≅π1(X,q)] - see the change of basepoint isomorphism (topology, fundamental group)