Difference between revisions of "Simply connected topological space"

Revision as of 15:27, 24 April 2017

Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, we say [ilmath]X[/ilmath] is simply connected if^[1]:

• [ilmath]X[/ilmath] is a path-connected topological space, and
• [ilmath]\pi_1(X) [/ilmath] is the trivial group^{[Note 1]}

See next

• [ilmath](X,\mathcal{ J })[/ilmath] is simply connected [ilmath]\iff[/ilmath] [ilmath]\forall p,q\in C([0,1],X)\big[\big(p(0)\eq q(0)\wedge p(1)\eq q(1)\big)\implies p\simeq q\ (\text{rel }\{0,1\}))\big][/ilmath] - note [ilmath]C([0,1],X)[/ilmath] is the set of all paths into [ilmath]X[/ilmath]
• If a topological space is simply connected then any retraction of that space is simply connected
  • and its contrapositive: If a retraction of a topological space is not simply connected then the space itself is not simply connected

Examples of simply connected spaces

• All convex subsets of [ilmath]\mathbb{R}^n[/ilmath] are simply connected topological subspaces
  • As [ilmath]\mathbb{R}^n[/ilmath] is itself convex, we see as a corollary: [ilmath]\mathbb{R}^n[/ilmath] is a simply connected topological space
• [ilmath]\mathbb{S}^n[/ilmath] is simply connected for [ilmath]n\ge 2[/ilmath]
• [ilmath]\mathbb{R}^n-\{0\} [/ilmath] is a simply connected topological subspace for [ilmath]n\ge 3[/ilmath]
  • Follows from [ilmath]\mathbb{R}^n-\{0\} [/ilmath] strongly deformation retracts to [ilmath]\mathbb{S}^{n-1} [/ilmath], which means [ilmath]\mathbb{R}^n-\{0\} [/ilmath] and [ilmath]\mathbb{S}^{n-1} [/ilmath] are homotopy equivalent topological spaces and then homotopy invariance of the fundamental group tells us [ilmath]\pi_1(\mathbb{R}^n-\{0\})\cong\pi_1(\mathbb{S}^{n-1})[/ilmath] and we know [ilmath]\mathbb{S}^n[/ilmath] is simply connected for [ilmath]n\ge 2[/ilmath] from above.
• [ilmath]\bar{\mathbb{B} }^n-\{0\} [/ilmath] is a simply connected topological subspace for [ilmath]n\ge 3[/ilmath]
  • Follows from [ilmath]\bar{\mathbb{B} }^n-\{0\} [/ilmath] strongly deformation retracts to [ilmath]\mathbb{S}^{n-1} [/ilmath], which means [ilmath]\bar{\mathbb{B} }^n-\{0\} [/ilmath] and [ilmath]\mathbb{S}^{n-1} [/ilmath] are homotopy equivalent topological spaces and then homotopy invariance of the fundamental group tells us [ilmath]\pi_1(\bar{\mathbb{B} }^n-\{0\})\cong\pi_1(\mathbb{S}^{n-1})[/ilmath] and we know [ilmath]\mathbb{S}^n[/ilmath] is simply connected for [ilmath]n\ge 2[/ilmath] from above.

Notes

1. ↑ Notice we do not specify the basepoint of the fundamental group here, that is we write [ilmath]\pi_1(X)[/ilmath] not [ilmath]\pi_1(X,x_0)[/ilmath] for some [ilmath]x_0\in X[/ilmath], that is because for a path-connected topological space all the fundamental groups are isomorphic
   • That is: [ilmath]\forall p,q\in X[\pi_1(X,p)\cong\pi_1(X,q)][/ilmath] - see the change of basepoint isomorphism (topology, fundamental group)

References

1. ↑ Introduction to Topological Manifolds - John M. Lee