Example:The Möbius band strongly deformation retracts onto its core circle
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Contents
Example
Let [ilmath](M,\mathcal{M})[/ilmath] be the topological space of the Möbius band[Note 1]. We claim that [ilmath]C[/ilmath] - the core circle - is a strong deformation retract of [ilmath]M[/ilmath].
Setup
We define the following:
- Let [ilmath]q:[-1,1]\times[-1,1]\rightarrow \frac{[-1,1]\times[-1,1]}{\sim}:\eq M [/ilmath] be the quotient map
We will seek to find:
- A retraction, [ilmath]r:M\rightarrow C[/ilmath]
- A homotopy [ilmath](\text{rel }C)[/ilmath] such that [ilmath]r\simeq_H\text{Id}_C\ \big(\text{rel }C\big)[/ilmath] - this is the deformation retraction itself.
Proof
We need to exhibit a homotopy [ilmath](\text{rel }C)[/ilmath], [ilmath]H:M\times I\rightarrow C\subseteq M[/ilmath] such that:
- Blah
Notes
- ↑ Considered as a quotient of [ilmath]\frac{[-1,1]\times[-1,1]}{\sim} [/ilmath] where [ilmath]\sim[/ilmath] is generated by [ilmath](-1,t)\sim(1,-t)[/ilmath]
