Example:The Möbius band strongly deformation retracts onto its core circle

Example

Let $(M,\mathcal{M})$ be the topological space of the Möbius band. We claim that $C$ - the core circle - is a strong deformation retract of $M$.

Setup

We define the following:

• Let $q:[-1,1]\times[-1,1]\rightarrow \frac{[-1,1]\times[-1,1]}{\sim}:\eq M$ be the quotient map

We will seek to find:

• A retraction, $r:M\rightarrow C$
• A homotopy $(\text{rel }C)$ such that $r\simeq_H\text{Id}_C\ \big(\text{rel }C\big)$ - this is the deformation retraction itself.

Proof

We need to exhibit a homotopy $(\text{rel }C)$, $H:M\times I\rightarrow C\subseteq M$ such that:

• Blah