Why does the boundary of mobius strip wrap twice around core circle but not any other line?

Question

A space $X$ deformation retracts onto a subspace $A$ if there exists continuous map $F:X\times [0,1]\rightarrow X$ such that $F(x,0)=x,F(x,1)\in A,F(a,t)=a$ $\forall a\in A$.

The mobius strip deformation retracts onto its core circle. But I don't understand how, under this deformation retraction,

1. The boundary circle wraps twice around the core circle
2. Any other line wraps only once around the core circle

I think 1 happens because at each step of the deformation retraction, the boundary goes around twice and so in the final step (i.e. $t=1$) it just becomes twice the circle. But I see this reason working equally well for any other line. After all, none of the lines actually coincide with the central line until the last step.

Any help would be appreciated.

Answer 1

The Möbius strip $M$ can be obtained by gluing two opposite edges of a rectangle by an orientation reversing homeomorpism. Explictly, we may define $$M = [0,1 ] \times [-1,1]/\sim$$ where $(0,t) \sim (1,-t)$. Let $p : [0,1 ] \times [-1,1] \to M$ denote the quotient map.

Then you get various types of embedded circles.

Type 1: The core circle $C = p([0,1] \times \{0\})$. It can be parameterized by $c : [0,1] \to M, c(x) = [x,0]$. The map $r : M \to C, r([x,t]) = [x,0]$, is a well-defined strong deformation retraction.

Type 2: Circles $C_t = p([0,1] \times \{-t,t\})$ for $t \in (0,1]$, especially the boundary circle $C_1$. These can be parameterized by $c_t(x) = [2x,-t]$ for $x \le 1/2$ and $c_t(x) = [2x-1,t]$ for $x \ge 1/2$. These circles do no intersect the core circle. The maps $r_t = r \mid_{C_t} : C_t \to C$ are $2$-$1$. In fact, they are covering maps with two sheets.

Type 3. Circles $C'_t = p(D_t)$ for $t \in [0,1]$, where $D_t = \{(x,t(2x-1)) \mid x \in [0,1]\}$. These can be parameterized by $c'_t(x) = [x,t(2x-1)]$. We have $C'_0 = C$ and for $t > 0$ these circles intersect the core circle in the single point $[1/2,0]$. The maps $r'_t = r \mid_{C'_t} : C'_t \to C$ are homeomorphisms.

There are of course many more embedded circles, but let us confine to the above.

It is now obvious that the circles $C_t$ wrap twice around the core circle, and that the circles $C'_t$ wrap once around the core circle.

Answer 2

I believe this picture answers your question. Every line that isn't the centre will wrap around twice. So "2. Any other line wraps only once around the core circle" isn't correct.

Credit: https://commons.wikimedia.org/wiki/File:Moebiusband-1s.svg by user Ag2gaeh. License CC BY-SA 4.0.