0

Right now I'm casually reading through Carson's "Topology of Surfaces, Knots, and Manifolds." I don't have a strong background in topology, and I was told that this was a very accessible and informative text.

The problem that I am looking at involves the four figures I have attached. I'm having problems finding way deform $X$ onto $Y$ and $Y$ onto $Z$. The problem further involves deforming $X$ and $Y$ onto $D$, but I will worry about that later. It's not a very rigorous.

I believe you can deform $X$ onto $Y$ by stretching $X$ into a rectangle. Linking two sides into a donut (making sure the interval is on the inside), and then shrinking the interval to a point.

Four shapes in quesition

emka
  • 6,494
  • Can you provide a definition from "deform onto"? I can make a pretty good guess based on your description but it'd be nice to be sure. In particular, it seems like we'd need some sort of ambient space defined if we plan to deform $Y$ onto $Z$ or $D$, since they aren't subsets of $Y$ in any obvious way. – MartianInvader Aug 29 '13 at 18:39
  • Basically I would like to, in pictures, deform X onto Y and Y onto Z in such a way to suggest that a continuous function exists between the two. So obviously no twisting or pinching one object into another. – emka Aug 29 '13 at 18:56
  • "A continuous function" - from your description it sounds like you want this function to be surjective, but not necessarily injective? Why no twisting or pinching? (These sound like continuous operations to me, unless I'm picturing them differently than you are). And if you're looking for a transformation more strict than a continuous map, are you sure it exists? Going from $Y$ to $Z$, for example, seems to at least require some folding. – MartianInvader Aug 29 '13 at 19:17
  • Actually you are right. I'm still not sure how to stretch $Y$ to make it deform into $Z$. – emka Aug 29 '13 at 20:16

1 Answers1

0

I looked at the book. I'll be a bit sloppy trying to describe the deformations (sorry for that).

To get $Y$ from $X$: stretch $X$ into a rectangle so that the thick line is one of the short edges, now bend it so that the two short edges (one thick and the other dashed) touch so that the "square bracket" remain in the "internal part". The "marked point" in the interior come from the "square bracket" in the arc on $X$.

To get $Z$ from $X$ you have to stretch again $X$ into a rectangle but now the thick line have to cover two edges: one long and one short. The "square bracket" has to lie on the corner of the short one. Now you can proceed as before bending so that the short edges touch.

To get $X$ from $Z$ you can do that: if there wasn't the "marked point" on the interior circle of $Z$ you can deform easily in $X\setminus0$: the additional point basically fill the puncture.

To get $D$ from $Y$ you can do the same.

Note that we have two continuous maps: one from $X$ to $Z$ and one from $Z$ to $X$ but one is not the inverse of the other (in fact the two spaces are not homeomorphic).

Dario
  • 5,749
  • 2
  • 24
  • 36