THese are the pictures you need for the pre-test homework. Your job for most of them will be to say which topological shape is represented in the picture. In most cases you are supposed to be looking at a surface; if it looks lik you are being shown a solid shape (e.g. a solid concrete ball) then please ignore the interior and focus only on the outer surface of that shape (in that case, the surface is a (hollow) sphere). If instead it looks to you like the picture indicates just a piece of string or something like that, then again I want you to think about the surface: even if it's a string, that string will have a surface -- the outermost layer of molecules on a string would form a very narrow tube (cylinder). Got it?

Question 2 The statue is hollow but sealed on the bottom. Here's a topologically equivalent version:

Question 3 Ignore the shape labeled "S". The other two shapes (labeled "S'") are topologically equivalent.

Question 4

Question 5 (It's not a ribbon -- it's a kind of hollow triangular tube, kind of like a long, curved prism)

Question 6 Your surface is the last one; the pictures illustrate that it's the result of glueing two pairs of pants together, just like I did two T-shirts in class.

Question 7 A hollow dodecahedron

Question 8

Question 9

Question 10 THis is made of varnished paper; it is a kind of squarish tube (i.e. a short section of this shape resembles a cylinder whose cross-section is a square instead of a circle).

Question 11 All three of these are topologically equivalent to each other. Note that there are no boundaries: you can see in the first picture that what might look like an edge in the green pictures is actually just a fold in the paper.

Question 12 Please disregard all the crinkly features of the surface; it's just supposed to be a computer geek's Christmas wreath!

Question 13 The left and right sides are supposed to be glued into separate little shapes as we have done before; each is then a surface without any boundaries except the little circles in the middles. Then the two little shapes are glued together by identifying those two circles.

Question 14 small statuette produced by a 3D printer.

Question 15

Question 16

Question 17 You can ignore the red and green stuff -- but do you see that there's an interesting game to play here?

Question 18 3D printer version of a classical statue of two kissing children. Notice the space under their chins.

Question 19

Question 20 It's a coffee mug -- isn't it? ...

Finally, here are some other illustrations I found online which I present just for your amusement.

A Klein Bottle in literature

Chicago's "Bean", which is topologically a sphere (i.e. there's no funny holes or anything underneath).