Questions handed in by students on 2017-10-04. Our friend the square - Many students asked questions about edge-wise square quotients. To save on drawing, I'll use the following notation. We order the edges of the square anticlockwise, starting at the bottom. We label an edge with a dash "-" if it is not part of a gluing. We label identified edges with a common letter if they are identified. The letter is lower-case if the arrow is anticlockwise and is upper-case if the arrow is clockwise. Examples: "----" gives a square without identifications "-a-a" gives the M\"obius strip "-a-A" gives the cylinder "abAB" was an example drawn in lecture "A--a" was an example drawn in lecture. Lecture - 1. How did you arrive at the cylinder and the M\"obius strip (in written form?) Ans: In lecture we didn't give a definition of the cylinder (C^2) or of the M\"obius strip (M^2). To _really_ answer the exercises given in lecture we would need, first, to settle on definitions of C^2 and M^2 and, second, to give homeomorphisms from the square quotients to the cylinder and M\"obius strip. 2. What is the equivalence relation for a klein bottle? Ans: In the notation above, one possibility is "abaB". 3. Where do the (x = x', y = y') points go? They form equivalence classes of size one. 4. Can you glue more than 2 sides of our friend the square together? [Draws a picture of "AA-a", in the notation above]. Ans: Edge-wise quotients were not explicitly defined in lecture. So you could allow this or disallow it. The problem is much easier if we only allow ourselves to glue edges in pairs (so: no self gluings and no triple or quadruple gluings). 5. Which topological space results if you consider the following [draws "abAb" and "AbAb"] Ans: These are the Klein bottle and the real projective plane respectively. 6. Why is "-a-A" a cylinder not a double twisted strip? [Draws a strip with two half-twists, and indicates that the ends are identified.] Ans: The cylinder (the "no-twisted band") and the "double half-twisted band" are homeomorphic. 7. Question about the exercise: This subspace of \RR^3 is homeo to M^2. [Draws the triple half-twisted band.] How exactly can we do this? The drawing is no clear definition for a subspace... Ans: One solution would be to convert our drawing into a parametrised surface. This is done for the once half-twisted band here: https://en.wikipedia.org/wiki/Mobius_strip#Geometry_and_topology Exercise: Generalise the parametrization found at Wikipedia to obtain an n-times half-twisted band in \RR^3. Plot the results for n = 0, 1, 2, ... on a computer. Another solution would be to revive the once-standard courses in descriptive geometry. :P https://en.wikipedia.org/wiki/Descriptive_geometry http://www.springer.com/gb/book/9780387345420 (available here - https://0-link-springer-com.pugwash.lib.warwick.ac.uk/content/pdf/10.1007%2F978-0-387-68120-7.pdf) 8. [Draws "ABaB"] <--- how to draw? [then draws the standard torus] <--- needs a twist * whats its name? Or is it just a torus? Ans: This is a Klein bottle, and it does not embed in \RR^3. Ask in lecture and I will draw some of the standard pictures. 9. Why {0} and {1} mentioned in disjoint union definition? Ans: Suppose that X = [0, 2] and Y = [1,3]. Then their union is X \cup Y = [0,3] and this has only one path component. But the disjoint union X \sqcup Y should have two path components.... 10. Difference between the Cartesian product of spaces and the Cartesian product of their basis? ie (Topological definition of X \cross Y) Ans: The Cartesian product of sets is a set. The product of spaces is a space, and so it needs a topology. A basis for the product topology was described in lecture. Overview - 1. Topology has been referred to as "rubber sheet geometry" - is that an accurate description of the content of this module? Ans: It describes topology very well. This module will go beyond just giving a description; we will also study the fundamental group and covering spaces. Above and beyond - 1. Have you ever played topological connect four? Ans: Interesting! No, I have not. I assume that this is played on a standard board, but we work in the edge-wise quotient which is a (vertical) cylinder? 2. What is P^2(\RR) homeomorphic to? [real 2-dimensional projective space] Also P^n(\RR) in general. Ans: These are new spaces, and are not homeomorphic to anything we have yet discussed. As a way to get started: you can obtain \RP^2 = P(\RR^3) = P^2(\RR) by gluing together a disk and a M\"obius strip. 3. I've heard that non-algebraic topology is pretty much figured out w/o much more to do. Is this true? Ans: No, it is not. See the dauntingly impressive scholarship of Van Mill, Reed, and Pearl: http://at.yorku.ca/topology/qintop.htm (scroll down to the section "Open problems in topology"). Admin - 1. Will the assignments be marked out of 2, like with Geometry? If so could we please change it I think most people find it quite weird and annoying. Ans: Assignments will be marked out of 2. This is done to reduce the workload on the TAs and it will not change in the immediate future.