Department of Mathematical Sciences University of Cincinnati
Algebraic Topology (15-Math-605-001) Winter Quarter 2007

Instructor	Prof David A Herron	810B Old Chem Bldg
Office Hours	M,W 12:30-1:30	556-4075
E-mail	David.A.Herron "at" UC.edu

Listed below is information regarding: some interesting *things*, the current week's hot topics, suggested problems, homework.

Textbooks There are some books on reserve (under my name) in the library. Below are the primary texts which I will use to generate my lectures; the latter two are somewhat more sophisticated than the rest. It's a good idea to look at more than one book, because often some author will say things in just the 'right' way.... The order here is somewhat indicative of how I will use the texts. Also, do not overlook the web as a resource; e.g., I just did a google search on "identification spaces" and found a number of interesting useful links.

• K Topology, An Introduction to Point Set and Algebraic Areas by Kahn
• L Introduction to Topological Manifolds by Lee
• Ma Algebraic Topology: An Introduction by Massey
• S An Introduction to Topology and Homotopy by Sieradski
• Mu Topology (second edition) by Munkres
• H Algebraic Topology by Hatcher
• B Topology and Geometry by Bredon

This course is an excursion into the realm of algebraic topology. Please take a few hours to review point-set topology; for the most part, chapters 3-6 of Kahn, or chapters 2 and 3 of Munkres, contain the prerequiste information. Be sure you understand quotient and adjunction spaces. (OK, yes we will begin by covering some point set ideas which you did not get to last quarter. :-)

I plan one in class midterm exam (date to be announced, but see the schedule below) and a comprehensive final exam on Monday 12 March. Homework will be assigned on a regular basis and collected daily. As soon as we schedule it, we will have a weekly problem session to work through that week's assignment. In order to make good use of this time, I ask that you please prepare ahead of time. The problems which we do not solve during this session will then be due in class the following week.

Roughly speaking, your final course grade will be an A if you are doing PhD level work or a B if you are doing graduate level work but not at the PhD level (or else a C). The final exam, midterm exam, and HW will all count about the same (roughly, 40%, 30%, 30%). I will grade each HW problem on a scale of 1-10 with 8-10 being A work and 5-7 being B work.

Topics Covered (although not necessarily in this order!)

• Construction of Spaces
  • identification and quotient spaces
  • disjoint unions, wedge products, adjunction spaces
  • simplicial complexes
• Manifolds
  • basic definitions and examples
  • Whitney's embedding theorem (baby version)
  • connected sums, handles, cross handles, cross caps
  • classification of closed surfaces
• Homotopy Theory
  • ideas and definition of homotopy, examples, essential maps versus null-homotopic ones
  • equivalence relation and basic properties (e.g., naturality)
  • relative homotopy
  • contractible spaces
  • deformation retractions and homotopy equivalence
• Fundamental Group
  • homotopy of paths, concatenation, algebraic structure
  • role of the basepoint
  • simple connectivity
  • fundamental group of circle
  • fundamental group of sphere
  • Seifert - van Kampen Theorem
  • fundamental groups of: wedges, compact surfaces
• Covering Spaces
  • ideas and definitions, examples
  • lifting problem: ULT, PLT, PHLT
  • the FG of a CS, the conjugacy property
  • lifting criterion
  • equivalence of CS
  • universal covering space
  • group action property (monodromy)
  • covering transformations
  • normal covering spaces
  • existence of covering spaces
• Applications

Here are some interesting *things* such as links, movies, etc.

• All you want to know about Möbius bands.
• An mpg showing how to cut open the Klein bottle.
• Some info about the Klein bottle.
• Outside In.
• Some history and kewl pix.
• Visualization of Sphere Eversion.
• The Optiverse.

3-5 Jan This week I hope to finish the relevant parts of chpt 6 in Kahn. In particular, we will investigate the notion of identification spaces.
8-12 Jan This week we finish quotient spaces and begin manifolds.
17-19 Jan This week we talk about partitions of unity and connected sums.
22-26 Jan This week we discuss the classification theorem for compact surfaces.
29-Jan-2 Feb This week we continue the classification theorem for compact surfaces.
5-9 Feb This week we finish the classification theorem for compact surfaces.
12-16 Feb Finally, we start homotopy theory! And take a midterm.
19-23 Feb This week we should begin the fundamental group.
26 Feb-2 Mar
5-9 Mar During our last week we
12-16 Mar Final Exam Week

Here is a table with the assigned homework and due dates. Please be sure to check out my suggestions for writing up HW solutions.

Due Dates	Assigned HW Problems			Problem Session
3,5 Jan		1st day	2
8,10,12 Jan	3,6	7	9	4-6,10,11
15,17,19 Jan	MLK	12,13(b),14(c),18(b)
22,24,26 Jan	19(c)		22(KB)	14-17,20
29,31 Jan 2 Feb	26	23(a,b,c)	27	22,24,28-31
5,7,9 Feb		37	38	28,34,35,36
12,14,16 Feb	26		Midterm	39,40,41
19,21,23 Feb		5(homeo),41	5(hpty),23(d),45,49,50	23(e),42,44,46,51-54
26,28 Feb 2 Mar			51,53,56,59,61	52,54,55,57,58,60,62
5,7,9 Mar
Mon 12 Mar	Final Exam 1:00-4:00

Here are suggested problems for each indicated section in Munkres.

• 51, Homotopy, p.330: 1-3
• 52, Fundamental Group, pp.354-355: 1,2,3,4,5,6
• 53, Intro to Covering Spaces, p.341: 1-6 all good
• 54, ULT, PLT, PHLT, pp.347-348: 1-3(do!),4,5,6-8 good too
• 55,56,57, Applications
• 58, Def Retracts and Homo Equiv, pp.366-367: 1&3(easy),2(fun!),4(do it!),5&6(easy),7,8
• 59, Fundamental Group of the Sphere, p.370: 1-2(good),3(easy),4good
• 60, Fundamental Groups of some Surfaces, p.375: 1-5 all good
• 67, Free Abelian Groups, p.412: 4
• 68, Free Products of Groups, p.412: 2,3
• 69, Free Groups, p.425: 1,3
• 70, Seifert-Van Kampen Theorem, p.433: 1,2,3
• 71, Wedge of Circles, p.438: 1,2,3,4,5
• 72,73,74, Compact Surfaces , p.445:1,2; pp.453-454:1-7
• 79, Equivalence of Covering Spaces, p.483:1-5