Department of Mathematical Sciences University of Cincinnati
Algebraic Topology (15-Math-605-001) Winter Quarter 2005

Instructor	Prof David A Herron	813 Old Chem Bldg
Office Hours	M,W 11-12	556-4075
E-mail	David.A.Herron "at" UC.edu

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

Textbooks There are some books on reserve (under my name) in the library. Here are the primary texts which I will use to generate my lectures. 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 text.

• Ma Algebraic Topology: An Introduction by Massey
• S An Introduction to Topology and Homotopy by Sieradski
• H Algebraic Topology by Hatcher
• Mu Topology (second edition) by Munkres
• 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 2 and 3 of Munkres contain the prerequiste information. Be sure you understand quotient and adjunction spaces.

I plan one in class midterm exam (date to be announced) and a comprehensive final exam. Homework will be assigned on a regular basis. Every Friday morning, 9-11, we will have a problem session to work through that week's assignment. In order to make good use of these two hours, 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 Monday.

Topics Covered (time amounts are tentative!)

• Construction of Spaces--2 lectures
  
  • manifolds-basic definitions and examples
  • connected sums, handles, cross handles, cross caps
  • classification of closed surfaces
  • CW complexes
  • wedges
• Homotopy Theory--2 lectures
  • 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--5 lectures
  • 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--7 lectures
  • 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--2 lectures

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
• ,
• , ,
• , ,

Here is the assigned homework with due dates.

Due Dates	Page:Problem
7,9 Jan		186:3	186:10
12,14,16 Jan	194:15	213:5	223:5
21,23 Jan		260:6	227:5
26,28 Jan	224:8	260:8
4,6 Feb		289:7	289:8
9,11,13 Feb	330:1	Midterm	330:3
16,18,20 Feb			335:5
23,25,27 Feb	335:3	335:6	341:6b
1,3,5 Mar	348:4,5	348:8	366:4
8,10,12 Mar	366:7	366:8	370:4
17 March	Final Exam 8-10

Here is the assigned homework with due dates.

Due Dates	Page:Problem
7,9 Jan
12,14,16 Jan
21,23 Jan
26,28 Jan
4,6 Feb
9,11,13 Feb		Midterm
16,18,20 Feb
23,25,27 Feb
1,3,5 Mar
8,10,12 Mar
17 March	Final Exam 8-10

we delve ever deeper into the mysteries surrounding the fundamental group. In regards to problem 3 (p.335 in Munkres) I'd like you to also/first answer the following:

• When will a given pair of paths joining two points induce the same isomorphism of the fundamental groups?
• When will all paths joining two points give induce the same isomorphism of the fundamental groups?