Math
797AT, Algebraic topology

Fall 2015

Fall 2015

Lectures: TuTh 1:00-2:30, LGRT 202.

Instructor: Mike Sullivan
Office: LGRT 1623E Phone: 545-0510
email: my last name <at> math <at> umass
<at> edu

Office hours: Tues and Thurs 10:30-12:00. Feel free to drop in anytime. I am almost always in 9:30-4:00 Tues, Thurs, Fri, and sometimes Mon and Weds,
except for certain times (12-100 lunch, TuTh 100-345 teach, Fri 230-330 seminar).

Text

Required: Algebraic Topology , by Allen Hatcher. It costs about
$22-$35 on Amazon.com, for example.
This text can also be legally downloaded for free at
Hatcher's website.

Description:
This course will be centered on Chapters 1 (fundamental group),
2 (homology) and 3 (cohomology) of the Hatcher text.
Chapter 1 will be more review since it is covered in Math 671.
Ideally, if time permits, there will be some topics/examples on
differential or low-dimensional topology, such as Morse homology or knot homologies.
The grade will be based on the timely completion of about 6 homeworks.
Prerequisites are Math 671 and 611, or equivalent.

Announcements:

9/29: I have to cancel my class and office hours Tues Oct 6 and Thurs Oct 15.
The make-up classes are TENTATIVELY scheduled for Thurs Oct 1 530-645pm (LGRT 1334) and Weds Oct 7 5:05-6:20 (LGRTT 1114). Make-up office hours is 2-5pm Weds Oct 7.

9/24: I won't be around Friday 9/25 because of a medical appt.

Homework

HW 6 Part I: Due Tuesday December 8 in class.

#5, #11 in Section 3.3 (For #11, the definition of degree appears in
#7 Section 3.3 For a hint, you may want to revisit Section 3.2 #1 from last HW.)

Part II: ``Due" Tuesday December 8. Skim sections 2 and 3 of
http://people.math.umass.edu/~sullivan/797SG/hutchings-morse.pdf
My plan is to start on differential topology after Thanksgiving, and hopefully
present Morse homology.

HW 5 Part I: Due Thursday November 12 in class.

#20, 22 in Section 2.2 and #5 in Section 2.C

Part II: Due TUESDAY November 24 in class.

#1, 11 in Section 3.2.

Extra problem: Give an example of a group G and two maps f, g: RP^2 -> S^2
such that f_* = g_* as maps on homology with integer coefficients,
but f_* and g_* are not equal as maps on homology with G-coefficients.
What does such an example illustrate in Theorem 3A.3?

Remark: In lecture I had said to prove a certain map labeled tau induced the identity on cohomology (this was in the proof that the cup product is
skew-commutative). I decided not to assign that problem, since it is too close to the proof in Hatcher.

HW 4 Part I: Due Thursday October 29 in class.

#2, 4, 17 in Section 2.2

Part II: Due Thursday November 5 in class.

Complete first 2 in-class Mayer-Vietoris problems if turning Parts 1 and 2 in on time.
Complete all 3 Mayer-Vietoris problems if turning it in one/both of the parts late.
(If turning in on-time, can complete 3rd problem for extra credit.)

HW 3: Due Thursday October 22 in class.

#8, 12, 14 (last question only), 17b (compute H_n(X,A) only), 18, 27
in Section 2.1.

Hint for 17b: the inclusion i:A ->X induces the zero map i_*:H_1(A) -> H_1(X).

HW 2: Due Thurs Oct 1 in class. (Extended to Oct 7)

#4, 12, 14, 18, 27 in Section 1.3

Read some source like
https://en.wikipedia.org/wiki/Topological_group
and prove that the fundamental group of a topological group,
with base point the identity element, is abelian.

HW 1: Due Thurs 9/24 in class.

#3, 6, 16 in Section 1.1 (of Hatcher, unless otherwise indicated)

#7, 10, 17 in Section 1.2