
MATH 7321 · Topology 3
Spring 2017
Tuesday & Thursday 4:105:40pm in 544 NI
Office hours: Tuesday & Thursday 3:004:00pm in 435 LA

This is a course in classical Algebraic Topology, and some of its applications. Topics we may cover include: Higher homotopy groups, cofibrations, fibrations, fiber bundles, homotopy sequences, homotopy groups of Lie groups and associated manifolds, cellular approximation, Hurewicz theorem, Whitehead theorem, EilenbergMacLane spaces, obstruction theory, Postnikov towers, Hspaces and Hopf algebras, Bockstein homomorphism, PoincaréLefschetz duality, Alexander duality, Euler class, Gysin sequence, cobordism, intersection form, signature, plumbing, cohomology of fiber bundles, classifying spaces, characteristic classes, spectral sequences, Steenrod squares.

We will cover material selected from the following textbooks:


Algebraic Topology, by Allen Hatcher, Cambridge University Press, 2002. MR.

Topology and Geometry, by Glen Bredon, GTM No. 139, SpringerVerlag, 1997.
MR.

Modern classical homotopy theory, by Jeffrey Strom, Graduate Studies in Mathematics, vol. 127,
American Mathematical Society, Providence, RI, 2011. MR.

Lecture Notes in Algebraic Topology, by James F. Davis and Paul Kirk, Graduate Studies in Mathematics, vol. 35, American Mathematical Society, Providence, RI, 2001. MR.

Here are some other useful references, and material for further reading:


Elements of Homotopy Theory, by George W. Whitehead, GTM No. 61, SpringerVerlag, 1979. MR.

Algebraic Topology, by Edwin H. Spanier, Corrected reprint, SpringerVerlag, 1981. MR.

Cohomology operations and applications in homotopy theory, by Robert Mosher and Martin Tangora, Harper and Row, New YorkLondon, 1968.
MR.

A user's guide to spectral sequences, by John McCleary, second edition, Cambridge Studies in Advanced Math, no. 58, Cambridge University Press, 2001. MR.

Characteristic Classes, by John W. Milnor and James Stasheff, Ann. Math. Studies, No. 76, Princeton University Press, 1973. MR.

The topology of fibre bundles, by Norman Steenrod, reprint of 1951 edition, Princeton University Press, 1999. MR.

Spectral Sequences in Algebraic Topology, by Allen Hatcher, draft book, 2004.

A Concise Course in Algebraic Topology, by J. Peter May, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1999.
MR.

Homework assignments

Homework 1: Hatcher, Chapter 3.3, pp. 257259: Problems 2, 3&4, 5, 6, 10, 24. Due January 19.

Homework 2: Hatcher, Chapter 3.3, pp. 258260: Problems 7&9, 25, 26, 28, 29, 32&33. Due February 2.

Homework 4: Hatcher, Chapter 4.1, p. 358, Problem 17; Chapter 4.2, p. 392, Problem 34; Chapter 4.G, p. 460, Problem 1; Chapter 4.H, p. 466, Problems 1, 2; Chapter 4.I, p. 469, Problem 1. Due March 2.
Class Materials

