Office: second floor of the Math Building, Room 2107.
Phone: 301-405-8175 (from any campus phone dial 58175) or 301-405-1412
Purpose of Math 730: This course is a basic introduction to algebraic topology. It helps prepare students for the Geometry/Topology graduate qualifying exam. Alternatively, it may be used as one of the four additional qualifying courses..
- definition, of topological spaces, continuous maps, homeomorphisms
- metric spaces, disks, spheres, simplexes, and manifolds
- subspaces, products
- paths, homotopy, homotopy equivalences, contractible spaces, deformation retracts
- what is algebraic topology?
Review of general topology
- the Hausdorff condition, compactness, local compactness
- compactly generated spaces, and continuous maps
- products and quotient spaces, and continuous maps
- second countability
- connectness, path connectedness
- local connectedness, local path connectedness, local contractiblity
CW complexes
- attaching a cell
- CW complexes and relative CW complexes, dimension
- graphs, spheres, projective spaces
- cones, suspensions, joins and smashes
- CW complexes are Hausdorff
- CW complexes are compact if and only if they are finite
-The homotopy extension property
-The product of CW complexes is a CW complex
The fundamental group
- definition and properties
- products
- example: the circle
- applications
- change of basepoint, normal sub groups, and homotopy equivalences
- free products of groups, free groups
- wedges and van Kampen's theorem
Covering spaces
- definition and unique path lifting
- construction
- normal covering spaces and deck transformations
- covering space of a CW complex
- fundamental group of a CWcomplex
- Riemann surfaces
- examples
Homology of a chain complex- definition and homology of a chain complex, naturality
- homotopy
- subcomplexes, relative homology
- the long exact sequence and the five-lemma for chain complexes and pairs
- Euler characteristic
Singular homology
- the singular chain complex, homology, naturality
- contractible spaces, points, reduced homology
- homotopy
- singular homology for pairs, the long exact sequence
- barycentric subdivision
- the subcomplex subordinate to an open cover
- excision, Mayer-Vietoris
- cellular homology
- singular chain complex of a covering space
- spheres, suspensions, degree of a map, projective spaces
Advanced topics
- fundamental group and homology
- homology of a polyhedron
- fundamental group of a CW complex
- Riemann surfaces geometrically
- homology with coefficients
- Borsuk-Ulam theorem
- Lefschetz fixed point theorem
www.math.cornell.edu/~hatcher
Dugundji, Topology, Allyn and Baker 1966
Kelley, General Topology, Springer Verlag GTM, ISBN 978-0-387-90125-1
May, A Concise Course in Algebraic Topology, 1999, Univ. of Chicago Press, ISBN 978-022651-132
Greenberg and
Harper, Algebraic Topology: A First Course, 1981, ISBN-13: 978-0805335576
Massey, Algebraic Topology: an Introduction, Springer Verlag, 1977, ISBN 978-0-387-90271-5
Instructor: Professor Steve Halperin
Contact Information:
Email: shalper@umd.edu.
Office: second floor of the Math Building, Room 2107.
Phone: 301-405-8175 (from any campus phone dial 58175) or 301-405-1412 (from any campus phone dial 51412) .
Lecture Classroom: Math 0104
Lecture:
MWF 9:00 - 9:50
Course webpage: http://www2.math.umd.edu/~shalper/Math730.html
Classroom rules: No cell phone conversations, no emails, no texting,
no web surfing.
Homework: Homework will be assigned by email with a specified due date, and is due in class that day. Solutions to problems must be clear and written in good mathematical language. Late homework will not be accepted! Homework must be typed or written in ink, or may be submitted as a pdf by email.
Midterm Schedule: Midterm 1: Fri. Oct 6 in class
Midterm 2: Fri. Nov.3 in class
Grades: Grades will be weighted as follows:
Final exam 40%
Final grades will be submitted using the plus/minus system.
Students may wish to check out university policies relevant to courses at http://www.ugst.umd.edu/
Academic integrity: Students are expected to know, understand and comply with the principles of academic integrity. Collaboration on homework is encouraged, copying is illegal.