Classes: Monday, Tuesday, Wednesday, Thursday, 9:30 - 10:18 a.m.
at Koffolt Lab 0136.
Grading: There will be biweekly homework assignments (70%) and take home final exam (30%).
Chain complexes. Simplicial homology. Cellular homology.
Singular homology. Chain homotopy and invariance of homology.
2. Exact sequences.
Relative homology. Long exact sequence. Mayer-Vietoris exact sequence.
Axioms for homology.
3. Homology and homotopy.
Betti numbers and Euler characteristic. Relation between fundamental group
and first homology. Higher homotopy groups. Hurewicz map.
4. Homology with arbitrary coefficients.
The universal coefficient theorem. The Künneth theorem.
Introduction to cohomology.
The additional topics which we may disscuss if the time will permit.
The Puancaré duality for manifolds.
Introduction to spectral sequences.
The Khovanov homology of knots.
Assignment #1 (due January 17, 2012)
Assignment #2 (due January 31, 2012)
Assignment #3 (due February 14, 2012)
Assignment #4 (due February 28, 2012)
Final assignment (due March 13, 2012)