Mathematical Notes for Algebraic Topology

Last revised 29 OCT 2010, by JMB
Maybe you really want the index page instead.

Most notes are available in both DVI and PDF formats.

Notes for 110.615-616 Algebraic Topology
Many of these notes fill in details that are only briefly discussed or omitted entirely in Hatcher's Algebraic Topology.

Pushouts and Adjunction Spaces (4 pages) Available in your choice of:
- Pushouts, in DVI format or
- Pushouts, in PDF format.
Attaching a 2-cell (2 pages) This note applies van Kampen's Theorem to compute the effect on the fundamental group of attaching a 2-cell to a space. Compare Proposition 1.26 in Hatcher. Available in your choice of:
- Attach 2-cell, in DVI format or
- Attach 2-cell, in PDF format.
van Kampen's Theorem (3 pages) This note presents an alternate proof of van Kampen's Theorem from the pushout point of view, for the case where the space is covered by two open sets. Available in your choice of:
- van Kampen, in DVI format or
- van Kampen, in PDF format.
The Torus Triangulated (2 pages) The most efficient triangulation of the torus is presented. Explicit cycles and cocycles are found whose classes generate the homology and cohomology. These may be used to compute cup products simplicially. Available in your choice of:
- Torus triangulated, in DVI format or
- Torus triangulated, in PDF format.
The Real Projective Plane Triangulated (2 pages) The most efficient triangulation of the real projective plane is presented. By removing one simplex, one obtains the Moebius band. The Klein bottle is displayed as the connected sum of two copies of the real projective plane. Available in your choice of:
- RP2 triangulated, in DVI format or
- RP2 triangulated, in PDF format.
Simplicial Complexes and Delta-Complexes (4 pages) This note compares simplicial complexes, ordered simplicial complexes, and delta-complexes. Available in your choice of:
- Delta Complexes, in DVI format or
- Delta Complexes, in PDF format.
Some Common Tor and Ext Groups (6 pages) This note computes all the groups G tensor H, Tor(G,H), Hom(G,H) and Ext(G,H) for G and H any of Z, Z/n or Q, including the difficult case Ext(Q,Z). Available in your choice of:
- Tor and Ext, in DVI format or
- Tor and Ext, in PDF format.
Field Coefficients (2 pages) The easy case of the Universal Coefficient Theorem when the ground ring is a field is discussed, starting from integer coefficients as in Munkres' book. Available only in:
- UCT over Field, in DVI format.
Homotopy Groups of Spheres (1 page) One pageful of homotopy groups of spheres, from Toda's book. It only goes up to the 12-sphere and the 23rd homotopy group. Available in your choice of:
- Sphere Homotopy, in DVI format or
- Sphere Homotopy, in PDF format.
Universal Coefficient Theorem for Homology (2 pages) This note presents a direct proof of the universal coefficient theorem for homology that is simpler and shorter than the standard proof. Available in your choice of:
- Homology UCT, in DVI format or
- Homology UCT, in PDF format.
Universal Coefficient Theorem for Cohomology (2 pages) This note presents a direct proof of the universal coefficient theorem for cohomology that is essentially dual to the proof for homology. Available in your choice of:
- Cohomology UCT, in DVI format or
- Cohomology UCT, in PDF format.