This page contains handouts for the whole year course 110.615-616 Algebraic Topology, in reverse chronological order, latest first. These notes are intended to supplement the text. They mostly provide alternate treatments of selected topics and cover some topics that are barely addressed, if at all, in the text.

Notes are available in both DVI and PDF formats. The DVI version is more reliable (being the original), but is readable only on systems (usually large) that fully support TeX. The PDF version is widely readable, but certain fonts may be incorrect or missing. Either version may be affected by the quality of the fonts used by your local system. Try both to see which works better for you.
The margins are arranged to allow three-hole punching for storage in a ring binder, assuming duplex printing is used.

• Graded Algebra (6 pages) - choose
  • Graded Algebra, in DVI format or
  • Graded Algebra, in PDF format.
    This note reinterprets cycles, cochains, cup products etc. in terms of chain maps of nonzero degree and chain homotopies between them.
• Alexander-Whitney chain map (3 pages) - choose
  • Alexander-Whitney, DVI format or
  • Alexander-Whitney, in PDF format.
    This note gives a detailed proof that the Alexander-Whitney formula does indeed define a chain map. This is essentially equivalent to the derivation formula for cup (or cross) products.
• Eilenberg-Zilber Theorem (5 pages) - choose
  • Eilenberg-Zilber, DVI format or
  • Eilenberg-Zilber, in PDF format.
    This note gives a proof of the Theorem.
• Some Common Tor and Ext Groups (6 pages) - choose
  • Tor and Ext groups, DVI format or
  • Tor and Ext groups, in PDF format.
    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).
• Universal Coefficient Theorem for Cohomology (2 pages) - choose
  • universal coefficient theorem for cohomology, in DVI format or
  • universal coefficient theorem for cohomology, in PDF format.
    This note presents a direct proof of the universal coefficient theorem for cohomology which is essentially dual to the proof for homology.
• Universal Coefficient Theorem for Homology (2 pages) - choose
  • universal coefficient theorem for homology, in DVI format or
  • universal coefficient theorem for homology, in PDF format.
    This note presents a direct proof of the universal coefficient theorem for homology that is simpler and shorter than the standard proof.
• Attaching a 2-cell (2 pages) - choose
  • attaching a cell, in DVI format or
  • attaching a cell, in PDF format.
    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.
• van Kampen's Theorem (3 pages) - choose
  • van Kampen, in DVI format or
  • van Kampen, in PDF format.
    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.
• The Homotopy Extension Property (4 pages) - choose
  • Homotopy Extension Property, in DVI format or
  • Homotopy Extension Property, in PDF format.
    This note presents basic facts about the Homotopy Extension Property (HEP), somewhat rearranged from Hatcher.
• Pushouts and Adjunction Spaces (4 pages) - choose
  • Pushouts, in DVI format or
  • Pushouts, in PDF format.
    This note defines pushouts and adjunction spaces in general, and reviews their elementary properties, with many examples. Most of the constructions in Chapter 0 of Hatcher's Algebraic Topology are examples of adjunction spaces.