\title Short Complete Proofs of the Serre\\ Spectral Sequence
Theorems\endtitle
A new improved "Simple complete proofs of the Serre spectral
sequence theorems".
by Edgar H. Brown, Jr.
April 1,1997
In (\cite{B}) I set forth a proof of the Serre calculation of $E^{p,q}$
and claimed among other things that unlike my previous attempts to prove
this in graduate course lectures, this proof was routine. On presenting
this material in class, I discovered it was not as routine as I had
imagined. With the help of Pallavi Jayawart and Saso Strle I have
drastically improved and simplified the presentation, reducing proofs of
the Serre Spectral Sequence (SSS) theorems (\cite{S}) to a collection of
lemmas provable by straightforward mechanical checking which is left to
the reader. In addition it offers some motivation for the definitions.
A knowledge of the standard material on singular homology and
cohomology, including the Eilenberg-Zilber theorem, is sufficient to
prove the lemmas. We do homology first and then add variations,
including cohomology, as exercises.