This book, published in 2002, is a beginning graduate-level textbook on algebraic
topology from a fairly classical point of view. To find out more or to download it in electronic form, follow this link to the download page.
Vector Bundles and K-Theory
This unfinished book is intended to be a fairly short introduction to topological K-theory, starting with the necessary background
material on vector bundles and including also basic material on characteristic classes. For further
information or to download the part of the book that is written, go
to the download page.
I have now returned to an earlier plan of having this material be an extra chapter of the Algebraic Topology book, rather than a separate book. The current version of this chapter is here. Its main focus is the Serre spectral sequence and its applications, but there is also some coverage of the Adams spectral sequence and, more briefly, a few other spectral sequences. In case you would like to see the older version that was planned as a separate book, this can be found on this page.
This is an undergraduate-level introduction to elementary number
theory from a somewhat geometric point of view, focusing on quadratic forms in two variables with integer coefficients. See the download page for more information and to get a pdf file of the book. Publication by the American Math Society is expected in 2022. The pdf version will remain freely available here after the print version appears.
- List of Published Papers
- "Moduli spaces of circle packings" (with William Thurston). pdf file (14 pages) posted March 2018. This is a partial write-up of a project that Bill and I worked on in 2008. The paper contains two theorems, but the proof of one of them is not complete. The remaining details could have been easily filled in by Bill, and my hope is that someone else will be able to do this now.
- "The Kirby torus trick for surfaces". pdf file (10 pages) posted January 2014 with small revisions in 2022. This is an expository account of two classical theorems in surface topology: Topological surfaces have unique smooth structures, and homeomorphisms of smooth surfaces are isotopic to diffeomorphisms. With the torus trick, almost no point-set topology is needed for the proofs.
- "A short exposition of the Madsen-Weiss theorem". pdf file (43
pages). This version posted February 2014. Appendices have been added giving the calculation of the stable rational homology, a proof of the Group Completion Theorem, and the Cerf-Gramain proof that the diffeomorphism groups of most surfaces have contractible components.
- "Simplifying and separating configurations of disjoint unlinked spheres in Euclidean space". pdf file posted in April 2022.
Papers published since 1996:
- "Tethers and homology stability for surfaces" (with Karen Vogtmann). Alg. & Geom. Topology 17 (2017), 1871-1916. pdf file.
- "Assembling homology classes in automorphism groups of free groups" (with Jim Conant, Martin Kassabov, and Karen Vogtmann). Commentarii Math. Helv. 91 (2016), 751-806. pdf file.
- "Configuration spaces of rings and wickets" (with Tara Brendle). Commentarii Math. Helv. 88 (2013), 131-162. pdf file
- "Generating the Torelli group" (with Dan Margalit).
l'Enseignement Math. 58 (2012), 165-188. pdf
- "Stabilization for mapping class groups of 3-manifolds" (with
Nathalie Wahl). Duke Math. J. 155 (2010), 205-269. Here is a pdf file of the version from October 2009
which is in some ways preferable to the published version.
- "Stabilization for the automorphisms of free groups with
boundaries" (with Nathalie Wahl). Geometry and Topology 9 (2005),
1295-1336. pdf file. See also the short Erratum that refers to our second
paper (listed above) for details.
- "Homology stability for outer automorphism groups of free groups"
(with Karen Vogtmann). Alg. & Geom. Topology 4 (2004), 1253-1272. pdf file. There is a correction:
- "Erratum to: Homology stability for outer automorphism groups of
free groups" (with Karen Vogtmann and Nathalie Wahl). Alg. & Geom.
Topology 6 (2006), 573-579. pdf file
- "Pants decompositions of surfaces" pdf file. This is a 12-page excerpt
from a joint paper with Pierre Lochak and Leila Schneps,"On the
Teichmuller Tower of Mapping Class Groups," J. reine angew. Math. 521
- "Rational homology of Aut(F_n)" (with Karen Vogtmann), Math
Research Letters 5 (1998), 759-780. pdf
- "The complex of free factors of a free group" (with Karen
Vogtmann), Oxford Quarterly 49 (1998), 459-468. Here is a corrected version posted in March 2022.
- "Cerf theory for graphs" (with Karen Vogtmann), J. London Math
Soc. 58 (1998), 633-655. pdf file There is also a short Addendum clarifying the proof of Proposition 6.2, written in 2018.
- "Finiteness of classifying spaces of relative diffeomorphism
groups of 3-manifolds" (with Darryl McCullough), Geometry and Topology
1 (1997), 91-109. pdf file
- "Isoperimetric inequalities for automorphism groups of free
groups" (with Karen Vogtmann), Pacific J. Math. 173 (1996), 425-441. pdf file
Updates of older papers:
- "Spaces of incompressible surfaces". This is an improved version
of a paper published in Topology in 1976 with the title:
"Homeomorphisms of sufficiently large P^2-irreducible 3-manifolds". pdf file (10 pages)
- "Triangulations of surfaces". The original version of this was
published in Topology and its Applications in 1991. pdf file (7 pages).
- "On the diffeomorphism group of S^1 x S^2". This is a revision,
written in 2003, of a paper originally published in the AMS Proceedings
in 1981. pdf file (6 pages).
- "Boundary curves of incompressible surfaces". A rewritten version
of a paper originally published in 1982 in the Pacific Journal. pdf file (5 pages).
- "Diffeomorphism groups of reducible 3-manifolds". This is an
incomplete draft, lacking some proofs, posted in August 2008, replacing
an older version from about 10 years earlier. pdf
file (10 pages)
- "Topological moduli spaces of knots". This is also unfinished,
but the aim is to describe the homotopy types of the components of the
space of all knots in the 3-sphere. pdf
file (12 pages) This version posted October, 2002.
- "Measured lamination spaces for 3-manifolds". An unfinished draft
of a paper written in the late 1980's, now converted into TeX and
mildly edited, but with the same mathematical content as the old paper.
pdf file (36 pages)
- "The cyclic cycle complex of a surface". pdf file (January 2008, 7 pages). This
has been largely superseded by section 2 of the paper "Generating the
Torelli group" with Dan Margalit listed above, which gives an
exposition of the construction of Bestvina-Bux-Margalit that was the
inspiration for "the cyclic cycle complex". Later I realized that the
cell complex constructed by BBM has a canonical triangulation which
makes it a subcomplex of the cyclic cycle complex, so for most purposes
it is probably better to use the smaller BBM complex.
- "Bianchi orbifolds of small discriminant" pdf file (6 pages). This is a brief
report I wrote in 1983 but never published. Here are some Pictures of Bianchi orbifolds.
- "Curves on the 4-punctured sphere: 31 Charts" (with Bill Floyd).
This is an unpublished paper from around 1980 that you can download as
a pdf file
(3.8MB) that was made by scanning the original typescript. You can also
view it onscreen here.
Slides of Talks:
Photos of Renwick Brook