Allen Hatcher Office: 553 Malott Hall Phone: (607)-255-4091 Math 6510 Spring 2013
Book Projects:	Algebraic Topology Vector Bundles and K-Theory Spectral Sequences in Algebraic Topology Topology of Numbers
Course Notes:	Basic 3-Manifold Topology Introductory Point-Set Topology
Papers
Other Things:	Pictures of stable homotopy groups of spheres Pictures of the cohomology of SO(n) List of recommended books in Topology (from 2003, needs updating) Books by other authors

Book Projects

Algebraic Topology This is the first in a planned series of three textbooks in algebraic topology having the goal of covering all the basics while remaining readable by newcomers seeing the subject for the first time. The first book contains the basic core material along with a number of optional topics of a relatively elementary nature. The second and third books are only partially written --- see below. To find out more about the first book or to download it in electronic form, follow this link to the download page.

Vector Bundles and K-Theory The intention is for this to be a fairly short book focusing on topological K-theory and containing also the necessary background material on vector bundles and characteristic classes. For further information, and to download the part of the book that is written, go to the download page.

Spectral Sequences in Algebraic Topology This idea is for this to be a readable introduction to spectral sequences, with emphasis on their applications to algebraic topology. What is written so far is mostly about the Serre spectral sequence. For more details and to download what is written so far, go to the download page. Update: My current plan is to make this an extra chapter of the algebraic topology book when I make a second edition of that book, rather than have it be a separate book.

Topology of Numbers This is intended to be an undergraduate textbook presenting a somewhat geometrically-flavored introduction to elementary number theory. See the download page for more information and to get a pdf file of the part of the book that has been written so far.

List of Published Papers

Slides of Talks:

• Stable homology of spaces of graphs. A talk at Banff in December 2012 about an analog of the Madsen-Weiss theorem for certain 3-manifolds. There is also an earlier version of this talk given at Stanford in July 2012.
• A 50-year view of diffeomorphism groups. A talk at the 50th Cornell Topology Festival in May 2012 sketching some highlights of what's known about the homotopy types of diffeomorphism groups of smooth manifolds. A full history would of course be impossible in an hour talk.
  
• Stable homology by scanning: Variations on a theorem of Galatius. A talk at Luminy in June 2010 outlining the current simplified form of the proof of Galatius' theorem on the stable homology of Aut(F_n), with comments on how the Madsen-Weiss theorem follows similarly, as do a couple analogs in dimension three involving handlebodies.
• The classification of 3-manifolds -- a brief survey. This is a written version (11 pages) of an expository talk at the 2004 Cornell Topology Festival. Also available is a pdf file of the transparencies for the talk itself.

Preprints:

• "An exposition of the Madsen-Weiss theorem". pdf file (34 pages). This is a revised version posted May 2013, with the main exposition rearranged and an appendix added giving the Cerf-Gramain proof that the diffeomorphism groups of most surfaces have contractible components.

Papers published since 1996:

• "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 file
• "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. (If you wish to maintain control on the final form of your published paper, do not submit it to the Duke Journal.)
• "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
• "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 (preprint 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 to this paper that is listed above.
• "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 (2000), 1-24.
• "Rational homology of Aut(F_n)" (with Karen Vogtmann), Math Research Letters 5 (1998), 759-780. pdf file
• "The complex of free factors of a free group" (with Karen Vogtmann), Oxford Quarterly 49 (1998), 459-468. pdf file
• "Cerf theory for graphs" (with Karen Vogtmann), J. London Math Soc. 58 (1998), 633-655. pdf file
• "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).

Unfinished drafts:

• "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)

Unpublished papers:

• "The cyclic cycle complex of a surface". pdf file (January 2008, 7 pages). This has been largely superceded 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 rather large pdf file (3.8MB) that was made by scanning the original typescript. You can also view it onscreen here.

Photos of Renwick Brook