Allen Hatcher Note: I have retired from teaching and advising students but am still active in research and writing. The best way to contact me is via email. If I do not respond in a timely fashion it is because I have nothing useful or definite to say.
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 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.

Spectral Sequences 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.

Topology of Numbers 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 webpage for the book for more information or to download an electronic version of the book. A print version has been published by the American Math Society in 2022, but the electronic version will remain freely available here.

List of Published Papers

Preprints:

• "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 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.
• "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 (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. 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 written in 2018 clarifying the proof of Proposition 6.2.
• "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 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:

• 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.

Photos of Renwick Brook