Math 990 Topics in Topology running topics page
Spring, 2008
Setting the stage: Lowdimensional topology and the fundamental group.
Article: Aschenbrenner, Friedl, Wilton, 3manifold.groups (2012)
[public: ArXiv]
In 3manifold topology, the fundamental group captures an immense amount of information about a manifold,
including irreducibility, fibering over the circle, fibering by circles (Seifert fibering), containing
essential tori, etc. When a new grouptheoretic concept comes along, a natural question to ask is: what
topological proerty does it reflect?
(Left) orderings on groups.
Text: Clay and Rolfsen, Ordered groups and Topology (2016)
Article: Sunic, .Orders on free groups induced by oriented words (2013) [public:ArXiv]
Basic concepts, positive cones; examples: Homeo(R), free group (Ping Pong Lemma); characterizations of orderability;
local indicability, Kaplansky's Conjecture; embedding LOgroups in Homeo(R).
Knot theory, 3manifolds, and left orderings.
Text: Clay and Rolfsen, Ordered groups and Topology (2016)
Text: Stillwell, Classical Topology and Combinatorial Group Theory (1980)
Knot groups, Wirtinger presentations, local indicability. Ordering 3manifold groups, leftorderable
quotients, convex subgroups. Dehn surgery/Dehn filling, Heegaard splittings, surgery description of a
3manifold.
Nonleftorderable groups.
Article: Clay and Watson, Leftorderable fundamental groups and Dehn surgery (2012)
Article: Boyer, Gordon, and Watson, On Lspaces and leftorderable fundamental groups (2011)
Dehn fillings of torus knot exteriors (Dehn surgery on a torus knot). Branched coverings, the nonLObility
of double branched covers over an alternating knot.
Foliations.
Text: Calegari, Foliations and the Geometry of 3manifolds (2007)
Article: Lickorish, A foliation for 3manifolds (1965)
Article: Calegari and Dunfield, Laminations and groups of homeomorphisms of the circle (2002)
Definitions, constructions, all 3manifolds foliate. Taut/Reebless foliations, the space of leaves,
Rcovered foliations. Foliated Seifertfibered spaces; Rcovered implies leftorderable fundamental group,
tautly foliated implies virtually leftorderable.
Big question: are tautly foliated and leftorderable
(essentially) the same? Experimental evidence/odd coincidences suggest yes!
Heegaard Floer homology.
Article: Ozsvath and Szabo, An Introduction to Heegaard Floer Homology
Article: Ozsvath and Szabo, Heegaard Floer homology and alternating knots (2003)
Article: Sahamie, Introduction to the basics of Heegaard Floer homology (2013)
Motivation: the seeming strong connections between LO, tautly foliated, and `nonsimple' HF homology.
Pointed Heegaard diagrams, equivalence, (tori in) symmetric product; Whitney disks, Spin^c structures, Morse
functions, the differential for HFhat. Knot Floer homology, categorification of the Alexander polynomial,
detection of knot genus. Knot surgery as 2handle addition, 3manifold triads, the surgery exact triangle.
Lspaces, Lspaces from Dehn surgery on knots, Lspaces from double branched covers of alternating knots.
Lspaces have no taut foliations!
Ordering braid groups.
Article: Fenn, Greene, Rolfsen, Rourke, and Wiest, Ordering the braid groups (1999)
Braid groups, Artin presentation, mapping class groups, arc diagrams; Dehornoy ordering, ipositive words.
