Miscellaneous lecture notes and computer programs
My main research interest is geometric topology, particularly low-dimensional topology. Most of my research ultimately concerns the study of fundamental groups of manifolds.
In my thesis and early work (articles [1], [2], [3], [4], [8] below), I studied link homotopy, that is maps of a disjoint union of spheres to a sphere with disjoint images, up to the equivalence relation of homotopy keeping the components disjoint. An especially interesting case was the problem of deciding whether a map S2 U S2 to S4 with disjoint images is nullhomotopic by a nullhomotopy keeping the images disjoint. I constructed some simple invariants based on the usual group ring Whitney trick obstruction. In particular, the article [4] contains the most elementary example (that I know of) of an immersed 2-sphere in a 4 manifold X (with fundamental group Z) whose Whitney trick obstruction vanishes but which is not homotopic to an embedding. In high dimensions I considered mostly the situation when one component has codimension 2.
I wrote 2 articles ([2],[7]) on cut-and-paste of 3-manifolds and Casson's invariant. The article [2] proves that mutation of a homology 3-sphere preserves Casson's invariant but can change the homeomorphism type. The article [4] (never published-Morita's work included this and more) shows that if T denotes the Torelli group, then the subgroup of Hom(T,Z) spanned by homomorphisms given by Casson's invariant has finite rank.My study of analysis on 3-manifolds exploring the relation between Floer's instanton homology and SU(2) character varieties of 3-manifold fundamental groups was carried out in a long-term collaboration with Eric Klassen in the articles [5], [6], [9], [10], [11], [12], [13], [14], [15], and [16]. Our motivating problem was to understand how to describe in terms of algebraic topology how the spectrum near zero of the odd signature operator on an odd dimensional manifold (typically a 3-manifold) varies as the flat connection the operator is coupled to varies. Much of this work was technical (i.e. analysis) becase of the delicate nature of cut-and-paste arguments and related boundary value problems in the spectral theory of Dirac operators. The article [11] used these idea to compute SU(2) spectral flow on torus bundles over the circle, verifying a conjecture in TQFT (more general results were obtained later in Ben Himpel's thesis). The article [14] identified the SU(n) spectral flow of the odd signature operator with Massey products with local coefficients on a closed manifold. Our ultimate goal (never fully realized) was to prove a theorem analogous to this for manifolds with boundary. In most applications the first obstruction (cup products) is sufficient.
This segued into work done carried out with Hans Boden and Chris Herald to construct and compute an SU(3) Casson invariant for 3-manifolds (articles [25], [30], [33], [34]). Singularitites of the SU(3) character variety require a careful analysis of how perturbations of flatness equations perturb away. Boden and Herald figured this out, and as a consequence were able to define a real valued invariant which we computed for certain Seifert-Fibered spaces. Later we defined an integer valued invariant and computed it in certain cases.
The results on SU(3) Casson invariants depended on understanding how spectral invariants of Dirac operators behave with respect to cut-and-paste constructions. This was the theme of my work with Klassen in the case of spectral flow, and work with with Mark Daniel on a splitting formula for the spectral flow of arbitrary paths of Dirac operators (article [24]). Later work with Matthias Lesch gave general splitting formulas for eta invariants and Atiyah-Patodi-Singer rho-invariants (articles [24], [28], [29], [31]). The article [24] provided a precise splitting formula that had been anticipated in work of others, notably Yoshida and later Cappell-Lee-Miller. It turned out to be the key to understanding what to do when the kernel of the tangential Dirac operator on the bounding manifold jumps. The article [28] with Lesch provided clean and useful picture of how eta-invariants of general Dirac operators behave with respect to cut-and-paste constructions. It also included a detailed analysis of the case of the odd signature operator twisted by a flat connection; in particular it gave a description of how their Calderon projectors behave with respect to the Hodge decomposition of a bounding manifold.
In another direction I have investigated problems in knot and link theory. Work emerging from my thesis concerned the problem of link homotopy in dimensions 4 and higher. My work in knot theory mostly concerns the study of knot concordance (articles [21], [22], [23], [26], [43], [44]) and the use of Reidemeister torsion as a concordance obstruction. This was carrried out in collaboration with Chuck Livingston. In the article [20] we showed that twisted Alexander polynomials are related to Casson-Gordon discriminants. This made possible calculations of sliceness obstructions which were critical in showing the knot 8_17 is not concordant to its reverse (article [21]), showing that there are infintely generated famiies of non-concordant knots which are mutants of slice knots (article [26]), determining (article [43], with Herald) which of the 12 crossing knots are slice (with one exception, 12a631), and proving (article [44], with Hedden) that the subgroup of the concordance group generated by a certain infinite family of links of algebraic singularities is free abelian and intersects the subgroup of algebraically slice knots in an infinitely generated free abelian group.
Livingston and I also studied finite-type invariants of knots and links in the classical dimension (articles [17], [18], [22]). In these articles type 1 invariants of links in a 3-manifold are defined (these are desuspensions of the invariants I defined in my thesis) and their properties investigated. The role of incompressible tori plays a critical role in determining how well defined these invariants are. The article [23], written with Zhenghan Wang, uses Reidemeister torsion to produce invariants of string links which interpolate between the Gassner representation for braids and the Alexander polynomial (for 1-string links, i.e. knots).
Fundamental groups of 4-manifolds and their impact on the geography problem is another area I have studied. I have been investigating the relationships between the fundamental group, Euler characteristic, and signature of a four-dimensional manifold with Chuck Livingston in the smooth and topological setting (articles [32], [37]), and with Scott Baldridge in the symplectic setting (articles [35], [36], [38], [39], [40], [41]).
The articles with Livingston are concerned with determining the minimal Euler characteristic among all closed 4-manifolds with a given fundamental group. We solved this for free abelian groups in [32]. A more refined question is to find the minimal Euler characteristic among those closed manifolds of signature n for any integer n. This is a open ended problem with many interesting questions, and we outline the basic ideas and bounds as well as calculations for some free abelian groups in [37]. One interesting observation is that for any finitely presented group G there are a finite collection of local minima to the function f(n)=minimal Euler characteristic among closed 4-manifolds with signature n and fundamental group G. Manifolds realizing these local minima are irreducible and form a distinguished subclass of 4-manifolds with fundamental group G.
The articles with Baldridge address similar questions for symplectic 4-manifolds. In light of Gompf's result that any finitely presented group is the fundamental group of a symplectic 4-manifold, we considered the problem of finding symplectic 4-manifolds with prescribed fundamental group and small Euler characteristic. In [35] we showed how to construct symplectic 4-manifolds whose euler characteristic and signature are determined by the presentation of the fundamental group. In the case of trivial fundamental group we were led to constructions of small exotic 4-manifolds and the technique of modifying fundamental groups of 4-manifolds using Luttinger surgery in [38], [39], [40], and [41].
The articles [45] and [46], written with Matt Hedden, revisit techniques of SO(3) instanton moduli spaces pioneered by Fintushel/Stern and Furuta (following Donaldson's groundbreaking work) to prove that certain infinite families of topologically slice knots, namely the untwisted doubles of certain torus knots, are linearly independent in the smooth knot concordance group. The mechanism of estimating lower bounds on Chern-Simons invariants of flat SO(3) connections on 3-manifolds to obstruct linear dependence in rational cobordism groups does not appear to have an analogue in Seiberg-Witten or Heegard-Floer theory, motivating the considerations of these articles.
Other collaborators include Ulrich Koschorke, Danny Ruberman, Doug Park, Anar Ahkmedov, and Inanc Baykur. I have been fortunate to work with such inspiring colleagues.
Mark Daniel (1997)
Benjamin Himpel (2004)Jonathan Yazinski (2009)
Current PhD students:Juanita Pinzon
Ash Lightfoot
47. Coisotropic Luttinger surgery and some new symplectic 6-manifolds with vanishing canonical class (with Scott Baldridge). Click here for a preprint.
46. Instantons, concordance, and Whitehead doubling (with Matt Hedden). Click here for a preprint.
45. Chern-Simons invariants, SO(3) instantons, and Z/2 homology cobordism (with Matt Hedden). In Chern-Simons gauge theory: 20 years after, AMS/IP Stud. Adv. Math., 50, Amer. Math. Soc., Providence, RI, 2011, pp 83-114. Click here for a preprint.
44. Non-slice linear combinations of algebraic knots (with Matt Hedden and Charles Livingston). To appear in Journal of the European Math. Soc. Click here for a preprint.
43. Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation (with Chris Herald and Charles Livingston). Math. Zeitschrift. Vol. 265 (2010) 925-949. Click here for more.
42. The impact of QFT on low-dimensional topology. Lecture notes for the 2007 summer school on Geometric and Topological methods for Quantum Field Theory. Villa de Leyva, Columbia July 2-20. In Geometric and topological methods for quantum field theory, Cambridge University press. Click here for more.
41. Simply connected minimal symplectic 4-manifolds with signature less than -1 (with A. Akhmedov, S. Baldridge, R. I. Baykur, and B. D. Park). Journal of the European Math. Soc. Volume 12, Issue 1, 2010. 133-161. Click here for more.
40. Constructions of small symplectic manifolds using Luttinger surgery (with S. Baldridge). Journal of Differential Geometry. Vol. 82, No. 2 (2009), 317-362. Click here for more.
39. A symplectic manifold homeomorphic but not diffeomorphic to CP2 # 3(-CP2). (with S. Baldridge). Geometry and Topology Vol 12, issue 2(2008), 919--940. Click here for more.
38. An interesting symplectic 4-manifold with small Euler characteristic (with S. Baldridge). Click here for a preprint.
37. The geography problem for 4-manifolds with specified fundamental group (with C. Livingston). Trans. Amer. Math. Soc. 361 (2009), 4091-4124. Click here for more.
36. Symplectic 4-manifolds with arbitrary fundamental group near the Bogomolov-Miyaoka-Yau line (with S. Baldridge). Journal of Symplectic Geometry 4 (2006), no. 1, 63--70. Click here for more.
35. On symplectic 4-manifolds with prescribed fundamental group (with S. Baldridge). Commentarii Math. Helvetica. 82 (2007), no. 4, 845--875. Click here for more.
34. The Calderon projector for the odd signature operator and spectral flow calculations in 3-dimensional topology (with H. Boden and C. Herald). 'Spectral geometry of manifolds with boundary and decomposition of manifolds,' 125--150, Contemp. Math., 366, Amer. Math. Soc., Providence, RI, 2005. Click here for a PDF version
33. The integer valued SU(3) Casson invariant for Brieskorn spheres (with H. Boden and C. Herald). Journal of Differential Geometry. 71 (2005) 23--83. Click here for more.
32. The Hausmann-Weinberger 4-manifold invariant of abelian groups (with C. Livingston). Proc. Amer. Math. Soc. 133 (2005), no. 5, 1537--1546 Click here for more.
31. Calderon projector for the Hessian of the perturbed Chern-Simons function on a 3-manifold with boundary (with B. Himpel and M. Lesch). Proceedings of the London Mathematical Society. 89 (2004), no. 1, 241--272. Click here for more.
30. On the integer valued SU(3) Casson invariant (with H. Boden and C. Herald). 2001 Georgia International Topology Conference, AMS Proceedings of Symposia in Pure Mathematics 71 (2003) 209-236.
29. On the Rho invariant for manifolds with boundary (with M. Lesch). Algebraic and Geometric Topology, Volume 3 (2003) No. 22, pages 623-675. Click here for more.
28. The eta-invariant, Maslov index, and spectral flow for Dirac-type operators on a manifold with boundary (with M. Lesch). Forum Math 16 (2004) 553-629. Click here for more.
27. An integer valued SU(3) Casson invariant (with H. Boden and C. Herald). Mathematical Research Letters (2001), Vol 8, Iss 5-6, pages 589-603. Click here for more.
26. Concordance and Mutation (with C. Livingston). Geometry and Topology, Vol. 5 (2001) Paper no. 26, pages 831--883. Click here for more.
25. Gauge theoretic invariants of Dehn surgeries on knots (with H. Boden, C. Herald, and E. Klassen). Geometry and Topology, Vol. 5 (2001) Paper no. 6, pages 143--226. Click here for more.
24. A general splitting formula for the spectral flow (with M. Daniel, appendix by K.P. Wojciechowski). Michigan Math J. Vol 46 (1999). Click here for more.
23. The Gassner representation for string links (with C. Livingston and Z. Wang). Communications in Contemporary Mathematics, Vol. 3, No. 1 (2001) pp. 87-136. Click here for more.
22. Knot invariants in 3-manifolds and essential tori. (with C. Livingston). Pacific J. of Math Vol 197, No. 1, 2001. Click here for more.
21. Twisted knot polynomials: inversion, mutation and concordance (with C. Livingston). Topology Vol 38, No. 3 (1999), pp. 663-671. Abstract.
20. Twisted Alexander invariants, Reidemeister torsion, and the Casson-Gordon invariants (with C. Livingston). Topology Vol 38, No.3, (1999), pp 635-661. Abstract .
19. The spectral flow of the odd signature operator on a odd dimensional manifold with boundary (with E. Klassen). Topology Appl. 116 (2001), no. 2, 199--226. . Abstract
18. Type 1 Knot invariants in 3-manifolds (with C. Livingston). Pacific J. of Math Vol 183, No 2. (1998) 305--331. Abstract
17. Vassiliev invariants of 2-component links and the Casson-Walker invariant (with C. Livingston). Topology, Vol 36, No. 6 (1997) pp.1333-1353. Abstract
16. Continuity and analyticity of families of self-adjoint Dirac operators on a manifold with boundary (with E. Klassen). Illinois J. of Mathematics Volume 42 Number 1 (1998) 123-138. Abstract
15. Analytic deformations of the spectrum of a family of Dirac operators on an odd-dimensional manifold with boundary (with E. Klassen). Memoirs of the American Mathematical Society Vol 124, Number 592 (1996) . Abstract
14. The spectral flow of the odd signature operator and higher Massey products (with E. Klassen). Mathematical proceedings of the Cambridge Philosophical society (121) (1997), pg 297-320. Click here for more.
13. SU(2) representation varieties of 3-manifolds, gauge theory invariants, and surgery on knots. In Proceedings of the GARC workshop on Geometry and Topology '93 ed. H.J. Kim, Lecture note series number 18, Seoul National University (1993). pg. 137-176. Abstract. Click here to download the paper in PDF format.
12. A local analytic splitting of the holonomy map on flat connections (with B. Fine and E. Klassen). In Math. Annalen 299 (1994) pg 171-189. Abstract
11. Computing spectral flow via cup products (with E. Klassen). In Journal of Differential Geometry 40 (1994), pg. 505-562. Abstract
10. Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space of T2 (with E. Klassen). In Communications in Mathematical Physics 153 (1993), pg. 521-557. Abstract
9. Splitting the spectral flow and the Alexander matrix (with E. Klassen and D. Ruberman). In Commentarii Math. Helvetica. 69 (1994) pg 375-416. Abstract
8. Generalized Seifert surfaces and linking numbers (with U. Koschorke). In Topology and its Applications 42, (1991), pg. 247-262. Abstract
7. Casson's invariant and the Torelli Group. unpublished manuscript Abstract. Click here to dowload a PDF version of the article
6. Chern-Simons invariants of 3-manifolds and representation spaces of knot groups (with E. Klassen). In Math. Annalen 287 (1990), pg. 343-367. Abstract
5. Representation spaces of Seifert-fibered homology spheres (with E. Klassen). In Topology Vol 30, No. 1 (1991), pg.77-95. Abstract
4. Link homotopy with one codimension two component. In Transactions of the American Mathematical Society Vol. 319, no. 2 (1990), pg. 663-688. Abstract
3. Embedded links with one codimension two component are nullhomotopic in the metastable range. In Journal of the London Mathematical Society, (2) 40 (1989), pg. 378-384. Abstract
2. Mutations of homology spheres and Casson's invariant. In Mathematical proceedings of the Cambridge Philosophical society, No 105, (1989), pg. 313-318. Abstract
1. Link maps in the four-sphere. In Differential Topology, Proceedings, Siegen 1987, ed. U. Koschorke. Springer-Verlag Lecture Notes in Mathematics. Vol. 1350 (1988), pg. 31-43. Abstract
A MAPLE program which computes twisted Alexander polynomials to obstruct sliceness. Click here. The documentation is "sketchy." The program takes a Wirtinger presentation of a knot and returns its twisted Alexander polynomial associated to prime power covers, factored to check if it is slice. It doesn't work for knots whose homology (as modules over the PID Q[\zeta]) has more than two cyclic summands. This program was used to obtain the results of [43] above.
Very brief intro to Seiberg-Witten invariants. This is a set of notes for a one-hour lecture whose purpose was to prepare working seminar participants to read articles that use Seiberg-Witten invariants, without spending any time defining Seiberg-Witten invariants. Click here.
Some aspects of Mathematics and Music. These are lecture notes from a summer program for high school teachers I taught at LSU in the summer 2010. They are meant to gently introduce the notions of music theorist Dmitri Tymoczko relating geometry and music. Click here.
Signature defects, twisted signature defects, and rho invariants for finite $G$. This is a note I wrote for myself, following Atiyah-Patodi-Singer, to explain the equivalence of rho invariants and G-signature defects for representations that factor through a finite group. Click here.