1. Photo credit: Cristina Taho Izuno

  3. Jeffrey D. Carlson

    (he/his)

    34 Evergreen St.
    Providence, RI 02906
    US

     

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Hi! I'm a topologist. I am currently on the job market following a visiting scholarship at Tufts University.

My work has started off in dynamics on surfaces, is mostly connected with the equivariant algebraic topology of manifolds, often in collaboration with geometers, and has expanded to have some overlap with sympectic geometry and A-infinity algebras, with some connections with Galois cohomology on the horizon.

In previous lives I've worked

 

Papers

Not everything is up on the arXiv. arXived versions may also not agree with published versions, but are invariably improvements.

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  • A ring structure on Tor

    Cartan computed the cohomology ring of a homogeneous space over the reals in 1949, setting off a search for cohomology groups over a more general field which more or less concluded in 1974. All these approaches looked at the Eilenberg–Moore spectral sequence of a fiber bundle where the base and total space had polynomial cohomology; Hans Munkholm's approach, however, considered the EMSS of a pullback square with polynomial cohomology more generally, and thus applies to Borel cohomology of a homogeneous space, cohomology of a biquotient, or cohomology of the free loop space of a space with polynomial cohomology over any principal ideal domain. The answer is that it is Tor of the cohomology rings of the span of input spaces. All these results were additive, but we show Munkholm's isomorphism is multiplicative, resolving a seventy-year old question. Under revision for Forum of Mathematics, Sigma.

  • Products on Tor

    We compare the product on Tor we construct to show Munkholm's isomorphism is additive with a product suggested by Munkholm and a product on the two-sided bar construction defined by Franz figuring in the biquotient paper.

  • The cohomology of homogeneous spaces in historical context

    This is a historical survey of Eilenberg–Moore collapse results establishing who did what when, which seemed appropriate to compile given the seemingly final closure of this family of questions. To be published in the Contemp. Math. volume Group Actions and Equivariant Cohomology.

    N.B.: The arXiv version is already better than the version to be published, and this gap is only expected to increase with subsequent updates.

  • The topology of Gelfand–Zeitlin fibers, with Jeremy Lane.

    Gelfand–Zeitlin systems are examples of classical interest in symplectic geometry satisfying many of the conclusions of standard theorems about toric integrable systems without satisfying their hypotheses, and so giving an interesting class of edge cases. In this work we find an closed-form description of the diffeomorphism type of any fiber of a GZ system and compute the cohomology ring and first three homotopy groups of a given fiber. Each of these expressions turns out to admit a transparent description in terms of the combinatorics of a certain graph associated to the fiber.

  • The cohomology of biquotients via a product on the two-sided bar construction, with an appendix joint with Matthias Franz.

    We compute the Borel equivariant cohomology ring of the left K-action on a homogeneous space G/H, where G and its subgroups H and K are compact, connected Lie groups and 2 and the torsion primes of the groups are units of the coefficient ring. Particularly, this computes the cohomology rings of biquotients H\G/K. (This had been done for H = 1 by Franz.) The extension depends on a version of the Eilenberg–Moore theorem developed in the appendix, where we show mod-2 formulas proposed by Franz give a sort of weak product structure on a two-sided bar construction B(A', A, A'') whenever A' ← A → A'' is a pair of maps of homotopy Gerstenhaber algebras. To be published in Algebraic & Geometric Topology.

    The proof of the main result relies among other things on a lemma proven in Joel Wolf's 1973 Brown dissertation but never published, which I wound up reproving for lack of access. (The seemingly one existing copy of this thesis is not available for loan.) On a recent trip to Providence, I reserved time in the rare books room at John Hay Library to digitize it, and it is now available to the public here.

  • Fixed points and bordism of semifree actions.

    We show the ABBV localization identities completely determine the possible discrete fixed-point sets of a semifree circle action on a stably complex manifold. This approach turns out to recover the cobordism ring of such actions (it's Z[S2]) through extremely naive (19th-century–style) manipulations of combinatorial identities in about a page. Following some correspondence we have included generous background on the problem.

  • Realization of abstract GKM isotropy data, with Elisheva Adina Gamse and Yael Karshon.

    The ABBV localization identities for a torus action put strong constraints on isotropy data and we show the converse, that such data must arise from a torus action on stably complex manifold. This comes with a concrete, visually appealing construction in dimensions up to four, and the proof otherwise relies on considerations connected with equivariant complex cobordism.

    N.B.: Still in progress, much changed, and approaching submittability; ask if you want a more recent version.

  • Equivariant formality of corank-one isotropy actions and products of rational spheres, with Chen He.

    We characterize the pairs of connected Lie groups G > K such that the difference of their ranks is one and the left action of K on G/K is equivariantly formal. The analysis requires us to correct and extend an existing partial classification of homogeneous quotients G/K with the rational homotopy type of a product of an odd- and an even-dimensional sphere. Published in Mathematische Zeitschrift.

  • My dissertation (now a book).

    This work was nominally about Borel equivariant cohomology, but in the writing became predominantly an account of the cohomology of homogeneous spaces. My opinion (perhaps self-serving) is that it is the fastest introduction to this theory. The exposition has some things that don't appear elsewhere.

    N.B.: This is not the final version. This book is to be the subject of a major revision at the behest of Springer's diligent referees.

  • K-theory and formality.

    The main result of this paper establishes as corollaries the equivariant K-theory rings in basically all the known equivariantly formal examples. We also show that under the same conditions, surjectivity of the forgetful map in K-theory is equivalent to weak equivariant formality in the sense of Harada–Landweber. The main tool is a map of spectral sequences from Hodgkin's Künneth spectral sequence to the analogous spectral sequence in Borel cohomology. Published in International Mathematics Research Notices.

  • Grassmannians and the equivariant cohomology of isotropy actions.

    Chen He had recently computed this through his development of GKM-theory and I noticed older methods gave a clean proof as well. This also gave me the chance to publish a proof of a structure result from a late, unpubished version of my thesis which was in danger of becoming folklore. Published in the Proceedings of the American Mathematical Society.

  • The K-theory of the conjugation action.

    This is a 1999 result of Brylinski and Zhang; my observation was simply that with some planning, much of the proof could be sidestepped, resulting in a proof of less than a page. Published in Comptes Rendus.

  • The K-theory of a cohomogeneity-one action.

    An outgrowth of the joint paper below. The appendices include lemmas providing a general framework for such computations in an arbitrary equivariant cohomology theory vanishing in certain degrees and a curious additional structure on the Mayer–Vietoris sequence no one seems to have discussed.

    N.B.: This work is under revision following substantial referee feedback at Advances in Mathematics and will only be updated on the arXiv when finalized.

  • The equivariant cohomology ring of a cohomogeneity-one action, with Oliver Goertsches, Chen He, and Liviu Mare.

    This one computes the object in the title, extending results of the first and third author. Cohomogeneity-one actions are an important and very well-studied class of examples in differential geometry whose topology is determined entirely by an inclusion diamond of four Lie groups, so this was a natural question to ask. Published in Geometriae Dedicata.

  • Equivariant formality of isotropy actions, with Chi-Kwong Fok.

    Also deals with equivariant formality in K-theory, and proves the underlying homogeneous space is formal and a regularity criterion for equivariant formality of an isotropy action. Published in the Journal of the London Mathematical Society.

    N.B.: Example 3.6 is incorrect, as Manuel Amann pointed out. This does not affect the other results of the paper, but requires a corrigendum and an arXiv update, both waiting on a reply from my coauthor.

  • Equivariant formality of isotropic torus actions.

    A development of my thesis research. Reduces (nearly) equivariant formality of an isotropy action to the case of a torus in a semisimple group, and provides a complete classification for circles. Appendices handle (partially) the case of a compact Hausdorff group and a disconnected group. Published in the Journal of Homotopy and Related Structures.

  • Commensurability of two-multitwist pseudo-Anosovs.

    This provides the first nontrivial examples of subgroups of mapping class groups of different surfaces simultaneously covering, element-wise, a subgroup of the mapping class group of a covered surface, and a lot of examples of invariants distinguishing noncommensurable elements.

  • Conceptions of topological transitivity, with Ethan Akin (arXived).

    Boris Hasselblatt once assigned a dynamics class I was sitting in on the problem of fixing a broken proof of a "folklore" result on topological transitivity from his and Katok's Introduction to the Modern Theory of Dynamical Systems, so I did. Boris suggested I verify with "a hardcore topological dynamicist" there were no published proofs of the desired result, and proposed Akin. Published in Topology and its Applications.

 

Miscellaneous notes

  • At one point in grad school, I decided that if I did all the exercises in Atiyah & Macdonald, I would become deeply knowledgeable about commutative algebra. Though that assessment turns out to have been optimistic, I am now the proud owner of a complete solution set (only slightly longer than the book itself!).
  • In the literature on cohomology of fibrations, there has long been in use a term coexact, which J. C. Moore insisted upon, by all available accounts, owing to categorical considerations that show the situation in question is dual to the familiar situation of an exact sequence. This notion doesn't seem to get a whole lot of press, so I wrote a brief note expounding on the notion and the difference from exactness, as well as conditions under which the two agree. It was originally a letter to Larry Smith, in case a "you" still appears anywhere in the revised note.
  • In Exercise 4.10.1 of their venerable Differential Forms in Algebraic Topology, Bott & Tu ask the reader to show that the image of a proper map is closed. While a pedant might object that the claim is not actually true, it does hold of most reasonably decent spaces. In this note I show that it suffices the codomain be Hausdorff and either first countable or locally compact, but without some sort of countability criterion, there still exist counterexamples even when the codomain is T5.

 

Teaching

    I taught commutative algebra at Imperial in Autumn 2021 and was compelled to transcribe complete lecture notes, reproduced here for your approval.

    I also once overwrote a number of word problems which were received with some mixture of enthusiasm and confusion in a vector calculus course at the University of Toronto at Mississauga.

 

CV and documentation