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Jeffrey D. Carlson

Publications and source records attributed to Jeffrey D. Carlson.

15 recordsLinked to original sources
arXiv2026

A ring structure on Tor

We prove that within a natural class of E_3-algebras, the graded Tor group induced by a span of E_3-algebra maps carries a graded algebra structure generalizing the classical structure when the algebras are genuine commutative differential graded algebras. We attempt to prove, as a topological corollary, that Munkholm's Eilenberg--Moore collapse result for pullbacks of spaces with polynomial cohomology can be enhanced to a ring isomorphism. This is not achieved, and in fact the claim as stated in the previous drafts is false. If additionally, 2 is assumed to be a unit of the base ring, then that claim is true (not that the results in this paper establish it) and is known due to previous work of the author and Franz, and also, as it turns out, to Huebschmann's unpublished 1983 habilitation work.

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arXiv2024

Equivariant formality of istropy actions

Let $G$ be a compact connected Lie group and $K$ a connected Lie subgroup. In this paper, we collect an assortment of results on equivariant formality of the isotropy action of $K$ on $G/K$. If the isotropy action of $K$ on $G/K$ is equivariantly formal, then $G/K$ is formal in the sense of rational homotopy theory. This enables us to strengthen a theorem of Shiga--Takahashi to a characterization of equivariant formality in this case. Using a K-theoretic analogue of equivariant formality introduced and shown by the second-named author to be equivalent to equivariant formality in the usual sense, we provide a representation-theoretic characterization for equivariant formality of the isotropy action and give a new, uniform proof of equivariant formality for some classes of homogeneous spaces for which it was previously known.

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arXiv2023

The cohomology of homogeneous spaces in historical context

The real singular cohomology ring of a homogeneous space $G/K$, interpreted as the real Borel equivariant cohomology $H^*_K(G)$, was historically the first computation of equivariant cohomology of any nontrivial connected group action. After early approaches using the Cartan model for equivariant cohomology with real coefficients and the Serre spectral sequence, post-1962 work computing the groups and rings $H^*(G/K)$ and $H^*_H(G/K)$ with more general coefficient rings motivated the development of minimal models in rational homotopy theory, the Eilenberg-Moore spectral sequence, and A-infinity algebras. In this essay, we survey the history of these ideas and the associated results.

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arXiv2023

Fixed points and semifree bordism

We apply fixed-point techniques to compute the coefficient ring of semifree geometric circle-equivariant complex cobordism with isolated fixed points, recovering a 2004 result of Sinha through 19th-century methods.

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arXiv2023

Products on Tor

In 1974 work establishing the collapse of certain Eilenberg-Moore spectral sequences, Munkholm constructs, in passing, a bilinear multiplication operation on Tor of a triple of $A_\infty$-algebras. In 2020, the present author, pursuing a multiplicative collapse result extending Munkholm's, studied a variant of this product, without actually showing it agrees with Munkholm's. In 2019, Franz had defined a weak product on the two-sided bar construction of a triple of $A_\infty$-algebras under similar hypotheses, with which this author proved a related collapse result, but without investigating the properties of the induced product on Tor. The present work demonstrates that the two products on Tor agree and are induced by the product of Franz.

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arXiv2023

Equivariant formality of isotropic torus actions

Considering the potential equivariant formality of the left action of a connected Lie group $K$ on the homogeneous space $G/K$, we arrive through a sequence of reductions at the case $G$ is compact and simply-connected and $K$ is a torus. We then classify all pairs $(G,S)$ such that $G$ is compact connected Lie and the embedded circular subgroup $S$ acts equivariantly formally on $G/S$. In the process we provide a proof of the structure (known to Leray and Koszul) of the cohomology rings $H^*(G/S;\mathbb Q)$.

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arXiv2023

Grassmannians and the equivariant cohomology of isotropy actions

Recent work of Chen He has determined through GKM methods the Borel equivariant cohomology with rational coefficients of the isotropy action on a real Grassmannian and an real oriented Grassmannian through GKM methods. In this expository note, we propound a less involved approach, due essentially to Vitali Kapovitch, to computing equivariant cohomology rings $H^*_K(G/H)$ for $G,K,H$ connected Lie groups, and apply it to recover the equivariant cohomology of the Grassmannians. The bulk is setup and commentary; once one believes in the model, the proof itself is under a page.

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arXiv2023

K-theory and formality

We compute the equivariant K-theory with integer coefficients of an equivariantly formal isotropy action, subject to natural hypotheses which cover the three major classes of known examples. The proof proceeds by constructing a map of spectral sequences from Hodgkin's Künneth spectral sequence in equivariant K-theory to that in Borel cohomology. A new characterization of equivariant formality appears as a consequence of this construction, and we are now able to show that weak equivariant formality in the sense of Harada--Landweber is equivalent with integer coefficients to surjectivity of the forgetful map under a standard hypothesis. The main structure theorem is formally similar to that for Borel equivariant cohomology, which appears in the author's dissertation/dormant book project and whose proof is finally made accessible in an appendix. The most generally applicable corollary of the main theorem for rational coefficients depends on a strengthening of the characterization of equivariant formality due to Shiga and Takahashi, which appears as a second appendix.

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arXiv2023

The cohomology of biquotients via a product on the two-sided bar construction

We compute the Borel equivariant cohomology ring of the left $K$-action on a homogeneous space $G/H$, where $G$ is a connected Lie group, $H$ and $K$ are closed, connected subgroups and $2$ and the torsion primes of the Lie groups are units of the coefficient ring. As a special case, this gives the singular cohomology rings of biquotients $H \backslash G / K$. This depends on a version of the Eilenberg-Moore theorem developed in the appendix, where a novel multiplicative structure on the two-sided bar construction $\mathbf{B}(A',A,A")$ is defined, valid when $A' \leftarrow A \to A"$ is a pair of maps of homotopy Gerstenhaber algebras.

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arXiv2023

The K-theory of the conjugation action

In 1999, Brylinski and Zhang computed the complex equivariant K-theory of the conjugation self-action of a compact, connected Lie group with torsion-free fundamental group. In this note we show it is possible to do so in under a page.

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arXiv2022

The topology of Gelfand-Zeitlin fibers

We prove several new results about the topology of fibers of Gelfand--Zeitlin systems on unitary and orthogonal coadjoint orbits, at the same time finding a unifying framework recovering and shedding light on essentially all known results. We find completely explicit descriptions of the diffeomorphism type of the fiber in many instances a direct factor decomposition of the fiber, and a torus factor corresponding to the action given by the Thimm trick. The new description also gives us a weak local normal form for a coadjoint orbit, which we use to define a topological toric degeneration, new in the orthogonal case. We also compute the first three homotopy groups (new in the orthogonal case) and cohomology rings of a fiber (new in both cases). All these descriptions can be read in a straightforward manner from the combinatorics of the associated Gelfand--Zeitlin pattern.

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arXiv2022

The equivariant K-theory of a cohomogeneity-one action

We compute the equivariant complex K-theory ring of a cohomogeneity-one action of a compact Lie group at the level of generators and relations and derive a characterization of K-theoretic equivariant formality for these actions. Less explicit expressions survive for a range of equivariant cohomology theories including Bredon cohomology and Borel complex cobordism. The proof accordingly involves elements of equivariant homotopy theory, representation theory, and Lie theory. Aside from analysis of maps of representation rings and heavy use of the structure theory of compact Lie groups, a more curious feature is the essential need for a basic structural fact about the Mayer--Vietoris sequence for any multiplicative cohomology theory which seems to be otherwise unremarked in the literature, and a similarly unrecognized basic lemma governing the equivariant cohomology of the orbit space of a finite group action.

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arXiv2012

Conceptions of Topological Transitivity

There are several different common definitions of a property in topological dynamics called "topological transitivity," and it is part of the folklore of dynamical systems that under reasonable hypotheses, they are equivalent. Various equivalences are proved in different places, but the full story is difficult to find. This note provides a complete description of the relationships among the different properties.

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arXiv2010

Commensurability of two-multitwist pseudo-Anosovs

This paper analyzes commensurability of the class of surface automorphism generated by two Dehn multitwists. We show pairwise noncommensurability between several classes arising from canonical curve configurations. In addition, we consider the Kenyon-Smillie invariant J of flat surfaces in this setting. We also introduce a general construction of infinite classes of commensurable pseudo-Anosov homeomorphisms.

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