Constanze undertook her doctoral studies at Bonn, and came to Kent in 2012 after fulfilling postdoctoral roles at Sheffield (2007-08) and Glasgow (2008-12).
Stable homotopy theory, in particular model categories and chromatic homotopy theory
Homological algebra and A-infinity algebras.
Please refer to Constanze's personal website
Patchkoria, I. and Roitzheim, C. (2020). Rigidity and exotic models for v1-local G-equivariant stable homotopy theory. Mathematische Zeitschrift [Online] 295:839-875. Available at: http://dx.doi.org/10.1007/s00209-019-02364-z.
We prove that the v1-local G-equivariant stable homotopy category for G a finite group has a unique G-equivariant model at p=2. This means that at the prime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p≥5 equivariant rigidity fails in general.
Muro, F. and Roitzheim, C. (2019). Homotopy Theory of Bicomplexes. Journal of Pure and Applied Algebra [Online] 223:1913-1939. Available at: https://doi.org/10.1016/j.jpaa.2018.08.007.
We define two model structures on the category of bicomplexes concentrated in the right half plane. The first model structure has weak equivalences detected by the totalisation functor. The second model structure's weak equivalences are detected by the E^2-term of the spectral sequence associated to the filtration of the total complex by the horizontal degree. We then extend this result to twisted complexes.
Ellis, E., Roitzheim, C., Scull, L. and Yarnall, C. (2019). Endomorphisms of Exotic Models. Glasgow Mathematical Journal [Online] 61:321-348. Available at: https://doi.org/10.1017/S001708951800023X.
We calculate the endomorphism dga of Franke’s exotic algebraic model for the K-local stable homotopy category at odd primes. We unravel its original abstract structure to give explicit generators, differentials and products.
Gutierrez, J. and Roitzheim, C. (2017). Bousfield localisations along Quillen bifunctors. Applied Categorical Structures [Online] 25:1113-1136. Available at: http://dx.doi.org/10.1007/s10485-017-9485-z.
Consider a Quillen adjunction of two variables between combinatorial model categories from C x D to E, and a set S of morphisms in C. We prove that there is a localised model structure L_S E on E, where the local objects are the S-local objects in E described via the right adjoint. These localised model structures generalise Bousfield localisations of simplicial model categories, Barnes and Roitzheim's familiar model structures, and Barwick's enriched Bousfield localisations. In particular, we can use these model structures to define Postnikov sections in more general left proper combinatorial model categories.
Roitzheim, C. and Gutierrez, J. (2016). Towers and fibered products of model categories. Mediterranean Journal of Mathematics [Online] 13:3863-3886. Available at: http://dx.doi.org/10.1007/s00009-016-0719-3.
Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products of model categories. As applications we consider Postnikov towers of model categories, chromatic towers of spectra and Bousfield arithmetic squares of spectra. For spectral model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization.
Roitzheim, C. (2015). A case of monoidal uniqueness of algebraic models. Forum Mathematicum [Online] 27:3615-3634. Available at: http://www.dx.doi.org/10.1515/forum-2013-0126.
We prove that there is at most one algebraic model for modules over the K(1)-local sphere at odd primes that retains some monoidal information.
Barnes, D. and Roitzheim, C. (2015). Homological Localisation of Model Categories. Applied Categorical Structures [Online] 23:487-505. Available at: http://link.springer.com/article/10.1007/s10485-013-9340-9.
One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate
for the E–localisation of this model category. We study the properties of this new construction and relate it to some well–known categories.
Barnes, D. and Roitzheim, C. (2014). Stable left and right Bousfield localisations. Glasgow Mathematical Journal [Online] 56:13-42. Available at: http://dx.doi.org/10.1017/S0017089512000882.
We study left and right Bousfield localisations of stable model categories which preserve stability. This follows the lead of the two key examples: localisations of spectra with respect to a homology theory and A-torsion modules over a ring R with A a perfect R-algebra.We exploit stability to see that the resulting model structures are technically far better behaved than the general case.We can give explicit sets of generating cofibrations, show that these localisations preserve properness and give a complete characterisation of when they preserve monoidal structures. We apply these results to obtain convenient assumptions under which a stable model category is spectral. We then use Morita theory to gain an insight into the nature of right localisation and its homotopy category. We finish with a correspondence between left and right localisation.
Gutierrez, J. and Roitzheim, C. (2014). Bousfield localisations along Quillen bifunctors and applications. arxiv.org [Online]. Available at: http://arxiv.org/abs/1411.0500.
We describe left and right Bousfield localisations along Quillen adjunctions of two variables. These localised model structures can be used to define Postnikov sections and homological localisations of arbitrary model categories, and to study the homotopy limit model structure on the category of sections of a left Quillen presheaf of localised model structures. We obtain explicit results in this direction in concrete examples of towers and fiber products of model categories. In particular, we prove that the category of simplicial sets is Quillen equivalent to the homotopy limit model structure of its Postnikov tower, and that the category of symmetric spectra is Quillen equivalent to the homotopy fiber product of its Bousfield arithmetic square. For spectral model categories, we show that the homotopy fiber of a stable left Bousfield localisation is a stable right Bousfield localisation.
Livernet, M., Roitzheim, C. and Whitehouse, S. (2013). Derived A-infinty algebras in an operadic context. Algebraic & Geometric Topology [Online] 13:409-440. Available at: http://dx.doi.org/10.2140/agt.2013.13.409.
Derived A-infinity algebras were developed recently by Sagave. Their advantage over classical A-infinity algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived A-infinity algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad dAs encoding bidgas, ie bicomplexes with an associative multiplication. This generalises the established result describing the operad A-infinity as a resolution of the operad As encoding associative algebras. We further show that Sagave’s definition of morphisms agrees with the infinity- morphisms of dA-infinity –algebras arising from operadic machinery. We also study the operadic homology of derived A-infinity algebras.
Roitzheim, C. and Whitehouse, S. (2011). Uniqueness of A-infinity structures and Hochschild cohomology. Algebraic & Geometric Topology [Online] 11:107-143. Available at: http://dx.doi.org/10.2140/agt.2011.11.107.
Working over a commutative ground ring, we establish a Hochschild cohomology criterion for uniqueness of derived A-infinity algebra structures in the sense of Sagave. We deduce a Hochschild cohomology criterion for intrinsic formality of a differential graded algebra. This generalizes a classical result of Kadeishvili for the case of a graded algebra over a field.
Barnes, D. and Roitzheim, C. (2011). Local Framings. New York Journal of Mathematics [Online] 17:513-552. Available at: http://nyjm.albany.edu/j/2011/17-22.html.
Framings provide a way to construct Quillen functors from simplicial sets to any given model category. A more structured set- up studies stable frames giving Quillen functors from spectra to stable model categories. We will investigate how this is compatible with Bousfield localisation to gain insight into the deeper structure of the stable homotopy category. We further show how these techniques relate to rigidity questions and how they can be used to study algebraic model categories.
Barnes, D. and Roitzheim, C. (2011). Monoidality of Franke’s Exotic Model. Advances in Mathematics [Online] 228:3223-3248. Available at: http://dx.doi.org/10.1016/j.aim.2011.08.005.
In 1996, Jens Franke proved the equivalence of certain triangulated categories possessing an Adams spectral sequence. One particular application of this theorem is that the K_(p)-local stable
homotopy category at an odd prime can be described as the derived category of an abelian category. We explain this proof from a topologist’s point of view.
Roitzheim, C. (2007). Rigidity and Exotic Models for the K-local Stable Homotopy Category. Geometry & Topology [Online] 11:1855-1886. Available at: http://dx.doi.org/10.2140/gt.2007.11.1855.
Can the model structure of a stable model category be recovered from the triangulated structure of its homotopy category? This paper introduces a new positive example for this, namely the K-local
stable homotopy at the prime 2. For odd primes, however, this is not true: we discuss a counterexample given by Jens Franke and show how such exotic models for the K-local stable homotopy category at odd primes can be detected.
Barnes, D. and Roitzheim, C. (2020). Foundations of Stable Homotopy Theory. [Online]. Cambridge University Press. Available at: https://www.cambridge.org/core/books/foundations-of-stable-homotopy-theory/791C9C413A83AD7094E055E5E818D33B.
The beginning graduate student in homotopy theory is confronted with a vast literature on spectra that is scattered across books, articles and decades. There is much folklore but very few easy entry points. This comprehensive introduction to stable homotopy theory changes that. It presents the foundations of the subject together in one place for the first time, from the motivating phenomena to the modern theory, at a level suitable for those with only a first course in algebraic topology. Starting from stable homotopy groups and (co)homology theories, the authors study the most important categories of spectra and the stable homotopy category, before moving on to computational aspects and more advanced topics such as monoidal structures, localisations and chromatic homotopy theory. The appendix containing essential facts on model categories, the numerous examples and the suggestions for further reading make this a friendly introduction to an often daunting subject.
Barnes, D. and Roitzheim, C. (2014). Rational Equivariant Rigidity. In: Ausoni, C., Hess, K., Johnson, B., Luck, W. and Scherer, J. eds. An Alpine Expedition through Algebraic Topology: Fourth Arolla Conference Algebraic Topology August 20–25, 2012 Arolla, Switzerland. Providence, RI; American Mathematical Society, pp. 13-30. Available at: http://dx.doi.org/10.1090/conm/617.
We prove that if G is S^1 or a profinite group, then all of the homotopical information of the category of rational G-spectra is captured by the triangulated structure of the rational G-equivariant stable homotopy category. That is, for G profinite or S1, the rational G-equivariant stable homotopy category is rigid. For the case of profinite groups this rigidity comes from an intrinsic formality statement, so we carefully relate the notion of intrinsic formality of a differential graded algebra to rigidity.
Roitzheim, C. (2005). Arbeitsgemeinschaft mit aktuellem Thema: Modern Foundations for Stable Homotopy Theory: Mathematisches Forschungsinstitut Oberwolfach Report No. 46/2005, organised/edited by John Rognes (Oslo) and Stefan Schwede (Bonn). In: Schwede, S. ed. Oberwolfach Reports. European Mathematical Society Publishing House, pp. 2603-2604. Available at: http://www.ems-ph.org/journals/show_abstract.php?issn=1660-8933&vol=2&iss=4&rank=4.
In recent years, spectral algebra or stable homotopical algebra over structured ring spectra has become an important new direction in stable homotopy theory. This workshop provided an introduction to structured ring spectra and applications of spectral algebra, both within homotopy theory
and in other areas of mathematics.