Periodic Reporting for period 4 - HToMS (Homotopy Theory of Moduli Spaces)
Période du rapport: 2023-04-01 au 2024-09-30
Moduli spaces are spaces which describe all mathematical objects of some type. Counterintiutively, moduli spaces of a given kind of mathematical objects tend---from a certain point of view---to become more comprehensible as the complexity of the given objects increases, and in the "infinite complexity" limit are often comprehensible. When this happens, it is important to understand the speed of convergence to this limit: the faster the convergence the better! The phenomenon of convergence is known as "homological stability".
An important example of this paradigm is when the mathematical objects involved are surfaces. In this case a good measure of complexity is the number of holes a surface has, known as its "genus". The cohomology of moduli spaces of surfaces was determined in the infinite genus limit in breakthrough work of Madsen--Weiss (2007), but the homological stability phenomenon had been understood much earlier starting with work of Harer (1985). There has been steady progress in understanding the speed of convergence. However, very little is known about the behaviour of these moduli spaces outside of the stable range. A major component of the project is to develop a new tool, known as cellular E_k-algebras, to analyse this moduli space of surfaces, as well as many others, outside of this stable range.
The second part of the project concerns a different kind of moduli spaces, spaces of positively curved metrics on a fixed manifold (here curvature means scalar curvature). A technically different but morally similar phenomenon to homological stability occurs, namely that the space of positively curved metrics on a manifold M is unchanged under quite drastic modifications (surgery) of M. The goal of this part of the project is to establish two major properties of such spaces.
The third part of the project has pivoted. It still concerns moduli spaces of smooth manifolds, but rather than investigating their tautological rings it is devoted to exploiting a powerful new tool: the "Weiss fibre sequence". This is a new technique for trying to reverse the operation of homological stability, by directly studying the difference between the finite and infinite complexity cases. When the technique can be made to work it can give us access to information about moduli spaces of manifolds that would have been unthinkable just a few years ago, but there are grave technical difficulties in getting it to work.
An important example of this paradigm is when the mathematical objects involved are surfaces. In this case a good measure of complexity is the number of holes a surface has, known as its "genus". The cohomology of moduli spaces of surfaces was determined in the infinite genus limit in breakthrough work of Madsen--Weiss (2007), but the homological stability phenomenon had been understood much earlier starting with work of Harer (1985). There has been steady progress in understanding the speed of convergence. However, very little is known about the behaviour of these moduli spaces outside of the stable range. A major component of the project is to develop a new tool, known as cellular E_k-algebras, to analyse this moduli space of surfaces, as well as many others, outside of this stable range.
The second part of the project concerns a different kind of moduli spaces, spaces of positively curved metrics on a fixed manifold (here curvature means scalar curvature). A technically different but morally similar phenomenon to homological stability occurs, namely that the space of positively curved metrics on a manifold M is unchanged under quite drastic modifications (surgery) of M. The goal of this part of the project is to establish two major properties of such spaces.
The third part of the project has pivoted. It still concerns moduli spaces of smooth manifolds, but rather than investigating their tautological rings it is devoted to exploiting a powerful new tool: the "Weiss fibre sequence". This is a new technique for trying to reverse the operation of homological stability, by directly studying the difference between the finite and infinite complexity cases. When the technique can be made to work it can give us access to information about moduli spaces of manifolds that would have been unthinkable just a few years ago, but there are grave technical difficulties in getting it to work.
The first part of this project concerned the development of a new perspective on the homological stability phenomenon. It takes as its basis the idea that collections of spaces which satisfy homological stability can often be assembled into a single object, an E_k-algebra, though homological stability uses only a small part of this structure. It proposes to fully exploit the rest of this structure to, e.g. (i) sharpen estimates on the speed of homological stability, or (ii) establish new kinds of homological stability results. The long foundational paper for this part of the project, "Cellular E_k-algebras", provides a broad collection of tools for working with E_k-algebras, adapted to homological stability applications.
The flagship application of this method concerns mapping class groups of surfaces, or equivalently the moduli spaces M_{g,1} of Riemann surfaces, as studied by Harer and Madsen--Weiss. In "E_2-cells and mapping class groups" we use the cellular E_k-algebra methods to discover a new form of stability in this setting. We call this "secondary homological stability", and it can be encapsulated in the slogan "the failure of homological stability is stable in its own right". Since its discovery as part of this project, the study of this new phenomenon has been taken up by several mathematicians.
The main further applications of cellular E_k-algebra methods as part of this project are its application to general linear groups of finite and infinite fields, where several quite long-standing problems were resolved.
The second part of this project concerned spaces of positive scalar curvature metrics. Two kinds of results have been obtained: computational ones and conceptual ones. On the computational side, in "Infinite loop spaces and positive scalar curvature in the presence of a fundamental group" we have been able to show that the topology of such spaces can be very rich. On the conceptual side, in "The positive scalar curvature cobordism category" we showed that despite their complexity such spaces often have extremely special properties: they are what is known as infinite loop spaces. From a geometric perspective this is highly surprising, and indeed the proof of this result uses sophisticated homotopy-theoretic ideas far from the geometric nature of spaces of positive scalar curvature metrics.
The third part of the project was originally about tautological rings of manifolds, but early on was pivoted to an emerging and exciting new research direction: exploiting the "Weiss fibre sequence". This is a basic new idea in the study of diffeomorphisms of manifolds which allows one to try to reverse the effect of stabilising manifolds. Combined with earlier results of Galatius and the PI, it in principle allows one to say something about diffeomorphisms of very simple manifolds. The PI and two RAs have worked extensively in this direction.
In "On diffeomorphisms of even-dimensional discs" this method is used to study diffeomorphisms of even-dimensional discs, so via smoothing theory to study homeomorphism of even-dimensional Euclidean space. As well as improving by several factors the range in which complete information is available about the rational homotopy groups of these spaces, we described a fundamental new pattern ("bands") underlying them.
In "A homological approach to pseudoisotopy theory. I" this method is used to rederive the relation between spaces of pseudoisotopies and algebraic K-theory, in the case of an even-dimensional disc, independently of the work of Waldhausen and Igusa.
In "Diffeomorphisms of discs and the second Weiss derivative of BTop(-)" this method is used to understand diffeomorphism groups of both even- and odd-dimensional discs, and so homeomorphism groups of even- and odd-dimensional Euclidean spaces.
In "On automorphisms of high-dimensional solid tori" this method is used on the non-simply-connected manifold S^1 x D^{2n-1}, to study the infinite-generation of the homotopy groups of its diffeomorphism groups in certain degrees.
The flagship application of this method concerns mapping class groups of surfaces, or equivalently the moduli spaces M_{g,1} of Riemann surfaces, as studied by Harer and Madsen--Weiss. In "E_2-cells and mapping class groups" we use the cellular E_k-algebra methods to discover a new form of stability in this setting. We call this "secondary homological stability", and it can be encapsulated in the slogan "the failure of homological stability is stable in its own right". Since its discovery as part of this project, the study of this new phenomenon has been taken up by several mathematicians.
The main further applications of cellular E_k-algebra methods as part of this project are its application to general linear groups of finite and infinite fields, where several quite long-standing problems were resolved.
The second part of this project concerned spaces of positive scalar curvature metrics. Two kinds of results have been obtained: computational ones and conceptual ones. On the computational side, in "Infinite loop spaces and positive scalar curvature in the presence of a fundamental group" we have been able to show that the topology of such spaces can be very rich. On the conceptual side, in "The positive scalar curvature cobordism category" we showed that despite their complexity such spaces often have extremely special properties: they are what is known as infinite loop spaces. From a geometric perspective this is highly surprising, and indeed the proof of this result uses sophisticated homotopy-theoretic ideas far from the geometric nature of spaces of positive scalar curvature metrics.
The third part of the project was originally about tautological rings of manifolds, but early on was pivoted to an emerging and exciting new research direction: exploiting the "Weiss fibre sequence". This is a basic new idea in the study of diffeomorphisms of manifolds which allows one to try to reverse the effect of stabilising manifolds. Combined with earlier results of Galatius and the PI, it in principle allows one to say something about diffeomorphisms of very simple manifolds. The PI and two RAs have worked extensively in this direction.
In "On diffeomorphisms of even-dimensional discs" this method is used to study diffeomorphisms of even-dimensional discs, so via smoothing theory to study homeomorphism of even-dimensional Euclidean space. As well as improving by several factors the range in which complete information is available about the rational homotopy groups of these spaces, we described a fundamental new pattern ("bands") underlying them.
In "A homological approach to pseudoisotopy theory. I" this method is used to rederive the relation between spaces of pseudoisotopies and algebraic K-theory, in the case of an even-dimensional disc, independently of the work of Waldhausen and Igusa.
In "Diffeomorphisms of discs and the second Weiss derivative of BTop(-)" this method is used to understand diffeomorphism groups of both even- and odd-dimensional discs, and so homeomorphism groups of even- and odd-dimensional Euclidean spaces.
In "On automorphisms of high-dimensional solid tori" this method is used on the non-simply-connected manifold S^1 x D^{2n-1}, to study the infinite-generation of the homotopy groups of its diffeomorphism groups in certain degrees.
The introduction of "cellular E_k-algebras" into the subject of homological stability has been a paradigm shift, and now forms the language and basic toolkit of much of the research in that subject. As with many changes of perspective, it has rendered some problems almost trivial, as the paper "E_\infty-cells and general linear groups of infinite fields" makes clear by reproving several classical results as well as unexpectedly solving problems of Rognes, Suslin, Mirzaii, and others.
The analysis of spaces of diffeomorphisms and pseudoisotopies of discs in this project are the flagship quantitative applications of the Weiss fibre sequence method. They have led to the first new quantitative results about such spaces for 40 years, and more importantly have led to a conjectural complete picture of the rational homotopy type of such spaces. This conjecture is now one of the main organising principles in the subject.
The analysis of spaces of diffeomorphisms and pseudoisotopies of discs in this project are the flagship quantitative applications of the Weiss fibre sequence method. They have led to the first new quantitative results about such spaces for 40 years, and more importantly have led to a conjectural complete picture of the rational homotopy type of such spaces. This conjecture is now one of the main organising principles in the subject.