Topology Seminar
Past Talks
Winter 2020

Niles Johnson (Ohio State)
Algebraic Ktheory for 2categories
Quillen recognized the higher algebraic Kgroups of a commutative ring R as homotopy groups of a certain topological space. This has both conceptual and computational advantages, which we review. We then describe how a 2categorical extension of this theory leads to a new model for K_3(R), together with more general applications. We will give a mild sampling of key technical details and close with some of the problems we're currently working on. The work we present is joint with GurskiOsorno, Fontes, and FontesGurski.

Michael Horst (Ohio State)
A Categorification of Group Cohomology
We present a categorification of group cohomology by way of Picard categories, which are to be thought of as categories that behave like Abelian groups. After briefly reviewing background information on Picard categories and free Picard categories, we will present a framework for a cohomology theory taking values not in Abelian groups but in Picard categories. We will motive the study of this theory by way of computing specific examples. The remainder of our time will involve discussing the theoretical underpinnings of this theory, including the particularities of the 2dimensional algebra required for it.

Jocelyne Ishak (Vanderbilt)
Rigidity of the K(1)local stable homotopy category
In some cases, it is sufficient to work in the homotopy category Ho(C) associated to a model category C, but looking at the homotopy level alone does not provide us with higher order structure information. Therefore, we investigate the question of rigidity: If we just had the structure of the homotopy category, how much of the underlying model structure can we recover?This question has been investigated during the last decade, and some examples have been studied, but there are still a lot of open questions regarding this subject. Starting with the stable homotopy category Ho(Sp), that is the homotopy category of spectra, it has been proved to be rigid by S. Schwede. Moreover, the E(1)local stable homotopy category, for p=2, has been shown to be rigid by C. Roitzheim. In this talk, we investigate a new case of rigidity, which is the localization of spectra with respect to the Morava Ktheory K(1), at p=2.

Prasit Bhattacharya (University of Virginia)
Revisiting Stable Adams conjecture
The Adams conjecture, perhaps one of the most celebrated result in the subject of stable homotopy theory, was resolved Quillen and Sullivan independently in 70s. Essentially, the Adams conjecture claims that that the Adams operation ψq:BU → BU composed with the Jhomomorphism J: BU → BGL1(S) is homotopic to J after localizing away from q. Here GL1(S) denotes the space of units of the sphere spectrum. The stable version of Adams conjecture states that J can be deformed to J ∘ ψq via the space of infinite loop space maps from BU1/q → BGL1(S)1/q. The stable version of Adams conjecture was resolved by Friedlander in 1980 and remains the only accepted proof. However, we suspect that there might be an error in Friedlander’s proof. In this talk, I will revisit the work of Friedlander, explain the error and a possible correction. This work is joint with N. Kitchloo.

Jonathan Campbell (Duke)
New Approaches to Hilbert's Generalized Third Problem
In this talk I'll explain how one might attack Hilbert's Generalized Third Problem via homotopy theory, and describe recent progress in this direction. Two ndimensional polytopes, P, Q are said to be scissors congruent if one can cut P along a finite number of hyperplanes, and reassemble the pieces into Q. The scissors congruence problem, aka Hilbert's Generalized Third Problem, asks: when can we do this? what obstructs this? In two dimensions, two polygons are scissors congruent if and only if they have the same area. In three dimensions, there is volume and another invariant, the Dehn Invariant. In higher dimensions, very little is known  but the problem is known to have deep connections to motives, values of zeta functions, the weight filtration in algebraic Ktheory, and regulator maps. I'll give a leisurely introduction to this very classical problem, and explain some new results.

Montek Singh Gill (University of Michigan)
Stabilizations of E_\infty operads and padic stable homotopy theory
I will describe a new class of dg operads, the stable operads, which are in particular sense, stabilizations of E_\infty operads, and an application of these operads to padic stable homotopy theory. Cochains on spaces yield examples of algebras over E_\infty operads, and work of May shows that the operations present in the homology of E_\infty algebras in this case yield exactly the Steenrod operations. Moreover, a result of Mandell says that, endowed with this algebraic structure, the cochains provide algebraic models of padic homotopy types. We show that cochains on a spectrum yield algebras over our stable operads. We calculate the homology such algebras, showing that, in the case of a point, we get a certain completion of the generalized Steenrod algebra. Finally, we show that, endowed with this algebraic structure, the spectral cochains provide algebraic models for padic stable homotopy types.

Katharine Adamyk (UC Boulder)
A classification of Q_0local A(1)modules
This talk will present a classification theorem for modules over A(1), a subalgebra of the mod2 Steenrod algebra. Applications of the classification theorem to lifting A(1)modules to modules over the Steenrod algebra will be discussed, as well as applications to the computation of certain localized Adams spectral sequences.

Candace Bethea (University of South Carolina)
RiemannHurwitz with Wild Ramification
In 2018 Marc Levine gave an arithmetic RiemannHurwitz formula for smooth projective varieties of the same dimension using tools from A^1 enumerative topology. This result requires a technical assumption on the residue fields. In joint work with Jesse Kass and Kirsten Wickelgren, we show by computation that the formula still holds with less restrictive assumptions for curves. In this talk I will explicitly show the result for a specific example suggested by Shuji Saito as a test case for the main theorem as well as outline the proof of the main theorem.

Christy Hazel (University of Oregon)
Equivariant fundamental classes in RO(C2)graded cohomology
Let C2 denote the cyclic group of order two. Given a manifold with a C2action, we can consider its equivariant Bredon RO(C2)graded cohomology. In this talk, we give an overview of RO(C2)graded cohomology in constant Z/2 coefficients, and then explain how a version of the Thom isomorphism theorem in this setting can be used to develop a theory of fundamental classes for equivariant submanifolds. We illustrate how these classes can be used to understand the cohomology of any C2surface in constant Z/2 coefficients, including the ring structure.

Tim Campion (Notre Dame)
Duality in homotopy theory
We explore some implications of a fact hiding in plain sight: Namely, the nsphere has the remarkable property that the “swap” map σ: Sn ∧ Sn → Sn ∧ Sn can be “untwisted”: it is homotopic to (1)n ∧ 1. This simple fact remains true in equivariant and motivic contexts. One consequence is a structural fact about symmetric monoidal ∞categories with finite colimits and duals for objects: it turns out that any such category splits as the product of three canonical subcategories (for instance, one of these subcategories is characterized by being stable). As another consequence, we show that for any finite abelian group G, the symmetric monoidal ∞category of genuine finite Gspectra is obtained from finite Gspaces by stabilizing and freely adjoining duals for objects. This universal property vindicates one motivation sometimes given for studying genuine Gspectra: namely that genuine Gspectra (unlike naive Gspectra or Borel Gspectra) have a good theory of SpanierWhitehead duality. We take steps toward a similar universal property for nonabelian groups and also in motivic homotopy theory.

Yu Zhang (Ohio State)
Topological Quillen localization of structured ring spectra
Homotopy groups and stable homotopy groups of spaces are central invariants in algebraic topology. Stable homotopy groups are comparatively easier to work with, at the expense of losing certain information. However, if we are working with nice spaces, nothing will be lost by working stably: A map between nilpotent spaces induces homotopy groups isomorphisms if and only if it induces stable homotopy groups isomorphisms. Structured ring spectra are spectra with certain algebraic structure encoded by the action of an operad O. For such Oalgebras, the analog of stable homotopy groups are played by Topological Quillen (TQ) homology groups. In this talk, we will draw the analogy between topological spaces and Oalgebras, discuss the TQlocalization of Oalgebras, and show the TQWhitehead theorem for homotopy pronilpotent Oalgebras. Part of the work in this talk is joint with John E. Harper.

Jun Hou Fung (Harvard)
Strict units of commutative ring spectra
Just as an ordinary commutative ring has a multiplicative group of units, a E_\inftyring spectrum R also has a spectrum of units gl_1 R, which plays an important role for example in orientation theory and twisted cohomology theories. However, these spectra are typically very large, and to understand twists by EilenbergMac Lane spaces or to isolate those units that come from geometry, it sometimes suffices to study the space of \emph{strict units} of R. Previously, Hopkins and Lurie have computed the strict units of Morava Etheories, but much remains unknown about them in general. In this talk, I will introduce these strict units and illustrate various methods for computing them, and sketch how these calculations relate to other interesting questions in homotopy theory.

Hood Chatham (MIT)
An Orientation Map for Higher Real Etheory
The real Ktheory spectrum KO is ``almost complex oriented''. Here are a collection of properties that demonstrate this: (1) KO is the C2 fixed points of a complex oriented cohomology theory KU. (2) Complex oriented cohomology theories have trivial Hurewicz image, whereas KO has a small Hurewicz image  it detects η and η2. (3) Complex oriented cohomology theories receive a ring map from MU. KO receives no ring map from MU but it receives one from MSU. (4) If E is a complex orientable cohomology theory, every complex vector bundle V is Eorientable. Not every complex vector bundle V is KOorientable, but V⊕ V and V⊗ 2 are. Higher real E theory EO is an odd primary analogue of KO. At p=3, EO is closely related to TMF. EO is defined as the Cp fixed points of a complex oriented cohomology theory, and it has a small but nontrivial Hurewicz image, so it satisfies analogues of properties (1) and (2). I prove that it also satisfies analogues of properties (3) and (4). In particular, I produce a unital orientation map from a Thom spectrum to EO and prove that for any complex vector bundle V the bundles pV and V⊗ p are complex oriented.

Ningchuan Zhang (UIUC)
Analogs of Dirichlet Lfunctions in chromatic homotopy theory
The relation between Eisenstein series and the Jhomomorphism is an important topic in chromatic homotopy theory at height 1. Both sides are related to the special values of the Riemann ζfunction. This relation is most clearly understood in the context of elliptic cohomology and topological modular forms. Number theorists have studied the twistings of the Riemann ζfunctions and Eisenstein series by Dirichlet characters. Motivated by the Dirichlet equivariance of these twisted Eisenstein series, we introduce the Dirichlet Jspectra in this talk. The homotopy groups of these Dirichlet Jspectra are related to the special values of the Dirichlet Lfunctions, and thus to congruences of the twisted Eisenstein series. If time allows, we will also explain the connection between Dirichlet Jspectra and the twisted Eisenstein series by generalizing Katz's algebrogeometric explanation of congruences of the (untwisted) Eisenstein series.

Zhouli Xu (MIT)
The intersection form of spin 4manifolds and Pin(2)equivariant Mahowald invariants
A fundamental problem in 4dimensional topology is the following geography question: "which simply connected topological 4manifolds admit a smooth structure?" After the celebrated work of KirbySiebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4manifold, the ratio of its secondBetti number and signature is least 11/8. Furuta proved the ''10/8+2''Theorem by studying the existence of certain Pin(2)equivariant stable maps between representation spheres. In this talk, we will present a complete solution to this problem by analyzing the Pin(2)equivariant Mahowald invariants of powers of certain Euler classes in the RO(Pin(2))graded equivariant stable homotopy groups of spheres. In particular, we improve Furuta's result into a ''10/8+4''Theorem. Furthermore, we show that within the current existing framework, this is the limit. For the proof, we use the technique of celldiagrams, known results on the stable homotopy groups of spheres, and the jbased AtiyahHirzebruch spectral sequence. This is joint work with Michael Hopkins, Jianfeng Lin and XiaoLin Danny Shi.