Topology Seminar
Past Talks
Fall 2019

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.