# University of Regina Topology Seminar

## Fall 2022

### Topic: Spectral sequences

Time: Mondays 1:00 - 2:15 PM, unless otherwise noted.
Location: CL 312

Most talks are held in person. Some talks are on Zoom:
https://uregina-ca.zoom.us/j/99127226830?pwd=bnFQR1R3UUdyWUxqSS9JMExMRlZwZz09

In this learning seminar, we will study spectral sequences and their applications in topology.

The lecture series will be interspersed with guest research talks, which are posted on our Youtube channel.

Date Speaker Title and abstract References
Friday September 16
1:00 PM
Room CL 410
Martin Frankland Introduction to spectral sequences

Hey, I've got this homology that's hard to compute, so what should I do? Let's try breaking up the problem into smaller computations and piecing together the answer from the smaller answers. A spectral sequence is just what I need.

Spectral sequences are a computational tool that (vastly) generalizes long exact sequences. They have many applications in topology and algebra. The seminar will survey some of the theory and applications of spectral sequences. In this first talk, I will present some motivation, small examples, and the basics of spectral sequences.

September 19 Sebastian Martensen
(NTNU Trondheim)
Triangulated categories and other n-angulations

Triangulated categories were introduced to capture some of the extra structure present in the derived category of a ring and the stable homotopy category, and today they are present wherever homological algebra plays a central role. In the case of derived categories, the triangulation captures certain “shadows” of short and long exact sequences, and so one may wonder: are there categories that capture only long exact sequences of a certain length? This led to the introduction of n-angulated categories in 2013 by Geiss, Keller, and Oppermann. Today, they find their use in representation theory, and it is a hope that they will one day play a role in topology as well. For this talk we will discuss triangulated categories, stable module categories, and introduce n-angulated categories along with their main class of examples.

Slides from the talk
September 26 Don Stanley The spectral sequence of a filtration

We will define spectral sequences and the spectral sequence of a filtration on a chain complex. We will then compute a few simple examples, including a two step filtration and the filtration coming from the cellular chains on a CW complex.

October 3
On Zoom
Matthew Alexander The Serre spectral sequence

The Serre spectral sequence is a tool for computing (co)homology using bundles (or, more generally, Serre fibrations). In this talk we will introduce the Serre spectral sequence and the path-loop fibration, and use them to compute the cohomology of several Eilenberg-MacLane spaces, and loop spaces of spheres.

October 10   Thanksgiving. No seminar.
October 17 Masahiro Takeda
(Kyoto University)
The Steenrod problem and some graded Stanley-Reisner rings

“What kind of rings can be represented as singular cohomology rings of spaces?” is a classic problem in algebraic topology, posed by Steenrod. When the rings are polynomial, this problem was especially well studied by various approaches, and finally solved by Anderson and Grodal in 2008. In this talk, we will consider what kind of graded Stanley-Reisner rings, as a generalization of polynomial rings, can be represented as cohomology rings by using a classical approach.

October 24   No seminar.
October 31 Arnaud Ngopnang Ngompe The Bockstein spectral sequence

In this talk, we describe the construction of the Bockstein spectral sequence through an exact couple. Also we present an application of the Bockstein spectral sequence in algebraic topology.

November 7   Reading Week. No seminar.
November 14
On Zoom
Sarah Petersen
(MPIM Bonn)
The $RO(C_2)$-graded homology of $C_2$-equivariant Eilenberg-MacLane spaces

This talk describes an extension of Ravenel-Wilson Hopf ring techniques to $C_2$-equivariant homotopy theory. Our main application and motivation for introducing these methods is a computation of the $RO(C_2)$-graded homology of $C_2$-equivariant Eilenberg-MacLane spaces. The result we obtain for $C_2$-equivariant Eilenberg-MacLane spaces associated to the constant Mackey functor $\underline{\mathbb{F}}_2$ gives a $C_2$-equivariant analogue of the classical computation due to Serre at the prime 2. We also investigate a twisted bar spectral sequence computing the homology of these equivariant Eilenberg-MacLane spaces.

Recording
November 21 Martin Frankland The first few stable homotopy groups of spheres

We will present the "method of killing homotopy groups" due to Cartan and Serre in the 1950s, a method to compute some homotopy groups of spheres. The method relies on the Hurewicz theorem and the Serre spectral sequence of the fibration that kills the bottom homotopy group of a space.

November 28 Gabriel Valenzuela Spectral sequences of algebras

Sometimes when computing spectral sequences, the target has some multiplicative structure and spectral sequences of algebras can help us determine it. The goal of the talk is to discuss when the spectral sequence converges as an algebra and how that helps us retrieve the multiplicative structure.

December 5 Manak Singh The Grothendieck spectral sequence

Suppose $F:\mathcal{A} \to \mathcal{B}$ and $G:\mathcal{B} \to \mathcal{C}$ are left exact functors between abelian categories, where $\mathcal{A}$ and $\mathcal{B}$ have enough injectives. For each object $A$ of $\mathcal{A}$, given some extra conditions on $F$ and $G$, there exists a convergent spectral sequence called the Grothendieck spectral sequence (GSS), which tells us how to compose the right derived functors of $F$ and $G$, in order to get the right derived functors of $G \circ F$. I prove the theorem by Grothendieck that encapsulates this property. I also briefly provide a context in which a GSS is constructed.