University of Regina Topology Seminar

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.

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.

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.

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.

“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.

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.

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.

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.

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.

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.