## Bousfield Localization of Spectra

This post is organized as a part of the UT Austin Spring 2020 DRP program.I’d like to thank my mentor Richard Wong for useful directions throughout the semester and introducing me to much of the foundational material in homotopy theory.

This post will be organized as follows:

First, I’ll present the basic definitions of the topic(spectra, homotopy groups,CW-spectra,triangulated categories etc).

Second, I’;ll introduce finite spectra, $Z_{p}$-localization of spectra aka. p-local spectra via Bousfield localization, thick subcategories.

Third, I’ll discuss the the Thick Subcategory Theorem but avoid technical details.

So, let’s begin.

A spectrum $X$ is a sequence of pointed spaces $\{X_{n} \}$ and structure maps $\sigma_{n}: \sum X_{n} \to X_{n+1}$. For example, one can take a pointed space $X$ and consider the suspensions $\sum^{n} X=S^{n} \wedge X$. This forms a spectrum with the structure maps the identity. This is the suspension spectrum of $X$.

One can consider this general spectrum $\{X_{n} \}$ and suspend it by the integer $i$ to get a spectrum $\sum^{i} X$ whose components are given by $(\sum^{i} X)_{n}=X_{n+i}$.

The homotopy groups $\pi_{k}(X)$ of a spectrum $X$ are defined as the colimit:

$\pi_{k}(X)=colim_{n} \pi_{n+k}(X_{n})$.

Since $\sum$ is a functor on $Top$, one is motivated to define maps between spectra $f:X \to Y$ as a a collection of maps $f_{n}:X_{n} \to Y_{n}$ such that the obvious diagrams commute.This may not actually be the best definition of maps between spectra but it doesn’t matter too much for now.

For example, one can consider $S^{0}$ and suspend this to get $\mathbb{S}$, the sphere spectrum, an example of a suspension spectrum.

Given the suspension and loop space functors $\sum, \Omega$, one has the loop-space adjunction for pointed topological spaces, namely,

$[\sum X,Y] \simeq [X,\Omega Y]$

Note that $\Omega Y=[S^{1},Y]$(with compact-open toplogy). This allows us to define $\Omega-spectrum$ as spectrum $X$ where the adjoint map $\tilde{\sigma_{n}}:X_{n} \to \Omega X_{n+1}$ is a weak homotopy equivalence.

Let’s look at an example.

### Eilenberg-MacLane Spectrum

Recall that an Eilenberg-Maclane space corresponding to $(G,n)$ where $G$ is an abelian group is a CW-complex $X$ such that $\pi_{n}(X)=G$ and $\pi_{k}(X)=0$ for $k \neq n$.Call this CW-complex $K(G,n)$

We know the following correspondence from algebraic topology : For any CW-complex $X,H^{k}(X;G) \simeq [X,K(G,n)]$.

Collect these spaces on the index $n$ to get $\{K(G,n) \}$. We claim this forms a spectra. What are the structure maps though?

In particular, note that we we have $\pi_{k}(\Omega K(G,n+1))=\pi_{k+1}(K(G,n+1))$. By uniqueness of the Eilemberg-McLane space upto homotopy equivalence, we have a homotopy equivalence $\tilde{\sigma}_{n}:K(G,n) \to \Omega K(G,n+1)$. Take the adjoint of this under the loopspace-adjunction to get the structure maps $\sigma_{n}:\sum K(G,n) \to K(G,n+1)$.

Hence, this is also an $\Omega-$spectrum.

### CW-spectra

Well, I suppose some analogy with the classical case is required here. Consider the Quillen-Serre model structure on pointed topological spaces(aka the usual one). Here, the cofibrations are retracts of generalized relative CW inclusions(see [1] for definitions). In particular, the CW complexes are cofibrant objects in this case. We’d like our so called CW complexed to be cofibrant objects in the stable model category on topological sequential spectra(these are the same things as the spectra defined above where $\sum X =S^{1} \wedge X$).

A CW spectra is a sequence $\{E_{n} \}$ of CW-complexes where the structure maps $\sigma_{n}:\sum E_{n} \to E_{n+1}$ map isomorphically into a subcomplex of $E_{n+1}$. TO prove the above fact, see Prop 0.49 of this.

## Hopf Fibrations-Construction and Quaternion Interpretations (I)

Before I begin discussing the Hopf fibration of the 3-spbhere, one of the simplest yet deeply profound example of a non-trivial fiber bundle, I’d like to recall the definition of a fiber bundle.

Let $E,B,F$ represent the entire space, base space and the fiber respectively where $E,B$ are connected. If $f:E \mapsto B$ is a continuous surjection onto the base space, then the structure $(E,B,F,f)$ is said to be a fiber bundle if for every $x\in E$, there exists a neighborhood $U \subset B$ of $f(x)$ such that there exists a homeomorphism $\psi:f^{-1}(U) \mapsto U \times F$ such that $\psi \circ \pi_{U}=f$.

What this basically means is that locally, the fiber bundle looks like the product $B \times F$ but globally, it may have different topological properties.

A trivial fiber bundle is a fiber bundle which in which the total space is $E=B \times F$. In fact, any fiber bundle over a contractible CW Complex is trivial.

## CW Complex Structure of Real Projective Space and Kusner’s Parametrization of Boy’s surface

The real projective space represented by $\mathbb{RP}^{n}$ is the space obtained from $\mathbb{R}^{n+1}$ under the equivalence relation $x \sim kx$ $\forall x \in \mathbb{R}^{n+1}$. Basically, $\mathbb{RP}^{n}$ is the set of lines which passed through the origin in $\mathbb{R}^{n+1}$. It can also be understood by identifying antipodal points(points which lie on opposite ends of the diameter) of a unit $n$-sphere,$S^{n}$.

One very basic yet deeply interesting example of these spaces is $\mathbb{RP}^{2}$, known as the real projective plane. While $\mathbb{RP}^{0}$ is a points and $\mathbb{RP}^{1}$ is homeomorphic to a circle with infinity, the real projective plane turns out to be far more interesting indeed. It can’t be embedded in $\mathbb{R}^{3}$ and its immersion/s such as Boy’s surface, Roman surface and Cross Cap have far stranger structures than a mere circle as in the case of $\mathbb{RP}^{2}$. In fact, I will delineate some of the calculations involved in the Kusner-Bryant 3-dimensional parametrization of Boy’s surface. It’s a little fascinating how much complexity can be added to mathematical structures upon generalization especially in the case of the projective space which I believe have a remarkably simple and ‘non-threatening’ definition.