## 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.

## Innocent Problem?

I thought of a random topological problem a week ago in my Analysis class and it has been bugging me for quite a while. I tried searching around but couldn’t find anything.

Consider a non-intersecting curve $\gamma: I=[0,1] \to \mathbb{R}^{3}$. It can even be a closed loop but it shouldn’t be ‘too straight’ else the problem is trivial. Considering a ‘thickening of the curve’. If $X(0)$ is the curve at time $t=0$ and $X(t)$ be obtained by adjoining closed disks of radius $t$ at every point of the curve $\gamma$(I won’t even bother to formalize it). Now, consider the relative complement homology(aka local at subspace) $\tilde{H}_{k}(\mathbb{R}^{3},\mathbb{R}^{3} \backslash X(t))$. The vague problem is to find nice examples of curves and how their homology groups vary with the parameter $t$. For example, if one considers an $8-loop$ except it doesn’t exactly intersect(instead, there is a small gap), then at $t=0$, we obviously have the homology of a circle. If we thicken it a little bit, we’ll get a 4-torus which gains information in $H_{2}$($\mathbb{Z}^{4}$) and thicken it even more, the entire thing collapses to 0. The same thing works with a circle(you get a torus on thickening). For a trivial example, if you just take a line segment then any thickening deformation retracts to the segment which contracts to a point.

I asked my Analysis professor about it, unfortunately, he didn’t have much of an idea and directed me to faculty who may know something about it(and who I’ve yet to contact). Then, I approached my linear algebra professor after class to ask if he knew anything about it(probably a bad idea since he works in representation theory), he just laughed and walked out. I have some stuff jotted down on the problem making little(obvious) progress but I wanted to gather more examples of classes of curves and some inkling of a theorem before I post anything.

I am currently working on the third part of the Derived Functors sequence, some interesting things in stable homotopy theory.

## Topological Constraints on the Universal Approximation of Neural Networks

The title may seem like a contradiction given that there is such a thing as the Universal Approximation Theorem which simply states that a neural network with a single hidden layer of finite width(i.e finite number of neurons) can approximate any function on a compact set of $\mathbb{R}^{n}$ given that the activation function is non-constant,bounded and continuous.

Needless to say, I haven’t found any kind of flaws in the existing proofs(see Kolmogrov or Cybenko). However, I thought of something a little simple and scoured the internet for an answer.

What if we allow an arbitrary number of hidden layers and bound the dimension of the hidden layers making them ‘Deep, Skinny Neural Networks’? Would that be a universal approximator?

## 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.

# Nets/Moore-Smith Sequences

Sequences are common objects in the field of topology. Often, sequences can help identify continuous functions, limit points and compact spaces in metric spaces.

Moore-Smith sequences(or nets) are essentially a generalization of the sequence for an arbitrary topological space and we can see that many foundational theorems of general topology can be stated in terms of nets.

So first, let’s recall what a partial order and direct set is:

A partial order is an order realtion $\leq$ satisfying:-

• $\alpha \leq \alpha$
• If $\alpha \leq \beta$ and $\beta \leq \alpha$, then $\alpha=\beta$.
• If $\alpha \leq \beta,\beta \leq \gamma$, then $\alpha \leq \gamma$.

A directed set $I$ is a set with a partial order relation $\leq$ such that for any $\alpha,\beta \in I$,there exists $\gamma$ such that $\alpha \leq \gamma,\beta \leq \gamma$.

It’s best to think of a directed set as a sort of analogue to an indexing set.