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, -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 is a sequence of pointed spaces and structure maps . For example, one can take a pointed space and consider the suspensions . This forms a spectrum with the structure maps the identity. This is the suspension spectrum of .

One can consider this general spectrum and suspend it by the integer $i$ to get a spectrum whose components are given by .

The homotopy groups of a spectrum are defined as the colimit:

.

Since is a functor on , one is motivated to define maps between spectra as a a collection of maps 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 and suspend this to get , the sphere spectrum, an example of a suspension spectrum.

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

Note that (with compact-open toplogy). This allows us to define as spectrum where the adjoint map is a weak homotopy equivalence.

Let’s look at an example.

### Eilenberg-MacLane Spectrum

Recall that an Eilenberg-Maclane space corresponding to where is an abelian group is a CW-complex such that and for .Call this CW-complex

We know the following correspondence from algebraic topology : For any CW-complex .

Collect these spaces on the index to get . We claim this forms a spectra. What are the structure maps though?

In particular, note that we we have . By uniqueness of the Eilemberg-McLane space upto homotopy equivalence, we have a homotopy equivalence . Take the adjoint of this under the loopspace-adjunction to get the structure maps .

Hence, this is also an 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 ).

A CW spectra is a sequence of CW-complexes where the structure maps map isomorphically into a subcomplex of . TO prove the above fact, see Prop 0.49 of this.

The cell of a CW complex is the collection where is a cell of and is not of the form for some cell where . Essentially, you package a cell of some component space in the spectrum and the cells in the higher components obtained from the structure maps such that the original itself is not obtained in this form.

A finite spectrum is a CW-spectrum which has a finite number of cells. Another important property of extends to the case of spectra:CW-Approximation.

A subspectrum of a CW spectrum is closed if is a union of cells and for any cell in with some suspension in , it is in , that is, is a CW spectrum. This will be important later in Bousfield localization.

We know that if is a topological space, then there exists a CW-Complex and a homotopy equivalence .

This extends as follows:

CW Approximation for Sequential Spectra:Let be a sequential spectrum then there exists a CW-Spectrum such that there is a weak homotopy equivalence between and .

#### Proof:

By CW-approximation on , consider a weak homotopy equivalence . Assume that there exists a collection of weak homotopy equivalences $f_{n}: \Gamma X_{k} \sim X_{k}$ for all which respects the structure maps, we have the following

Consider the composite . By CW-Approximation on , the second map factorizes as where is a weak homotopy equivalence and the first one is a relative CW inclusion.

## Triangulated Category

Consider an additive category an automorphism on (called a shift functor). A triangle in this category is a sequence:

A morphism of triangles and with component-wise maps that commute in the obvious diagram.

We have a distinguished class of these triangles called exact triangles/distinguished triangles with respect to which the following axioms are satisfied:

#### TR1(Cofiber, triangles are replete)

The triangle is an exact triangle.

For any map , there is an object and an exact triangle of the form:

Third, every triangle isomorphic to an exact triangle is exact.

#### TR2(Shift exact triangle)

If

is an exact triangle if and only if the following triangle is exact:

#### TR3(Maps between exact triangles)

If are two exact triangles and if we’re given maps between the first two components which makes the square commute, then there exists a third map $latex $Z \to Z’$ which makes the entire diagram commute.

#### TR4(Octahedral axiom)

This is quite technical condition so I’ll just leave this here for self-explanation:

## Homotopy Category of Stable Model Category

The homotopy category of a stable model category, in our case, the category of spectra. This means that loop and suspension functors are equivalences. Note that this isn’t the case in . The projective and injective model structures on chain complexes on an abelina category are also a stable model category.

## Smash Product of Spectra

So let be the stable homotopy category discussed above. We’d like to define a smash product on homotopy category of sequential spectra equipped with its stable model structure, defined above. We want a symmetric monoidal product on this category, the smash product, in this case. The situation is a little complicated in general but one could define a naive smash product of sequential sepctra as follows:

where the structure maps are:

where the last map is the smash product of the maps and and:

To define a symmetric monoidal product, we need a unit. This is the sphere spectrum .

## Ring spectra

A ring spectrum is a spectrum equipped with a multiplication and unit map and such that the following diagram commutes with these monad-like axioms of associativity:

## Bousfield Localization

We wish for a way to add more weak equivalences to a model category which are reflected in the stable homotopy category. To this effect, in a sense, we add more cofibrations to the data of the model category . We’d also want to get atleast some sort of a functor with some results on existence/uniqueness where is the class of maps that we wish to localize on.

Just like in the case of algebraic localization ,we expect to ‘kill off’ more than just under the localization process. A basic example is the localization with respect to a homology theory. The upshot is that the localizing at primes will prove to be computationally better to handle as we’ll see.

We’re just going to be looking at localization of CW-spectra with respect to a homology theory here.. Let be the category of CW-spectra.

Let . One can prove that is a generalized homology theory. Throughout, we’ll write this as for simplicity.

Also, fix the notation .

A morphism of spectra is an if the map is an isomorphism in , that is the induced map, is an isomorphism

A spectrum is acyclic if =0$.

A spectrum is local if for every equivalence , the induced map is a bijection.

Whithead Theorem:

Let be local and is an $E-$equivalence then in .

#### Proof:

The definitions give the equivalences and which gives the stable weak equivalence

Ok, we ‘re now going to get to building this localization functor. Based on everything so far, let be the localization functor we’ll construct;For a spectrum , we want to be and a natural transformation .Here, represents the number of cells of CW spectrum .

Lemma 1:

Let be a CW-spectrum and be a proper closed subspectrum of with . Let be a infinite cardinal number. Then, there is a closed subspectrum with at most cells such that and

#### Proof:

Start off with some such that and . Proceed inductively. Given this , choose for each , a subspectrum such that in . This is possible by the hypothesis . Anyways, define . We get that for all ,.

Choose to finish the proof.

Lemma 2:

There exists an spectrum such that the a spectrum is if and only if .

#### Proof:

By the CW approximation theorem(proven above), let be a set of CW spectrum representatives of the weak equivalence classes of spectra with atmost cells. Consider . If is $E-local$ then .

Now, if then for any formed from the under weak equivalence, shift, homotopy cofiber, wedges, summands. Let be this class of spectra. We’ve to show that all spectra lie in .

Let be an spectra, upto weak equivalence consider the representative . By the previous Lemma(Lemma 1), we start from and get the sequence:

where:

is an closed subspectrum$ **and**

where is the closed subspectrum from Lemma 1 such that and **and **

For a limit ordinal,

Now, let be the CW representative of . One can see that . Assume that .

By shifting, we get the cofiber sequence:

hence .

For a limit ordinal, consider the triangle:

where the first map is where is the inclusion .Transfinite induction gives that .

Localization theorem:

Every spectrum sits in a homotopy cofiber sequence of the form

which is natural in such that is $E-acyclic$ and is

#### Proof:

The proof can be found using the previous lemma in Proposition 2.5 of this followed by Lemma 2.6. It uses a slightly technical ‘small object’ argument followed by transfinite induction.

## Moore spectrum of an Abelian Group

We’re slowly approaching our goal of .

Consider an abelian group , then one has a Moore spectrum is characterized as having the homotopy groups:

for for and where is the Eilenberg-Moore spectra of the integers.

Moreover, this is a ring spectrum

For a spectrum , define where is an abelian group. We have the following Universal Coefficient Theorem:

Universal Coefficient Theorem:

There exists natural exact sequences:

Two abelian groups have the same acyclicity type if:

- is a torsion group iff is a torsion group
- For each prime , is uniquely p-divisible iff is uniquely p-divisible.

So, both and are determined by the acyclicity type of which gives the the following equivalence of localizations:

Proposition:

have same acyclicity type iff give equivalent localization functors.

Thus to determine the localization functors is equivalent to considering the cases when and where is a set of primes and are the integers localized at . The case gives us the so-called p-local spectra.

From ([1], Prop 2.4,Prop2.6), one can determine the following when :

If then

If then . In this case, we have a split exact sequence:

. Also, if is finitely generated for all , then one gets

Moreover, is if and only if are uniquely p-divisible for all .

## Thick Subcategory Theorem

Ok, so one could perform Bousfield localization with respect to more general generalized cohomology theories. For example, one could consider , Morava K-theory. We’re not going to get into any of the actual details. It gives an interesting answer as to what the thick subcategories of the category of finite p-local spectra are.

A full subcategory is said to be thick if:

- For every , any spectra weakly equivalent to are also in
- Given , a cofibration in and two of are in then third is also.
- A retract of an object in is also in .

One can prove that if $X \in F_{p}$ and then for . We say that is of type if but .Let be the category of spectra in of type . A major theorem in homotopy theory which can be proven using a weaker version of the nilpotence theorem is the Thick Subcategory Theorem:

Thick Subcategory Theorem:

The thick subcategories of are exactly for .

One can find a proof of this fact in Lurie’s notes.