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.