Mrthfx

Let $X,Y$ be objects of some category $mathcal C$, and suppose I want to study homotopy classes of maps from $X$ to $Y$ (almost everything one does in algebraic topology can be viewed this way). It seems as though there are two classes of methods available to me: I can approach this via some kind of obstruction theory, or else I can use some kind of spectral sequence.

But what makes a method of computation "obstruction-theoretic" versus "spectral sequence-theoretic"? It seems as though both types of method typically take as their starting point some kind of filtration of one of the objects involved (or else a filtration of whatever data defines an object of $mathcal C$), and proceed by analyzing the problem in terms of the layers of the filtration -- they yield the obstruction groups / the starting page of the spectral sequence. Perhaps the methodologies diverge in an essential way after that? For instance, spectral sequences seem like a much more "structured" methodology -- are they available only in a more restricted class of situations?

Questions:

1. What is the relationship between obstruction theory and spectral sequences in general? Is there a systematic way to pass back and forth between the two viewpoints?




2. For example, does every spectral sequence have an associated obstruction theory or vice versa?




3. For a very concrete example, suppose that by "obstruction theory", I mean the most classical possible thing -- computing the space of lifts in a commutative square of spaces via a Postnikov tower. Is there an associated spectral sequence (perhaps under certain conditions)? How does one pass back and forth?




4. Conversely, take your favorite spectral sequence. Is there an associated obstruction theory?

This is a partial answer, but every obstruction theory (in some precise sense) provides you with a spectral sequence (in fact several). Let me clarify what do I mean with obstruction theory. All this material is a modern reinterpretation of Bousfield's amazing paper Homotopy Spectral Sequence and Obstructions, although I am being a little more general than he is because I want the Goerss-Hopkins-Miller obstruction theory to be a special case of what I'm writing here.

Suppose you have a moduli space $mathcal{M}$ of some kind of objects you wish to study, which has a forgetful map to some moduli space of objects you understand $mathcal{M}_0$. Typically $mathcal{M}$ is going to be a moduli space of "topological" objects, while $mathcal{M}_1$ is a moduli space of "algebraic" objects, but of course this is not required. For example $mathcal{M}$ could be the space of sections of some fibrations, and $mathcal{M}_1$ the groupoid of the possible lifts at the level of fundamental groupoids. Then an obstruction theory is a factorization of $mathcal{M}to mathcal{M}_1$ as a tower
$$mathcal{M}to cdots to mathcal{M}_2to mathcal{M}_1$$
where $mathcal{M}cong lim mathcal{M}_n$ and for every $ige 0$ there's a cartesian square
$$require{AMScd}
begin{CD}
mathcal{M}_{i+1} @>>> BG_i\
@VVV @VVV \
mathcal{M}_i @>>> (A_i)_{hG_i}
end{CD}$$
where $G_i$ is some discrete group and $A_i$ is some group-like $E_infty$-space equipped with a $G_i$-action, such that the vertical map is the one induced by the zero map $*to A_i$.

For example, if $Eto B$ is a map, $mathcal{M}=mathrm{Map}_B(B,E)$ is the space of sections, we can let $Eto E_nto B$ be the $n$-th relative Postnikov truncation and $mathcal{M}_n=mathrm{Map}_B(B,E_n)$, so that $mathcal{M}_1$ is the space of lifts at the level of fundamental groupoids.

Why am I calling this an obstruction theory? Well, the reason is simple: suppose we have a point $x_1in mathcal{M}_1$ and we want to lift it to $mathcal{M}$. First we need to lift it to $mathcal{M}_2$. This is the same as lifting its image in $(A_1)_{hG_1}$ to $BG_1$, wich is possible iff its image in $pi_0(A_1)_{hG_1}=(pi_0A_1)/G_1$ is zero. Moreover the space of possible lifts is parametrized by the fiber, which is $Omega A_1$, and in particular the equivalence classes of lifts are parametrized by $pi_1A_1$. Iterating this procedure we obtain an infinite sequence of obstruction classes. If all these obstruction classes vanish, we have reached a point of $mathcal{M}$. This in the above example recovers the classical obstruction theory for the sections of a fibration.

Ok, we know what we mean with an "obstruction theory" now, but where do spectral sequences come in? The point is that it is all well and good to be able to construct points of $mathcal{M}$, but it would be nice to have a better description of its homotopy types (i.e. how many equivalence classes are in $mathcal{M}$ and what are their automorphism groups). Let us fix a basepoint $xinmathcal{M}$, constructed with the above procedure. Then, Bousfield and Kan taught us that to every tower of fibrations there is an associated fringed spectral sequence. Note that the pullback square forces the fiber of $mathcal{M}_{i+1}tomathcal{M}$ to be equivalent to $Omega A_i$, so we can write the spectral sequence as
$$pi_{t+1}A_sRightarrow pi_{s-t}mathcal{M}$$
where for convenience of notation I write $A_0=mathcal{M}_1$ (i.e. I'm fixing $mathcal{M}_0=*$). Note that this spectral sequence is "fringed" (i.e. the low degree terms are not abelian groups), and so it needs to be handled with care, but it still very useful. An example is the Goerss-Hopkins-Miller obstruction theory, where they compute the moduli space of $E_∞$-structures on Morava E-theory and show it is essentially algebraic.