the twisted eilenberg zilber theorem

Recently I was curious about the origins and initial applications of the homological perturbation lemma (more on that below). Turns out it’s a beautiful story involving the homology of simplicial fibrations (due to [1] and [2]) that I had somehow managed to avoid learning about. In this brief note I want to sketch this arc of ideas.

For spaces \(B\) and \(F\), the Eilenberg-Zilber theorem [3] constructs chain equivalences

\begin{equation*} \begin{xy} \xymatrix{ C_*(B\times F) \ar@(ur, ul)[r]^{} & C_*(B) \otimes C_*(F) \ar@(dl, dr)[l]^{} } \end{xy} \end{equation*}

computing the singular chains of the trivial bundle \(B \times F\) as a tensor product of the chains on \(B\) and the chains on \(F\).¹ The twisted Eilenberg-Zilber theorem is the generalization to an arbitrary bundle \(E=B \times F\). We will work in the setting of simplicial sets instead of topological spaces. In this setting, the combinatorics of simplicial sets conspire to make Kan fibrations amenable to straightforward calculation. The key idea, due to [4], is that of a minimal Kan fibration. We won’t worry about the definition of minimality, but the following two results are important:

• every Kan fibration fiberwise strong deformation retracts to a minimal Kan fibration
• every minimal Kan fibration is a fiber bundle.

The details of the combinatorics can be found in [5]; see, in particular, Proposition 10.3 and Corollary 10.7.

Simplicial fiber bundles have a particularly simple structure: they can be written as twisted Cartesian products (TCP) \(B\times_\tau F\). That is, the underlying set of \(n\)-simplices \((B\times_\tau F)_n=B_n\times F_n\) is that of the trivial bundle, but the zeroth face map \(d_0\) is modified:

\begin{equation*} \begin{cases} d_i(b, f) = (d_ib, d_if) & i > 0 \\ d_0(b, f) = (d_0b, \tau(b)(d_0f)) & i=0 \\ s_i(b, f) = (s_ib, s_if) & i\geq0 \end{cases} \end{equation*}

for some \(\tau: B\to \Aut(F)\) (see [4], Section 3, Definition 4.2, and section 5, as well as [6], pp. 1-10). In other words, TCPs are just Cartesian products where the zeroth face map is twisted according to simplicial automorphisms of the fiber.

The following simple example demonstrates the intuition behind TCPs. Let \(B=S^1\) and \(F=\Z\). The trivial bundle \(B\times F=S^1\times\Z\) is the disjoint union of countably many circles. We can construct a twisted Cartesian product from the automorphism \(n\mapsto n+1\) of \(\Z\). In this way each copy of the interval \(\Delta^1\) has its \(t=1\) endpoint (picked out by the zeroth face map \(d_0\)) glued to \(n+1\) if the \(t=0\) endpoint is glued to \(n\). The result is the helix \(p:\R \to S^1\).

The story so far, then, is that every Kan fibration is fiberwise homotopy equivalent to a TCP. It is in this sense that the following is a generalization of the Eilenberg-Zilber theorem:

This was first proved in the category of spaces using the method of acyclic models [2]. A concise proof was provided in [7] using essentially what is now the standard formulation of the homological perturbation lemma, which was inspired by the formulas in [1].² The basic unit of homological perturbation theory (see [8] for an introduction) is the strong deformation retraction:

We often draw this data diagrammatically as

\begin{equation*} \begin{xy} \xymatrix{ (C,d_C) \ar@(ur, ul)[r]^{i} & (D,d_D)\ar@(ul, ur)[]|{h} \ar@(dl, dr)[l]^{p} } \end{xy} \end{equation*}

Suppose now that we have a perturbation \(\delta\) of the differential \(d_D\) on the right-hand side of an SDR. That is, a differential \(\delta:D\to D\) such that \((d+\delta)^2 =0\), i.e.

\begin{equation*} [d,\delta]=0. \end{equation*}

We ask, moreover, that \(\delta\) is a small perturbation. That is, we ask that the degree zero operator

\begin{equation*} (1-\delta h)^{-1} = \sum_{j=0}^\infty (\delta h)^j \end{equation*}

be well-defined on \(D\). Under this technical condition, the perturbation can be “transferred” to the left-hand side to obtain a new, perturbed SDR:

We can now sketch the proof of the twisted Eilenberg-Zilber theorem. The (untwisted) Eilenberg-Zilber theorem of [9] gives us an SDR³

\begin{equation*} \begin{xy} \xymatrix{ C_*(B)\otimes C_*(F) \ar@(ur, ul)[r]^{i} & C_*(B\times F)\ar@(ul, ur)[]|{h} \ar@(dl, dr)[l]^{p} } \end{xy} \end{equation*}

where the differential on the left is the usual differential on the tensor product. The differential on the right — call it \(d^\id\) — is the standard differential on the simplicial chains of \(B\times F\). Notice, now, that the underlying graded module of the complex \(C_*(B\times_\tau F)\) is just \(C_*(B\times F)\). The differential, however, is different — call it \(d^\tau\). The resulting perturbation satisfies the technical condition described above:

Hence we may apply the homological perturbation lemma to transfer the perturbation to the left and obtain a new SDR:

\begin{equation*} \begin{xy} \xymatrix{ C_*(B)\otimes_\tau C_*(F) \ar@(ur, ul)[r]^{i’} & C_*(B\times_\tau F)\ar@(ul, ur)[]|{h’} \ar@(dl, dr)[l]^{p’} } \end{xy} \end{equation*}

where the complex of the left is just

\begin{align*} (C_*(B)\otimes_\tau C_*(F), d) =&\, (C_*(B) \otimes C_*(F), d_{C_*(B)\otimes C_*(F)} \\ &+ p(1-(d^\tau-d^\id)h)^{-1}(d^\tau -d^\id)i). \end{align*}

Thus we see that the chains \(C_*(B\times_\tau F)\) on the twisted Cartesian product \(B\times_\tau F\) can be written as a “twist”, or perturbation, of the tensor product \(C_*(B)\otimes C_*(F)\). This completes the proof of the twisted Eilenberg-Zilber theorem (we refer to Chapter II of [1] for the full proof).

So we’ve seen that homological perturbation theory was born from the study of Kan fibrations in algebraic topology. Since then, however, there have been all sorts of applications across algebra, geometry, and physics. One thing in particular that I’ve been meaning to understand for a long time is its relationship with Feynman diagrams (see, for instance, [10]). Hopefully I’ll get around to learning about that soon!