the twisted eilenberg-zilber theorem

a sketch proof note to myself

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

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. ⁴

The twisted Eilenberg-Zilber theorem is the generalization to an arbitrary bundle E\to B. 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 ⁵, 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 ⁶; 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 \gt 0 \\ d_0(b, f) = (d_0b, \tau(b)(d_0f)) & i=0 \\ s_i(b, f) = (s_ib, s_if) & i\geqslant 0 \end{cases} \end{equation*} for some \tau: B\to \mathrm{Aut}(F) (see ⁵, Section 3, Definition 4.2, and section 5, as well as ⁷, 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=\mathbb{Z}. The trivial bundle B\times F=S^1\times\mathbb{Z} is the disjoint union of countably many circles. We can construct a twisted Cartesian product from the automorphism n\mapsto n+1 of \mathbb{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 of the t=0 endpoint is glued to n. The result is the helix p:\mathbb{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:

Theorem. Let B and F be simplicial sets and let B \times_\tau F be a twisted Cartesian product. Then there is an equivalence \begin{equation*} C_*(B) \otimes_\tau C_*(F) \longrightarrow C_*(B \times_\tau F) \end{equation*} where the complex on the left is the usual tensor product with a modified differential which we describe below (the twisted tensor product of ²).

This was first proved in the category of spaces using the method of acyclic models ². A concise proof was provided in ⁸ using essentially what is now the standard formulation of the homological perturbation lemma, which was inspired by the formulas in ¹.⁹ The basic unit of homological perturbation theory (see ¹⁰ for an introduction) is the strong deformation retraction:

Defintion. A strong deformation retraction (SDR) is the following data:

• complexes (C, d_C) and (D, d_D),
• chain maps i:C\to D and p:D\to C such that \begin{equation*} p\circ i = \mathrm{id}_D, \end{equation*}
• a chain homotopy h:D\to D such that \begin{equation*} \mathrm{id}_D - i\circ p = [d_D, h], \end{equation*} satisfying the "side conditions" \begin{equation*} hi = 0, \qquad ph = 0, \qquad h^2=0. \end{equation*}

We often draw this data diagrammatically as

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:

Theorem (Homological Perturbation Lemma). Let \delta be a small perturbation of d_D. Then, given an SDR in the notation from above, there exists a perturbed SDR

Proof. Straightforward, if tedious, computation. See, for instance, ¹⁰.

We can now sketch the proof of the twisted Eilenberg-Zilber theorem. The (untwisted) Eilenberg-Zilber theorem of ¹¹ gives us an SDR ¹²

where the differential on the left is the usual differential on the tensor product. The differential on the right -- call it d^\mathrm{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:

Lemma. The perturbation d^\tau - d^\mathrm{id} of the differential d^\mathrm{id} on C_*(B\times F) is small.

Proof. We will indicate the idea of the proof. Filtering B by skeleta we obtain filtrations on all the algebraic gadgets above. Lemma 2 in Chapter II Section 2 of ¹ shows that d^\tau-d^\mathrm{id} decreases filtration degree. Hence the infinite sum (1- (d^\tau-d^\mathrm{id})h)^{-1} collapses to a finite sum and the perturbation is small.

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

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^\mathrm{id})h)^{-1}(d^\tau -d^\mathrm{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, ¹³)...