The Mandell Theorem

Recently, I learned about a wonderful result, Mandell’s theorem. We use algebraic invariants like (co-)homology, Massey products, Steenrod squares, etc, to prove two spaces, say, $X$ and $Y$ to be different. Anyway, each topological space can be encoded by means of some algebraic data, i. e., Postnikov’s system and $k$ invariants belonging to some cohomology groups with local coefficients — some sort of higher data. The rational part of the one, in the case of simply connected (more generally, nilpotent) spaces, can be suppressed into a commutative differential graded algebra (Dennis Sullivan’77). In general case, there are many results, contained, e. g., in the books “Rational Homotopy Theory I, II” by Y. Felix, S. Halperin, J.-C. Thomas.

All data of $X$ being nilpotent, not only the rational one, recovers from the cochain complex $C^\ast(X)$ with integer coefficient, considering as $E_{\infty}$ algebra (Michael Mandell’06). It means that whenever $C^{\ast}(X, \mathbb{Z}) \simeq_{E_{\infty}} C^\ast(Y, \mathbb{Z})$ — a weak equivalence in the model category of $E_{\infty}$ algebras for nilpotent $X$ and $Y$ — we have a homotopy equivalence $X\simeq Y$. And that’s wonderful!

Recall, an $E_{\infty}$ algebra is an algebra over an $E_{\infty}$ operad. The last one, in turn, is an operad $\mathcal{O}$, each operation space $\mathcal{O}(n)$ being a contractible set with a free action of the symmetric group on $n$ elements $\Sigma_{n}$ — this is equivalent to saying that an operad $\mathcal{O}$ is a fibrant object in the model structure on operads (which comes from the forgetful functor adjunction, see for details). And an algebra over an operad $\mathcal{O}$ is an object together with a sequence of maps $\mathcal{O}(k)\otimes X^{\otimes k}\to X$ with some natural conditions.

As an example of an $E_{\infty}$ algebra, consider an algebra over the commutative operad $\mathcal{C}om,$ where $\mathcal{C}om(n) = k$ with the trivial $\Sigma_{n}$ structure. In this setting, a $\mathcal{C}om$ algebra is just a commutative differential graded algebra.

The fact that the cochain complex functor $C^{\ast}$ has the category of $E_{\infty}$ algebras as a target is a statement. It can be shown via the Eilenberg-Zilber operad or surjective operad.

One can introduce a more general notion of a cochain theory. Let $T: \mathsf{Top}\to \mathsf{M}$ be a contravariant functor from spaces to graded differential $k$ modules for some commutative ring $k.$ If $T$ satisfies properties of an excision, product, dimension, and invariance under weak equivalences then $T$ is called a cochain theory. In this setting, suppose that $T: \mathsf{Top}\to \mathsf{E}$ is a contravariant functor from spaces to $E_{\infty}$ algebras over $k.$ Then $T$ is naturally quasi-isomorphic in $\mathsf{E}$ to the singular cochain functor $C^{\ast}$ if and only if the underlying functor $T: \mathsf{Top}\to \mathsf{M}$ is a cochain theory (Mandell’02).

Now, look at the case of Mandell’s theorem where $k$ is a field $\mathbb{F}_{p}$ and give a sketch of proof:

Theorem (Mandell’01)

Two nilpotent $p$ complete spaces $X$ and $Y$ of finite $p$ type are weakly homotopy equivalent if and only if

\begin{equation*}C^{\ast}(X, k){\simeq}_{E_{\infty}} C^{\ast}(Y, k),\end{equation*}

i. e., their singular cochain complexes with $k$ coefficients are weakly equivalent as ${E}_{\infty}$ algebras.

Sketch of proof. One direction is clear. For the other one, assume that

as $E_{\infty}$ algebras over $k$. It suffices to consider the case over the algebraic completed field $\overline{k}$ as the map $C^{\ast}(X, k)\otimes_{k}\overline{k}\to C^{\ast}(X, \overline{k})$ is a weak equivalence.

As it was mentioned above, the category of $\mathcal{O}$ algebras (here, $\mathcal{O}$ denotes an operad in the category of unbounded chain complexes over $k)$ has a model structure, with weak equivalences and fibrations being quasi-isomorphisms and surjections respectively¹.

Definition

A space $X$ is called resolvable if for any cofibrant $E_\infty$ algebra $A$ whenever $A\to C^\ast(X)$ is a quasi-isomorphism of $E_\infty$ algebras, the adjoint map $X\to \langle A \rangle$ is a weak equivalence of simplicial sets. Here $\langle - \rangle$ denotes a right adjoint to the cochain functor $C^\ast : s\mathsf{Set}\to \mathcal{E}\mathsf{alg}^{\mathrm{op}},$ where $\mathcal{E}$ is an $E_\infty$ operad.

It is to see that two any resolvable spaces $X$ and $Y$ are homotopy equivalent if and only if the map $C^\ast(X)\to C^\ast(Y)$ is a weak homotopy equivalence of $E_\infty$ algebras. Indeed, if $C^\ast(X)\to C^\ast(Y)$ is a weak homotopy, there is a zig-zag $C^\ast X\leftarrow A\to C^\ast$. Hence, by the adjunction and definition we get a zig-zag $X\to \langle A\rangle\leftarrow Y$ of weak equivalences, and $X\simeq Y$.

So, we will be done if we prove that any nilpotent space is resolvable over the algebraic closure $\mathbb{F}_p.$ The rest of proof is given inductively from Postnikov system considerations. More concretely, one can show that spaces $K(\mathbb{Z}/p^m, n)$ and $K(\mathbb{Z}_p, n)$ are resolvable over any field $k$ of characteristic p with $\Phi - 1: k\to k$ being onto. Here, $\Phi: x\mapsto x^p$ is the Frobenius homomorphism. Now, we use that the Eilenberg-MacLane spaces are gluing blocks of any nilpotent space.

References

1. Alexander Berglund, $E_\infty$ algebras and Mandell’s theorem, notes for a PhD course in Stocholm, 2016
2. Michael A. Mandell, E-infinity Algebras and p-Adic Homotopy Theory, Topology 40 (2001), no. 1, 43-94.
3. Michael A. Mandell., Cochains and Homotopy Type, Publ. Math. IHES, 103 (2006), 213-246.
4. The Mandell’s theorem project