Characteristic classes for block bundles

Question

Block bundles are PL analogs of vector bundles, see e.g. Rourke-Sanderson's article in Bulletin of AMS, 1966. There is a classifying space $B\widetilde{PL}_q$ for rank $q\ $ block bundles, which is a PL analog of $BO_q$, and there are similar classifying spaces $B\widetilde{SPL}_q$, $BSO_q$ for oriented bundles.

Question. Is the rational homotopy type of $B\widetilde{SPL}_q$ recorded in the literature?

Long time ago Colin Rourke explained to me how to compute the rational homotopy type of $B\widetilde{SPL}_q$, but I cannot find the correspondence. If memory serves, the result was as follows.

1. If $q\ge 3$ is even, then $H^*(B\widetilde{SPL}_q;\mathbb Q)$ is a polynomial algebra on the Pontryagin classes and the Euler class. The Euler class occurs in degree $q$, while the Pontryagin classes occur in all degrees divisible by $4$ (which is different from $BSO_q$ where there is no Pontryagin classes in degrees $\ge 2q$).

2. If $q\ge 3$ is odd, then $H^*(B\widetilde{SPL}_q;\mathbb Q)$ a polynomial algebra on the Pontryagin classes, which occur in all degrees divisible by $4$, and a new class in degree $2q-2$ which arises from works of Haefliger and Hirsch.

3. If $q\le 2$, then $BSO_q\to B\widetilde{SPL}_q$ is a rational homotopy equivalence.

As usual, the Pontryagin classes are stable, i.e. they survive in $B\widetilde{SPL}\approx BPL$, while the Euler class and the Haefliger-Hirsch classes die in $B\widetilde{SPL}_{q+1}$. I recall that the proof of 1-2 was not too hard, and I can probably reconstruct it, but it would be much nicer if it were written somehwere.

Answer 1

I don't know where the results are written down in one place (perhaps in the book of Madsen and Milgram?), but see the the end of this post for a list of references.

In any case, here is a proof of the result for $q \ge 3$.

By a result of Haefliger, there is a homotopy cartesian square $$ B\widetilde{PL}_q \quad \to \quad BG_q $$ $$ \downarrow \qquad \qquad \quad \downarrow $$ $$ B\widetilde{PL} \quad \to \quad BG $$ where $BG_q$ classifies oriented $(q-1)$-spherical fibrations, $BG$ classifies stable oriented spherical fibrations and $B\widetilde{PL}$ classifies stable block bundles. Rationally, $BG$ is trivial (since its homotopy groups are the shifted stable homotopy groups of spheres), and $B\widetilde{PL}\simeq BPL$ is rationally weak equivalent to $BO$.

Consequently, there is a rational equivalence $$ B\widetilde{PL}_q \simeq BO \times BG_q . $$ It suffices to identify the rational homotopy type of $BG_q$.

Note that $G_q$ is the topological monoid self-equivalences of $S^{q-1}$. Let $SG_q \subset G_q$ be the submonoid of degree one self maps. Then $BSG_q \to BG_q$ is a rational equivalences as well (they have the same rational homotopy groups). It therefore suffices to identify $BSG_q$ rationally (note: the advantage of $SG_q$ over $G_q$ is that the former is connected).

Case 1, $q$ is even: If $q$ is even, then $S^{q-1}$ is rationally equivalent to an Eilenberg-Mac Lane space $K(\Bbb Q,q-1)$. Using the fiber sequence $SF_{q-1} \to SG_q \to S^{q-1}$ (where $SF_{q-1}$ is the topological monoid of degree one pointed self maps of $S^{q-1}$) and the fact just noted, we see that $SF_{q-1}$ is rationally trivial, so $SG_{q}$ is rationally $K(\Bbb Q,q-1)$.

Consequently, $BSG_q$ is rationally $K(\Bbb Q,q)$ when $q$ is even, so we get a rational equivalence $$ B\widetilde{PL}_q \simeq BO \times K(\Bbb Q,q) $$ when $q \ge 3$ is even.

Case 2, $q$ is odd: In this instance $S^{q-1}$ is not rationally an Eilbenberg-Mac Lane space. But there is a rational fiber sequence $$ S^{q-1} \to K(\Bbb Q,q-1) \to K(\Bbb Q,2q-2) . $$ Arguing similarly to case 1, we see that $SG_{q-1}$ is rationally $K(\Bbb Q,2q-3)$. Hence $BSG_q$ is rationally $K(\Bbb Q,2q-2)$ and we obtain a rational equivalence $$ B\widetilde{PL}_q \simeq BO \times K(\Bbb Q,2q-2) . $$

Addendum

(1). Haefliger's theorem can be found in

Haefliger, André : Differential embeddings of $S^n$ in $S^{n+q}$ for $q>2$. Ann. of Math. 83 (1966), 402–436.

The proof uses embedded framed surgery.

(2). In Wall's book, he says that the case $q=2$ follows from

Wall, C.T.C.: Locally flat PL submanifolds with codimension two. Proc. Cambridge Philos. Soc. 63 (1967) 5–8.

(3) The proof that $B\widetilde{PL}\simeq BPL$ is a consequence of Rourke and Sanderson's paper on block bundles:

Rourke, C. P.; Sanderson, B. J.: Block bundles. Bull. Amer. Math. Soc. 72 (1966) 1036–1039.

(4). The proof that $BO \to B\widetilde{PL}$ is rational requivalence is a consequence of Kervaire and Milnor's work (which amounts to the Browder-Novikov sequence for a sphere), since $\pi_n(\widetilde{PL}/O)$ is the group of exotic homotopy $n$-spheres (at least if $n \ge 5$), and this is a finite group.

Answer 2

Jacob Lurie's notes appear to indicate that BSPL is rationally homotopy equivalent to BSO.