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 $> q/2$).

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.

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.

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 $q$ 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 $> q/2$).

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.

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.

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 $> q/2$).

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.