[107, Theorem 6.1] For $n\geq 1$ there exists a genus $g(n)\geq 2$, where
$g(n)\leq 6n$, and canonical relations on the graded polynomial ring
${\mathbb{Q}}[\kappa_{1},\ldots,\kappa_{g-2}]$ such that the map

(53) ${\mathbb{Q}}[\kappa_{1},\ldots,\kappa_{g-2}]/{\rm relations}\to
H^{*}(B{\rm Diff}^{+}(\Sigma_{g});{\mathbb{Q}})$

is injective up to degree $2n$ for all $g\geq g(n)$.

A celebrated paper by Madsen and Weiss proved the remarkable result:

######  THEOREM 10.13.

[90] For $n\geq 1$ and $g$ sufficiently large, there is an isomorphism

(54) $\\{{\mathbb{Q}}[\kappa_{1},\ldots,\kappa_{g-2}]/{\rm
relations}\\}^{2n}\cong H^{2n}(B{\rm Diff}^{+}(\Sigma_{g});{\mathbb{Q}})$

This result was given alternative, “more geometric” proofs by Eliashberg,
Galatius and Mishachev [29] and Hatcher [59]. In contrast to the conclusion of
Theorem 10.13, Morita showed in the celebrated work [107, 108] that the higher
MMM-classes always vanish in $H^{*}(B{\rm
Diff}^{+}(\Sigma_{g})^{\delta};{\mathbb{Q}})$.

######  THEOREM 10.14.

The class $\kappa_{\ell}\in H^{6\ell}(B{\rm
Diff}^{+}(\Sigma_{g})^{\delta};{\mathbb{Q}})$ is trivial for $\ell\geq 3$.

The proof that $\kappa_{\ell}=0$ follows from the Bott Vanishing Theorem.
Observe that $e_{g}^{2}=p_{i}$ is the first Pontrjagin class of the tangent
bundle to the fibers, and so $e_{g}^{\ell+1}\in H^{6\ell+2}(B{\rm
Diff}^{+}(\Sigma_{g})^{\delta};{\mathbb{Q}})$ vanishes for $\ell\geq 3$. It is
unknown whether $\kappa_{2}=0$, as it is one of the intriguing open problems
whether the cube of the Euler class $e_{g}^{3}\in
H^{6}(B\Gamma_{2}^{+};{\mathbb{Q}})$ must vanish.

In contrast to this, Kotschick and Morita proved in [85] the following result,
which was given an alternate proof by Nariman [113, Theorem 0.12]:

######  THEOREM 10.15.

There is an inclusion ${\mathbb{Q}}[\kappa_{1}]\subset H^{2\ell}(B{\rm
Diff}^{+}(\Sigma_{g})^{\delta};{\mathbb{Q}})$ stably. That is, given $\ell\geq
3$ there exists $g_{\ell}$ so that for $g\geq g_{\ell}$ the class
$\kappa_{1}^{\ell}\neq 0$ in $H^{2\ell}(B{\rm
Diff}^{+}(\Sigma_{g})^{\delta};{\mathbb{Q}})$.

As discussed by Margalit [91, Section 9], the cohomology above the range of
Theorem 10.13 contains many classes not of the form of the MMM-classes, and in
fact the constructions of these classes show they grow exponentially as $g$
increases. The general problem is thus:

######  PROBLEM 10.16.

For fixed $g\geq 2$, determine which classes in
$H^{*}(B{{\mathcal{M}}{\mathcal{C}}{\mathcal{G}}}_{g};{\mathbb{Q}})$ have non-
zero image under the map $\iota_{\delta}^{*}\colon
H^{*}(B{{\mathcal{M}}{\mathcal{C}}{\mathcal{G}}}_{g};{\mathbb{Q}})\cong
H^{*}(B{\rm Diff}^{+}(\Sigma_{g});{\mathbb{Q}})\to H^{*}(B{\rm
Diff}^{+}(\Sigma_{g})^{\delta};{\mathbb{Q}})$.

There are further results related to Problem 10.16 in the works by Kotschick
and Morita in [85], Bowden in [18] and Nariman in [112, 113].

##  11\. Characteristic classes for dimension 3

Let $M$ be a connected oriented $3$-manifold. The tangent bundle to $M$ is
trivial, so the methods in Section 4 can be applied to construct families of
classes in $H_{*}(B\overline{{\rm Diff}_{c}(M)};{\mathbb{Q}})$. Moreover, for
$M$ of dimension $q\geq 3$, there are additional families of spherical classes
in $H^{*}(B\Gamma_{q}^{+};{\mathbb{Q}})$ beyond the Godbillon-Vey type classes
in $H^{2q+1}(B\overline{\Gamma}_{3};{\mathbb{R}})$. For the case when $q=3$,
these new spherical classes appear in degree
$H^{10}(B\overline{\Gamma}_{3};{\mathbb{R}})$ which give rise to even more
families of homology classes in $H_{*}(B\overline{{\rm
Diff}_{c}(M)};{\mathbb{Q}})$. First, we recall the Vey basis [44]:

$\displaystyle H^{*>6}(WO_{3})$ $\displaystyle=$
$\displaystyle\bigwedge(y_{1}c_{1}^{3},\ y_{1}c_{1}c_{2},\ y_{1}c_{3},\
y_{1}y_{3}c_{1}^{3},\ y_{1}y_{3}c_{1}c_{2},\ y_{3}c_{3})$ $\displaystyle
H^{*}(W_{3})$ $\displaystyle=$ $\displaystyle\bigwedge(y_{1}c_{1}^{3},\
y_{1}c_{1}c_{2},\ y_{1}c_{3},\ y_{2}c_{2},\ y_{1}y_{2}c_{1}^{3},\
y_{1}y_{2}y_{3}c_{1}c_{2},\ y_{1}y_{2}y_{3}c_{1}^{3},\ y_{1}y_{2}c_{1}c_{2},\
y_{3}c_{3})\ .$

Theorem 7.1 implies the classes
$\\{\Delta(y_{1}c_{1}^{3}),\Delta(y_{1}c_{1}c_{2}),\Delta(y_{1}c_{3})\\}\in
H^{7}(B\Gamma_{3}^{+};{\mathbb{R}})$ are spherical classes, as was previously
observed in [65, 66]. The linear independence of the classes in the image of
$\Delta\colon H^{7}(W_{3})\to H^{7}(B\overline{\Gamma}_{3};{\mathbb{R}})$
follows from the works cited for the proof of Theorem 6.3.

In addition, for dimension $q=3$, Theorem 7.2 yields additional families of
spherical classes as follows. The first Pontrjagin class $p_{1}\in H^{4}(B{\rm
SO}(3);{\mathbb{R}})$ is spherical, detected by a map
$\xi_{4}\colon{\mathbb{S}}^{4}\to B{\rm SO}(3)$. As $B\overline{\Gamma}_{3}$
is $4$-connected, there is a lift $\widehat{\xi}_{1}\colon{\mathbb{S}}^{4}\to
B\Gamma_{3}^{+}$ with $\nu\circ\widehat{\xi}_{1}=\xi_{1}$. The brace product
with the class $[\widehat{\xi}_{1}]\in\pi_{4}(B\Gamma_{3}^{+})$ as defined in
(70) gives a mapping

(55)
$\\{[\widehat{\xi}_{1}],\cdot\\}\colon\pi_{j}(B\overline{\Gamma}_{3})\to\pi_{j+3}(B\overline{\Gamma}_{3})\
.$

The classes
$\\{\Delta(y_{1}c_{1}^{3}),\Delta(y_{1}c_{1}c_{2}),\Delta(y_{1}c_{3})\\}\subset
H_{7}(B\overline{\Gamma}_{3};{\mathbb{R}})$ are spherical, so using the
mapping (55) the classes
$\\{\Delta(y_{1}y_{2}c_{1}^{3}),\Delta(y_{1}y_{2}c_{1}c_{2}),\Delta(y_{1}y_{2}c_{3})\\}\subset
H^{10}(B\overline{\Gamma}_{3};{\mathbb{R}})$ are also spherical. Using the
procedures of the previous sections, we then obtain:

######  THEOREM 11.1.

The classes
$\\{\Delta(y_{1}c_{1}^{3}),\Delta(y_{1}c_{1}c_{2}),\Delta(y_{1}c_{3})\\}\in
H^{7}(B\Gamma_{3}^{+};{\mathbb{R}})$ yield a surjection

(56)
$\\{\Delta(y_{1}c_{1}^{3}),\Delta(y_{1}c_{1}c_{2}),\Delta(y_{1}c_{3})\\}\colon\pi_{7}(B\overline{\Gamma}_{3})\to\pi_{7}(B\Gamma^{+}_{3})\to{\mathbb{R}}^{3}\
.$

######  PROPOSITION 11.2.

Choose a Hamel basis $\\{a_{i}\mid i\in{\mathcal{I}}\\}\subset{\mathbb{R}}$ of
${\mathbb{R}}$ over ${\mathbb{Q}}$. Then there exists a collection of maps
$\\{f_{i,j,k}\colon{\mathbb{S}}^{7}\to B\overline{\Gamma}_{3}\mid
i,j,k\in{\mathcal{I}}\\}$ such that

$\langle\Delta(y_{1}c_{1}^{3}),[f_{i,j,k}]\rangle=a_{i}\ ,\
\langle\Delta(y_{1}c_{1}c_{2}),[f_{i,j,k}]\rangle=a_{j}\ ,\
\langle\Delta(y_{1}c_{3}),[f_{i,j,k}]\rangle=a_{k}\ .$

In particular, the classes $\\{[f_{i,j,k}]\mid
i,j,k\in{\mathcal{I}}\\}\subset\pi_{7}(B\overline{\Gamma}_{3})$ are linearly
independent over ${\mathbb{Q}}$.

######  THEOREM 11.3.

The cohomology ring $H^{*}(\Omega^{3}B\overline{\Gamma}_{3};{\mathbb{Q}})$
contains a free exterior algebra with basis an uncountable collection
$\\{\phi_{i,j,k}\mid i,j,k\in{\mathcal{I}}\\}\subset
H^{4}(\Omega^{3}B\overline{\Gamma}_{3};{\mathbb{Q}})$.

######  COROLLARY 11.4.

The groups $H^{4\ell}(B\overline{{\rm
Diff}_{c}({\mathbb{R}}^{3})};{\mathbb{Q}})$ are uncountable for all $\ell>0$.

######  PROBLEM 11.5.

Show that $\sigma_{3}\colon
H_{4\ell}(\Omega^{3}B\overline{\Gamma}_{3};{\mathbb{Q}})\to
H_{4\ell+3}(B\overline{\Gamma}_{3};{\mathbb{Q}})$ is non-trivial for $\ell>1$.

Recall that $\widetilde{\nu}_{M}\colon M{\rm Diff}_{c}^{+}(M)^{\delta}\to
B\overline{\Gamma}_{3}^{+}$ classifies the foliation on the total space over
$B{\rm Diff}_{c}^{+}(M)^{\delta}$. Define the generalized Godbillon-Vey
classes in $H^{7}(M{\rm Diff}_{c}^{+}(M)^{\delta};{\mathbb{R}})$:

(57) $gv_{1}=\widetilde{\nu}_{M}^{*}(\Delta(h_{1}c_{1}^{3}))\ ,\ \
gv_{2}=\widetilde{\nu}_{M}^{*}(\Delta(h_{1}c_{1}c_{2}))\ ,\ \
gv_{3}=\widetilde{\nu}_{M}^{*}(\Delta(h_{1}c_{3}))\ .$

Define classes $\\{\alpha_{1},\alpha_{2},\alpha_{3}\\}\subset H^{4}(B{\rm
Diff}_{c}^{+}(M)^{\delta};{\mathbb{R}})$ by integration over the fiber:

(58) $\alpha_{1}=\int_{M}\ gv_{1}\ ,\ \alpha_{2}=\int_{M}\ gv_{2}\ ,\
\alpha_{3}=\int_{M}\ gv_{3}\ .$

Next, $\\{f_{i,j,k}\colon{\mathbb{S}}^{7}\to B\overline{\Gamma}_{3}\mid
i,j,k\in{\mathcal{I}}\\}$ be the collection of maps constructed in Proposition
11.2. By Proposition 8.3, for each $i,j,k\in{\mathcal{I}}$, there exists a
compact oriented $4$-manifold $N_{i,j,k}$ and map $g_{i,j,k}\colon
N_{i,j,k}\to B\overline{{\rm Diff}_{c}(M)}$ such that the classes
$\\{[g_{i,j,k}(N_{i,j,k})]\mid i,j,k\in{\mathcal{I}}\\}\subset
H_{4}(B\overline{{\rm Diff}_{c}(M)};{\mathbb{Q}})$ are rationally linearly
independent in the image of
$\vec{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3})\colon H_{4}(B\overline{{\rm
Diff}_{c}(M)};{\mathbb{Q}})\to{\mathbb{R}}^{3}$. By the comments at the end of
Section 8, the image of these classes inject into $H_{4}(B{\rm
Diff}_{c}^{+}(M)^{\delta};{\mathbb{Q}})$:

######  THEOREM 11.6.

For a connected oriented $3$-manifold $M$, the cycles

(59) $\\{[g_{i.j,k}(N_{i,j,k})]\mid i,j,k\in{\mathcal{I}}\\}\subset
H_{4}(B{\rm Diff}_{c}^{+}(M)^{\delta};{\mathbb{Q}})$

are linearly independent, and evaluation of the classes
$\\{\alpha_{1},\alpha_{2},\alpha_{3}\\}\subset H^{4}(B{\rm
Diff}_{c}^{+}(M)^{\delta};{\mathbb{R}})$ on these cycles yields a surjection
onto ${\mathbb{R}}^{3}$.

Next, as $M$ has trivial tangent bundle, Theorem 4.4 yields additional classes
in $H^{*}(B\overline{\Gamma}_{q}^{+};{\mathbb{R}})$. For the section
$\sigma\colon B\overline{{\rm Diff}_{c}(M)}\to B\overline{{\rm
Diff}_{c}(M)}\times M$, define classes in $H^{7}(B\overline{{\rm
Diff}_{c}(M)};{\mathbb{R}})$:

(60) $\beta_{1}=\sigma^{*}(gv_{1})\ ,\ \beta_{2}=\sigma^{*}(gv_{2})\ ,\
\beta_{3}=\sigma^{*}(gv_{3})\ .$

Then modify the maps $\\{f_{i,j,k}\colon{\mathbb{S}}^{7}\to
B\overline{\Gamma}_{3}\mid i,j,k\in{\mathcal{I}}\\}$ constructed in
Proposition 11.2, using the Mather-Thurston Theorem, to obtain:

######  THEOREM 11.7.

For $i,j,k\in{\mathcal{I}}$, there exists a compact oriented $7$-manifold
$N^{\prime}_{i,j,k}$ and map $\widehat{f}_{i,j,k}^{\prime}\colon
N^{\prime}_{i,j,k}\to B\overline{{\rm Diff}_{c}(M)}$ such that the cycles
$\\{[\widehat{f}_{i,j,k}^{\prime}(N^{\prime}_{i,j,k})]\mid
i,j,k\in{\mathcal{I}}\\}\subset H_{7}(B\overline{{\rm
Diff}_{c}(M)};{\mathbb{Q}})$ are linearly independent, and evaluation of the
classes $\\{\beta_{1},\beta_{2},\beta_{3}\\}\subset H^{7}(B\overline{{\rm
Diff}_{c}(M)};{\mathbb{R}})$ on these cycles yields a surjection onto
${\mathbb{R}}^{3}$.

Using the brace product and proceeding as before, we obtain:

######  THEOREM 11.8.

$\\{\Delta(y_{1}y_{2}c_{1}^{3}),\Delta(y_{1}y_{2}c_{1}c_{2}),\Delta(y_{1}y_{2}c_{3})\\}\in
H^{10}(B\overline{\Gamma}_{3};{\mathbb{R}})$ yield a surjection

(61)
$\\{\\{\Delta(y_{1}y_{2}c_{1}^{3}),\Delta(y_{1}y_{2}c_{1}c_{2}),\Delta(y_{1}y_{2}c_{3})\\}\colon\pi_{10}(B\overline{\Gamma}_{3})\to\pi_{10}(B\Gamma^{+}_{3})\to{\mathbb{R}}^{3}\
.$

######  THEOREM 11.9.

For $i,j,k\in{\mathcal{I}}$, there exists a compact oriented $10$-manifold
$N^{\prime\prime}_{i,j,k}$ and map $\widehat{f}_{i,j,k}^{\prime\prime}\colon
N^{\prime\prime}_{i,j,k}\to B\overline{{\rm Diff}_{c}(M)}$ such that the
cycles $\\{[\widehat{f}_{i,j,k}^{\prime\prime}(N^{\prime\prime}_{i,j,k})]\mid
i,j,k\in{\mathcal{I}}\\}\subset H_{10}(B\overline{{\rm
Diff}_{c}(M)};{\mathbb{Q}})$ are linearly independent, and evaluation of the
classes

$\\{(\iota_{0}\circ\sigma)^{*}(\Delta(y_{1}y_{2}c_{1}^{3})),(\iota_{0}\circ\sigma)^{*}(\Delta(y_{1}y_{2}c_{1}c_{2})),(\iota_{0}\circ\sigma)^{*}(\Delta(y_{1}y_{2}c_{3}))\\}\subset
H^{10}(B\overline{{\rm Diff}_{c}(M)};{\mathbb{R}})$

on these cycles yields a surjection onto ${\mathbb{R}}^{3}$.

Using the methods of Theorem 4.7, when $H^{1}(M;{\mathbb{Q}})$ is non-trivial,
additional families of classes can be constructed in $H_{6}(B\overline{{\rm
Diff}_{c}(M)};{\mathbb{Q}})$ and $H_{9}(B\overline{{\rm
Diff}_{c}(M)};{\mathbb{Q}})$, for each class $0\neq Z\in
H^{1}(M;{\mathbb{Z}})$ defined by a mapping $M\to{\mathbb{S}}^{1}$. Details
are left to the reader.

Note that the cycles constructed in the proofs of Theorems 11.7 and 11.9,and
using the methods of Theorem 4.7 are all supported in the interior of $M$. A
$3$-manifold admits a decomposition into its irreducible components

(62) $M=M_{1}\ \\#_{{\mathbb{S}}^{2}}\ M_{2}\ \\#_{{\mathbb{S}}^{2}}\ \cdots\
\\#_{{\mathbb{S}}^{2}}\ M_{k}\ .$

Then we can apply the above results to construct families of cycles for each
irreducible component $M_{\ell}$ and they inject into $H_{*}(B\overline{{\rm
Diff}_{c}(M)};{\mathbb{Q}})$. One can thus apply these procedures in a “mix-
and-match” construction.

The classes obtained using Theorem 11.6 are non-trivial in $H_{4}(B{\rm
Diff}_{c}^{+}(M)^{\delta};{\mathbb{Q}})$, but for the other classes
constructed above in $H_{*}(B\overline{{\rm Diff}_{c}(M)};{\mathbb{Q}})$, it
seems to be completely unknown whether any of them survive for the map to
$H_{*}(B{\rm Diff}_{c}^{+}(M)^{\delta};{\mathbb{Q}})$.

Finally, assume that $M$ is compact, and consider the basic fibration sequence
(1) and its associated Leray-Serre spectral sequence, with $E_{2}$-term

(63) $E_{2}^{r,s}(B{\rm Diff}^{+}(M)^{\delta};{\mathbb{Q}})\cong H^{r}(B{\rm
Diff}^{+}(M);H^{s}(B\overline{{\rm Diff}(M)});{\mathbb{Q}})\ .$

The topological type of ${\rm Diff}^{+}(M)$ has been determined for many
compact $3$-manifolds [60], so an analysis of (63) may be feasible for such
manifolds. We consider some select cases.

Let $M={\mathbb{S}}^{3}$, then the inclusion ${\rm SO}(4)\subset{\rm
Diff}^{+}({\mathbb{S}}^{3})$ is a homotopy equivalence by Hatcher [58]. Thus
$H^{*}(B{\rm
Diff}^{+}({\mathbb{S}}^{3});{\mathbb{Q}})\cong{\mathbb{Q}}[e_{4},p_{1}]$ is a
polynomial ring with generators the first Pontrjagin class $p_{1}$ and the
Euler class $e_{4}$, both of degree $4$. Let
$\widetilde{p}_{1}=\iota_{\delta}^{*}(p_{1})$ and
$\widetilde{e}_{4}=\iota_{\delta}^{*}(e_{4})$ in $H^{4}(B{\rm
Diff}^{+}({\mathbb{S}}^{3})^{\delta};{\mathbb{Q}})$. Nariman has proved the
following remarkable result:

######  THEOREM 11.10.

[116, Theorem 1.2] There is an inclusion
${\mathbb{Q}}[\widetilde{e}_{4},\widetilde{p}_{1}]\subset H^{4}(B{\rm
Diff}^{+}({\mathbb{S}}^{3})^{\delta};{\mathbb{Q}})$.

Let $M={\mathbb{S}}^{1}\times{\mathbb{S}}^{2}$, Hatcher showed in [57] that
the group ${\rm Diff}({\mathbb{S}}^{1}\times{\mathbb{S}}^{2})$ has the
homotopy type of ${\rm O}(2)\times{\rm O}(3)\times\Omega{\rm O}(3)$, though
the H-space structure is not preserved.

######  PROBLEM 11.11.

Determine the image of $\iota_{\delta}^{*}\colon H^{*}(B{\rm
Diff}({\mathbb{S}}^{1}\times{\mathbb{S}}^{2});{\mathbb{Q}})\to H^{*}(B{\rm
Diff}({\mathbb{S}}^{1}\times{\mathbb{S}}^{2})^{\delta};{\mathbb{Q}})$.

Let
$M={\mathbb{T}}^{3}={\mathbb{S}}^{1}\times{\mathbb{S}}^{1}\times{\mathbb{S}}^{1}$,
Hamstrom showed in [55] that the homotopy groups $\pi_{\ell}({\rm
Diff}({\mathbb{T}}^{3}))$ are trivial for $\ell>1$. On the other hand, there
are inclusions ${\mathbb{T}}^{3}\subset{\rm Diff}_{0}({\mathbb{T}}^{3})$ and
${\rm SL}(3,{\mathbb{Z}})\to{\rm Diff}^{+}({\mathbb{T}}^{3})$, and their
images generate a subgroup isomorphic to the semi-direct product
$G={\mathbb{T}}^{3}\rtimes{\rm SL}(3,{\mathbb{Z}})$. The following problem
seems approachable.

######  PROBLEM 11.12.

Determine the image of $\iota_{\delta}^{*}\colon H^{*}(B{\rm
Diff}({\mathbb{T}}^{3});{\mathbb{Q}})\to H^{*}(B{\rm
Diff}({\mathbb{T}}^{3})^{\delta};{\mathbb{Q}})$.

Finally, let $M_{\kappa}$ denote a compact oriented $3$-manifold without
boundary, and constant curvature $\kappa=\pm 1$. Then Bamler and Kleiner
determine the homotopy type of ${\rm Diff}^{+}(M_{\kappa})$ in [3]; see also
the monograph [63]. Again, it seems that nothing is known about the image of
the map $\iota_{\delta}^{*}$ in this case.

##  12\. Characteristic classes for higher dimensions

Let $M$ be an oriented connected manifold of dimension $q>3$. The
constructions of classes in $H^{*}(B{\rm Diff}_{c}^{+}(M);{\mathbb{R}})$ and
$H_{*}(B\overline{{\rm Diff}_{c}(M)};{\mathbb{Q}})$ proceeds using a
combination of the methods described in Sections 6, 7 and 8, but with added
complexity due to the increasing number of linearly independent spherical
classes in $H^{2q+1}(B\Gamma_{q}^{+};{\mathbb{R}})$ and
$H^{2q+1}(B\overline{\Gamma}_{q};{\mathbb{R}})$. In addition, there are
multiple brace product operations on $\pi_{*}(B\overline{\Gamma}_{q})$ that
create additional spherical classes. We give a broad description of the
constructions that are possible, but more precise statements require fixing
the dimension and topological type of the manifold $M$.

For manifolds of dimension $q\geq 5$ there also arise a variety of non-trivial
classes in $H^{*}(B{\rm Diff}_{c}^{+}(M);{\mathbb{Q}})$ constructed using
pseudo-isotopy theory. The relationship of these with $H^{*}(B{\rm
Diff}_{c}^{+}(M)^{\delta};{\mathbb{Q}})$ is completely unknown. It would be
especially interesting to understand the images under $\iota_{\delta}^{*}$ of
the exotic classes described in the works [36, 87, 138].

First, we describe the general form of the results obtained in previous
sections. Let ${\mathcal{V}}_{q}$ be the collection of elements $y_{i}c_{J}\in
WO_{q}$ of the Vey basis for $H^{2q+1}(WO_{q})$ which are independently
variable as in Theorems 6.1 and 6.2, [62]. Let $v_{q}$ denote the cardinality
of the set ${\mathcal{V}}_{q}$.

Theorem 7.1 implies that $\Delta({\mathcal{V}}_{q})$ is a linearly independent
set of spherical classes in $H^{2q+1}(B\Gamma_{q}^{+};{\mathbb{R}})$, whose
evaluation on $H_{2q+1}(B\Gamma_{q}^{+};{\mathbb{Q}})$ define surjections onto
${\mathbb{R}}^{v_{q}}$.

Define $\alpha_{i,J}=\int_{M}\ \widetilde{\nu}^{*}(\Delta(y_{i}c_{J}))\in
H^{q+1}(B{\rm Diff}_{c}^{+}(M)^{\delta};{\mathbb{R}})$ for
$y_{i}c_{J}\in{\mathcal{V}}_{q}$.

By Proposition 8.3, there are $v_{q}$ families of uncountably generated
subgroups of $H_{q+1}(B\overline{{\rm Diff}_{c}(M)};{\mathbb{Q}})$, and the
evaluation of $\vec{\alpha}$ on the images of these classes in
$H_{q+1}(B\overline{{\rm Diff}_{c}(M)};{\mathbb{Q}})$ yields a surjective map
onto ${\mathbb{R}}^{v_{q}}$. In particular, these classes inject into
$H_{q+1}(B{\rm Diff}_{c}^{+}(M)^{\delta};{\mathbb{Q}})$.

Next, consider the application of the brace product as in Theorem 7.2. Let
$\kappa_{q}$ be the greatest integer such that $4\kappa_{q}\leq q+1$. The
first solution $\kappa_{3}=1$ appears when $q\geq 3$, and a second solution
$\kappa_{7}=2$ appears when $q\geq 7$, and so forth. Then for each
$1\leq\ell\leq\kappa_{q}$ there exists
$\xi_{\ell}\colon{\mathbb{S}}^{4\ell}\to B\Gamma_{q}^{+}$ such that the pull-
back $\xi_{\ell}^{*}\circ\nu^{*}(p_{\ell})\neq 0$, where $p_{\ell}\in
H^{4\ell}(B{\rm SO}(q);{\mathbb{Z}})$ is the $\ell$-th Pontrjagin class. Then
the brace product defines a mapping
$\\{\xi_{\ell},\cdot\\}\colon\pi_{j}(B\overline{\Gamma}_{q})\to\pi_{j+4\ell-1}(B\overline{\Gamma}_{q})$.

Define the extended subset of the Vey basis for $H^{*}(WO_{q})$:

(64) $\widehat{{\mathcal{V}}}_{q}=\\{y_{I}y_{I^{\prime}}c_{J}\mid
y_{I}c_{J}\in{\mathcal{V}}_{q}\ ,\
I^{\prime}=(i_{1}^{\prime}<\cdots<i_{r}^{\prime})\ {\rm each}\ i_{\ell}\ {\rm
even\ with}\ 2i_{r}^{\prime}\leq\kappa_{q}\\}$

Let $\widehat{{\mathcal{V}}}_{q,k}\subset\widehat{{\mathcal{V}}}_{q}$ denote
the terms of degree $k$, and let $\widehat{v}_{q,k}$ denote the cardinality of
the set $\widehat{{\mathcal{V}}}_{q,k}$. Then as in previous sections, we have
the non-triviality results as follows.

######  THEOREM 12.1.

For $k\geq 2q+1$, the classes $\Delta(\widehat{{\mathcal{V}}}_{q,k})\in
H^{k}(B\overline{\Gamma}_{q};{\mathbb{R}})$ yield a surjection

(65)
$\Delta(\widehat{{\mathcal{V}}}_{q,k})\colon\pi_{k}(B\overline{\Gamma}_{q})\to{\mathbb{R}}^{\widehat{v}_{q,k}}\
.$

######  PROPOSITION 12.2.

Choose a Hamel basis $\\{a_{i}\mid i\in{\mathcal{I}}\\}\subset{\mathbb{R}}$ of
${\mathbb{R}}$ over ${\mathbb{Q}}$. Then for each $k\geq 2q+1$ with
$\widehat{v}_{q,k}>0$, there exists a collection of maps
$\\{f_{I,I^{\prime},J,i}\colon{\mathbb{S}}^{k}\to B\overline{\Gamma}_{q}\mid
y_{i}y_{I^{\prime}}c_{J}\in\widehat{{\mathcal{V}}}_{q,k}\ ,\
i\in{\mathcal{I}}\\}$ such that
$\langle\Delta(y_{i}y_{I^{\prime}}c_{J}),[f_{I,I^{\prime},J,i}]\rangle=a_{i}$,
and evaluates to zero on these spherical cycles otherwise. In particular, the
classes $\\{[f_{I,I^{\prime},J,i}]\\}\subset\pi_{k}(B\overline{\Gamma}_{q})$
are linearly independent over ${\mathbb{Q}}$.

As before, this implies that the cohomology ring
$H^{*}(\Omega^{q}B\overline{\Gamma}_{q};{\mathbb{Q}})$ contains a free
exterior algebra with basis indexed by the set
$\widehat{{\mathcal{V}}}_{q,k}\times{\mathcal{I}}$, and correspondingly the
groups $H^{*}(B\overline{{\rm Diff}_{c}({\mathbb{R}}^{q})};{\mathbb{Q}})$ are
uncountable.

In this manner, corresponding to the variable spherical classes in
$H^{2q+1}(W_{q})$ we obtain families of infinite dimensional subspaces of
$\pi_{k}(B\overline{\Gamma}_{q})$ for $k=2q+4,2q+8,2q+11,2q+12,\ldots$, and
thus subspaces of $H_{k-q}(B\overline{{\rm Diff}_{c}(M)};{\mathbb{Q}})$ with
an uncountable number of rationally independent generators. This just accounts
for the classes obtained by integration over the fiber method in Theorem 4.2.
There are additional homology classes corresponding to the methods in Theorems
4.4 and4.7.

######  REMARK 12.3.

The results on the homology groups $H_{*}(B\overline{{\rm
Diff}_{c}(M)};{\mathbb{Q}})$ and $H_{*}(B{\rm
Diff}_{c}^{+}(M)^{\delta};{\mathbb{Q}})$ show that they contain many non-
trivial cycles, and each of these cycles is represented by a map $g\colon N\to
B\overline{{\rm Diff}_{c}(M)}$ which is detected by a secondary class in
$H^{*}(B\overline{\Gamma}_{q};{\mathbb{R}})$ or in
$H^{*}(B\Gamma_{q}^{+};{\mathbb{R}})$. Each such cycle induces a
representation $g_{\\#}\colon\pi_{1}(N,x_{0})\to{\rm Diff}_{c}(M)$. It is
unknown what is the “geometric meaning” of the non-triviality of the homology
class of $g(N)$, or of its pairing with a secondary class used in its
construction? The author’s works [70, 72] related the vanishing of the
secondary invariants with ergodic properties of the action of
$\pi_{1}(N,x_{0})$ on $M$ induced by $g_{\\#}$. However, the construction of
the cycles $N$ using foliated surgery, as in Meigniez’s proof of the Mather-
Thurston Theorem in [97], suggests that the groups $\pi_{1}(N,x_{0})$ are
absolutely enormous, and in particular not amenable, so the question remains
essentially open, as to what the constructions in this paper mean in a
geometric or dynamical context.

We conclude with a discussion of another direction of research, which is very
much in the spirit of the recent works by Galatius, Kupers, Nariman and
Randal-Williams. Let $M$ be compact, and Consider the basic fibration sequence
$B\overline{{\rm Diff}_{c}(M)}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}B{\rm
Diff}_{c}^{+}(M)^{\delta}\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}B{\rm
Diff}_{c}^{+}(M)$ and its associated Leray-Serre spectral sequence, with
$E_{2}$-term

(66) $E_{2}^{r,s}(B{\rm Diff}_{c}^{+}(M)^{\delta};{\mathbb{Q}})\cong
H^{r}(B{\rm Diff}_{c}^{+}(M);H^{s}(B\overline{{\rm
Diff}_{c}(M)});{\mathbb{Q}})\ .$

The first step in analyzing (66) is to understand the groups
$H^{*}(B\overline{{\rm Diff}_{c}(M)});{\mathbb{Q}})$, but this appears to be
an almost completely unknown for dimensions $q\geq 4$. For example, according
to Kupers [86], apparently nothing is known about $H^{*}(B\overline{{\rm
Diff}({\mathbb{S}}^{4})});{\mathbb{Q}})$. Even the simplest case when
$M={\mathbb{D}}^{q}$ for $q\geq 4$ is wide open. Kupers [86] writes:

> Question 1.1.3. What is the homotopy type of ${\rm
> Diff}_{c}({\mathbb{D}}^{q})$? For example, what are its path components?
> What are its homology groups? What are its homotopy groups? Is it homotopy
> equivalent to a Lie group? Are its homotopy groups finitely generated? Can
> we relate it to other objects in algebraic topology? Can we relate it to
> other objects in manifold theory?

One of the original motivations for the work in [69] was a result of Farrell
and Hsiang [30, 64]:

######  THEOREM 12.4.

For $\ell\ll q$ and $\ell\equiv 3$ mod 4 and $q$ odd:

(67) $\pi_{\ell}({\rm
Diff}_{0}({\mathbb{T}}^{q}))\otimes{\mathbb{Q}}={\mathbb{Q}}\ .$

######  PROBLEM 12.5.

Show that the Farrell-Hsiang classes in Theorem 12.4 vanish in the image of
$\iota_{\delta}^{*}\colon H^{k}({\rm
Diff}_{0}({\mathbb{T}}^{q});{\mathbb{Q}})\to H^{k}({\rm
Diff}_{0}({\mathbb{T}}^{q})^{\delta};{\mathbb{Q}})$.

The vanishing of classes in the image of $\iota_{\delta}^{*}$ implies the
existence of non-trivial classes in some groups $H_{\ell}(B\overline{{\rm
Diff}({\mathbb{T}}^{q})};{\mathbb{Q}})$ for $1<\ell\ll q$, and hence by the
Mather-Thurston Theorem to the existence of non-trivial classes in
$H_{k}(B\overline{\Gamma}_{q};{\mathbb{Q}})$ in the “unknown range” $q+1<k\leq
q$, so is of great interest.

The works of Galatius and Randal-Williams greatly extend this seminal result
of Farrell and Hsiang in their works [35, 36, 38, 37, 39], and the techniques
used their have intriguing aspects analogous to ideas in foliation theory. The
(stabilized) homotopy groups of ${\rm Diff}_{0}(M)$ can be calculated in terms
of surgery groups and the Waldhausen K-theory of $M$, as described for example
by Hatcher in [56, 60] and Loday [89]. The standard method of calculating the
Waldhausen K-theory groups introduces the notion of _block automorphisms_ ,
denoted $\widetilde{{\mathcal{A}}}(M)$, that were introduced by Burghelea in
[19]. A discussion of the relation between block automorphisms and the study
of diffeomorphism groups is given in Section 2 of Hatcher [56]. (See also the
works of Burghelea [20, 21, 22], Cohen [26] and Weiss and Williams [144].) At
an intuitive level, the notion of a block automorphism is analogous to the
fracturing technique in the proof of the Mather-Thurston Theorem, as described
say in Mather’s paper [95]. It seems a very intriguing question whether this
analogy has a formal mathematical explanation. This suggests a far-reaching
and speculative research program:

######  PROBLEM 12.6.

Find geometric or algebraic correspondences between the Waldhausen K-theory
groups of a manifold $M$ and the groups ${\mathcal{K}}_{*}(M)$.

##  Appendix A Basic homotopy techniques

We recall some basic results from homotopy theory that are frequently used in
our constructions.

###  A.1. Hurewicz Theorems

The Hurewicz Theorem relates the homotopy and homology groups.

######  THEOREM A.1.

Let $X$ be an $r$-connected space for $r\geq 1$. That is, $X$ is a simply
connected space with $\pi_{k}(X)=0$ for $0\leq k\leq r$. Then the Hurewicz
homomorphism $h_{*}\colon\pi_{k}(X)\to H_{k}(X;{\mathbb{Z}})$ is an
isomorphism for $0<k\leq r+1$.

The Rational Hurewicz Theorem relates $\pi_{*}(X)\otimes{\mathbb{Q}}$ with the
groups $H_{*}(X;{\mathbb{Q}})$. Klaus and Kreck give an elementary proof of
this result in [84], and provide additional interesting discussions on this
fundamental theorem.

######  THEOREM A.2.

Let $X$ be a simply connected space with $\pi_{k}(X)\otimes{\mathbb{Q}}=0$ for
$0\leq k\leq r$. Then the Hurewicz map induces an isomorphism

(68) $h\colon\pi_{k}(X)\otimes{\mathbb{Q}}\longrightarrow
H_{k}(X;{\mathbb{Q}})$

for $1\leq k\leq 2r$, and a surjection for $k=2r+1$.

###  A.2. Integration over the fiber

The operation of “integration over the fiber” is often used in the context of
sphere bundles, but it applies much more generally. It was first introduced in
a paper by Chern [23], and stated more formally by Borel and Hirzebruch in [8,
Section 8], where they explain the operation in terms of the differential on
the $E_{2}$-term of the spectral sequence associated to the fibration. This
definition was related to the definition using differential forms by Auer [1].
Bott and Tu discuss integration over the fiber for vector bundles in [16,
Section 6].

######  THEOREM A.3.

Let $p\colon E\to B$ be a fibration with fiber $X=p^{-1}(b)$, where $B$ is
connected and $X$ is a compact oriented manifold of dimension $q$. Then for
$n\geq q$, there exists a natural homomorphism

(69) $\int_{X}\ \colon H^{n}(E;{\mathbb{R}})\to H^{n-q}(B;{\mathbb{R}})\ ,$

called _integration over the fiber_. For $\alpha\in H^{n}(E;{\mathbb{R}})$ and
$\omega\in H^{\ell}(B;{\mathbb{R}})$,
$\omega\cdot\int_{X}(\alpha)=\int_{X}(p^{*}\omega\cdot\alpha)$.

###  A.3. Brace products

The _brace product_ was introduced by James in [76].

######  PROPOSITION A.4.

Let $\displaystyle
F\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}E\stackrel{{\scriptstyle
p}}{{\longrightarrow}}B$ be a fibration, and suppose there exists a section
$\sigma\colon B\to E$. Then there is a well-defined product, called the _brace
product_ ,

(70)
$\\{\cdot,\cdot,\\}_{\sigma}\colon\pi_{i}(E)\times\pi_{j}(F)\to\pi_{i+j-1}(F)\
.$

where the subscript $\sigma$ indicates that the product depends upon the
choice of the section $\sigma$.

###### Proof.

Given classes $\alpha\in\pi_{i}(E)$ and $\beta\in\pi_{j}(F)$ the Whitehead
product $[\sigma_{\\#}(\alpha),\iota_{\\#}(\beta)]\in\pi_{i+j-1}(E_{q})$ is
the image of a unique class denoted
$\\{\alpha,\beta\\}_{\sigma}\in\pi_{i+j-1}(B\overline{\Gamma}_{q})$. ∎

The brace product vanishes when the space $E$ is a product fibration, so the
non-triviality of a pair $\\{\alpha,\beta\\}_{\sigma}$ is a measure of the
“homotopy twisting” of the fibration. Kallel and Sjerve [78] relate the brace
product to the structure of free loop space $\Lambda E$.

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