2.1 Block diffeomorphisms

In this subsection, we give a short overview of block bundles and diffeomorphisms, and we explain how to compare them to honest fiber bundles and diffeomorphisms. For $p\ge 0$ , let $\Delta ^p$ denote the standard topological p-simplex.Footnote ³

The set of all block diffeomorphisms forms a semisimplicial group denoted by whose p-simplices are the block diffeomorphisms of $\Delta ^p\times M$ . The space of block diffeomorphisms is defined as the geometric realization of , and the associated classifying space is denoted by . This space classifies block bundles. Let us recall the definition of a block bundle over a simplicial complex.

By [Reference Ebert and Randal-WilliamsER14, Proposition 3.2], a block bundle $\pi \colon E\to B$ has a stable analogue of the vertical tangent bundle (i.e., there exists a stable vector bundle $T_\pi ^sE\to E$ which is stably isomorphic to the vertical tangent bundle $T_\pi E$ provided that E is an actual fiber bundle). If furthermore B is a manifold, the total space E is again a manifold and there is a stable isomorphism $T_{\pi }^sE \oplus \pi ^*TB \cong _{\mathrm {st}} TE$ (cf. [Reference Ebert and Randal-WilliamsER14, Lemma 3.3]).

Next, let us consider the semisimplicial subgroup of those block diffeomorphisms that commute with the projection $\Delta ^p\times M\to \Delta ^p$ . This gives precisely the p-simplices of the singular semisimplicial group . We have an inclusion , and since the geometric realization of $\operatorname {Sing}_\bullet (X)$ is homotopy equivalent to X for any space X ([Reference BauesBau95, pp. 8]), we get an induced map

Let denote the group-like topological monoid of (orientation preserving) homotopy equivalences of M with classifying space . Again, let be the realization of the semisimplicial group of block homotopy equivalences defined analogously to block diffeomorphisms and let be the corresponding classifying space. By [Reference DoldDol63, Thm 6.1], the inclusion is a homotopy equivalence. Consider the following maps induced by inclusions:

and let and denote the respective homotopy fibers. Note that (resp. ) classifies M-bundles (resp. M-block bundles) together with a fiberwise (resp. blockwise) homotopy equivalence to the product M-bundle – that is, fiber homotopy trivial M-bundles (resp. blockwise homotopy trivial M-block bundles). We make the following definition: A characteristic class $c\in H^{d+k}(\operatorname {BO}(d);\mathbb {Q})$ is called block-spherical (resp. block- h-spherical) if there exists an M-block bundle $E\to S^k$ (resp. a blockwise homotopy trivial one) with $ \left \langle {c(TE),[E]} \right \rangle \not =0$ . We have implications:

and the following comparison result which follows from [Reference Burghelea and LashofBL82].

Lemma 2.3. If $k\le \min (\frac {d-1}{3},\frac {d-5}{2})$ , then the natural map

is surjective.

Before we dive into the proof, let us recall the stable range: For a manifold M, possibly with boundary, let

be the space of pseudoisotopies. There is a canonical map $e\colon C(M)\to C(M\times I)$ , and we define

which is called the pseudoisotopy stable range. By a classical theorem of Igusa, $\phi (d)\ge \min \left (\frac {d-4}3, \frac {d-7}2\right )$ [Reference IgusaIgu88].

Proof of Lemma 2.3.

For $\omega \in \mathbb {N}$ , let be the $\omega $ -th stage Postnikov tower of ; that is, the map is $\omega $ -connected and higher homotopy groups of vanish. Furthermore, let be the localization away from $2$ ; that is, its homotopy groups are $\mathbb {Z}\left [\frac {1}2\right ]$ -modules. We use analogous notations for and . By [Reference Burghelea and LashofBL82, Theorem C7], there is a map , such that the map induced by projection factors as

where the vertical map is induced by the inclusion , provided that $\omega \le \phi (d)+1$ . After completing this to a square and taking homotopy fibers over the image of the identity, we obtain a map of fiber sequences

Note that the left square is a homotopy pullback square since the induced map on its homotopy fibers is a weak equivalence. We map this square to the trivial homotopy pullback square

and obtain the following homotopy pullback square on homotopy fibers:

Hence, the map admits a split, and we get a (split-)surjection

as long as $\ell \le \phi (d)+1 = \min (\frac {d-1}{3},\frac {d-5}{2})$ .

Therefore, an element of yields an M-bundle $E\to S^k$ that is fiber homotopy trivial, provided that the dimension of M is high enough. The advantage of working with instead of stems from the fact that the former is accessible through surgery theory as we will review in the Section 2.2.

Remark 2.4. Another approach to compare and is by using Morlet’s lemma of disjunction as in [Reference Krannich, Kupers and Randal-WilliamsKKR21, Lemma]. Consider the following diagram of (homotopy) fibrations:

If M is $\ell $ -connected with $\ell \le d-4$ , then the induced map on homotopy fibers is $(2\ell -2)$ -connected by Morlet’s lemma of disjunction (cf. [Reference Burghelea, Lashof and RothenbergBLR75, Corollary 3.2 on page 29]). Now is isomorphic to the finite group of exotic spheres in dimension $(d+k-1)$ .

For $k=1$ , we note that both and are connected and that their fundamental groups are isomorphic.

For $k\ge 2$ , we need to distinguish cases depending on the parity of d: If d is even, then is rationally $(d-1)$ -connected by [Reference Kupers and Randal-WilliamsKR24, Theorem A]. However, if d is odd and $k-1\le d-4$ is not divisible by $4$ , then is trivial by [Reference Krannich and Randal-WilliamsKR21, Theorem A]. We note that both [Reference Kupers and Randal-WilliamsKR24] and [Reference Krannich and Randal-WilliamsKR21] are generalizations of a classical result due to Farrell–Hsiang [Reference Farrell and HsiangFH78].

Therefore, in all these cases, is trivial, and the same is true for , provided $k\le 2\ell -1$ . This implies that the map

is surjective. By the five-lemma, the same holds for the induced map

We summarize this discussion about unblocking in the following definition and lemma.

Lemma 2.5. Let M be simply connected and let k be in the unblocking range for M. Then the following map is surjective:

2.2 Surgery theory

Let X be a simply connected manifold of dimension at least $5$ with boundary $\partial X$ . The structure set $\mathcal {S}(X,\partial X)$ of $(X,\partial X)$ (sometimes written as $\mathcal {S}_\partial (X)$ ) is defined to be the set of equivalence classes of tuples $(W,\partial W,f)$ where W is a manifold with boundary $\partial W$ and f is an orientation preserving homotopy equivalenceFootnote ⁴ that restricts to a diffeomorphism on the boundary. Two such tuples $(W_0,\partial W_0,f_0)$ and $(W_1,\partial W_1,f_1)$ are equivalent if there exists a diffeomorphism $\alpha \colon W_0\to W_1$ such that $f_0=f_1\circ \alpha $ on the boundary $\partial W_0$ and on the whole $W_0$ , the map $f_1\circ \alpha $ is homotopic to $f_0$ relative to $\partial W_0$ ([Reference Lück and MackoLM24, Definition 11.2]). It is a consequence of the h-cobordism theorem that for $\dim (M)\ge 5$ , we have the following isomorphism ([Reference Berglund and MadsenBM13, Section 3.2, pp.33]):

The main result of surgery theory is that the structure set $\mathcal {S}_\partial (D^k\times M)$ fits into an exact sequence of sets known as the surgery exact sequence (cf. [Reference Lück and MackoLM24, Theorem 11.22 and Remark 11.23]):

(1) $$ \begin{align} L_{k+d+1}(\mathbb{Z})\longrightarrow\mathcal{S}_\partial(D^k\times M) \longrightarrow \mathcal{N}_\partial(D^k\times M)\overset{\sigma}{\longrightarrow} L_{k+d}(\mathbb{Z}). \end{align} $$

Here, $\mathcal {N}_\partial (D^k\times M)$ is the set of normal invariants which is given by equivalence classes of tuples $(W, f,\hat f, \xi )$ , where W is a $(d+k)$ -dimensional manifold with (stable) normal bundle $\nu _W$ , $\xi $ is a stable vector bundle over $D^k\times M$ and $f\colon W\to D^k\times M$ is a map of degree $1$ that restricts to a diffeomorphism of the boundary and which is covered by a stable bundle map $\hat f\colon \nu _W\to \nu _{D^k\times M}\oplus \xi $ . The equivalence relation is given by bordism of manifolds with cylindrical ends of degree $1$ normal maps (see [Reference Lück and MackoLM24, Definition 11.7]).

Since we only consider simply connected manifolds, the relevant L-groups are $4$ -periodic and given by (cf. [Reference Lück and MackoLM24, Theorem 8.99])

$$\begin{align*}L_n(\mathbb{Z}) \cong\begin{cases} \mathbb{Z} &\quad\text{ if } n\equiv0\;(4)\\ \mathbb{Z}/2 &\quad\text{ if } n\equiv2\;(4)\\ 0 &\quad\text{ otherwise}. \end{cases} \end{align*}$$

In particular, the map $\mathcal {S}_\partial (D^k\times M) \hookrightarrow \mathcal {N}_\partial (D^k\times M)$ if $(k+d)$ is even. The map $\sigma $ in the surgery exact sequence (1) is the so-called surgery obstruction map, which in degrees $d+k\equiv 0\;(4)$ with $k\ge 1$ and for simply connected M is given by

where $\operatorname {sign}$ denotes the signature (cf. [Reference Lück and MackoLM24, Theorem 8.173, Exercise 8.191]). The signature of $W'$ can be computed via Hirzebruch’s signature theorem, which constructs a power series with rational coefficients

$$ \begin{align*} \mathcal{L}(x_1,x_2,\dots) ={}&1 + s_1x_1 + \dots + s_ix_i + \dots + s_{i,j}x_i\cdot x_j + \dots\\ & + s_{i_1,\dots,i_n}x_{i_1}\cdots x_{i_n}+ \dots \end{align*} $$

such that $\operatorname {sign}(W')= \left \langle {\mathcal {L}(p_1(TW'),p_2(TW'),\dots ),\ [W']} \right \rangle $ . Here $p_i(TW')$ are the Pontryagin classes of  $W'$ .

In order to further analyze $\mathcal {N}_\partial (D^k\times M)$ , let us define $G(n)=\{f\colon S^{n-1}\to S^{n-1} \text{homotopy equivalence}\}$ and . Note, that the index shift stems from the fact that one wants to have an inclusion $O(n)\subset G(n)$ of the orthogonal group. Analogously, let . Note that $\operatorname {BG}$ is the classifying space for stable spherical fibrations, whereas $\operatorname {BO}$ is the classifying space for stable vector bundles. The inclusion induces $J(n)\colon \operatorname {BO}(n)\hookrightarrow \operatorname {BG}(n)$ which in the colimit yields a map $J\colon \operatorname {BO}\to \operatorname {BG}$ , and we denote its homotopy fiber by $\operatorname {G/O}$ . By [Reference Lück and MackoLM24, Equation 11.11], there is an identification

$$\begin{align*}\mathcal{N}_\partial(D^k\times M) \cong [(D^k,S^{k-1})\times M, (\operatorname{G/O},\ast)],\end{align*}$$

where $[\_,\_]$ denotes homotopy classes of maps of pairs. For our purpose, we need a more explicit description of this identification; in particular, we want to pay attention to the vector bundle data. We follow [Reference Lück and MackoLM24, Theorem 7.10 and above] for this. A map into $\operatorname {G/O}$ consists of a map $\gamma $ into $\operatorname {BO}$ and a homotopy h in $\operatorname {BG}$ from $J\circ \gamma $ to the constant map. Since $D^k\times M$ is compact, $\gamma $ and h actually land in finite stages $\operatorname {BO}(\ell )$ and $\operatorname {BG}(\ell )$ . We obtain a vector bundle $\gamma ^* U_\ell \to (D^k\times M)/(S^{k-1}\times M)$ , where $U_\ell \to \operatorname {BO}(\ell )$ is the universal vector bundle and a trivialization $\overline h\colon S(\gamma ^* U_d)\to \underline {S^{\ell -1}}$ as spherical fibrations. Next, we choose an embedding such that $S^{k-1}\times M$ embeds into $\{x_1=0\}$ . The relative Pontryagin Thom construction yields a map $(D^N,S^{N-1})\to ( \operatorname {Th}(S(\nu _{M\times D^k})),\operatorname {Th}(S(\nu _{M\times S^{k-1}})))$ into the Thom spaces of the respective sphere bundles. Using the map $\overline h$ from above, we can define a map

$$ \begin{align*} (D^{N+\ell},S^{N+\ell-1}) &= (D^\ell,S^{\ell-1})\times (D^N, S^{N-1}) \longrightarrow (D^\ell,S^{\ell-1})\times \operatorname{Th}(S(\nu_{D^k\times M}))\\ \longrightarrow\quad{}\ &(\operatorname{Th}(S(\nu_{ D^k\times M})\ast \underline{S^{\ell-1}}),\operatorname{Th}(S(\nu_{ S^{k-1}\times M})\ast \underline{S^{\ell-1}})) \\ \overset{\operatorname{Th}(\operatorname{id}\ast\overline h^{-1})}\longrightarrow {}&(\operatorname{Th}(S(\nu_{D^k\times M} \oplus \gamma^*U_\ell)),\operatorname{Th}(S(\nu_{S^{k-1}\times M} \oplus \gamma^*U_\ell))), \end{align*} $$

where the middle map is induced by the projection $D^\ell \times (D(\nu _{D^k\times M})/S(\nu _{D^k\times M})) \to D(D^\ell \times \nu _{D^k\times M})/S(D^\ell \times \nu _{D^k\times M})$ . The reverse of the Pontryagin–Thom-construction yields an element $(W, \hat f, f, \gamma ^*U_\ell )\in \mathcal {N}_\partial (D^k\times M)$ . Note that the bundle $\xi $ is precisely given by the (stable) vector bundle classified by $\gamma \colon D^k\times M\to \operatorname {BO}(\ell )\to \operatorname {BO}$ .

Next, we identify $S^k\wedge M_+ = (S^k\times M_+)/(\{1\}\times M_+\cup S^k\times \{+\}) \cong (D^k\times M)/ (S^{k-1}\times M)$ , where $M_+$ is M with a disjoint base point and $\wedge $ denotes the smash product of pointed spaces. The functor $S^k\wedge {(\_)}$ is adjoint to the k-fold loop space functor $\Omega ^k(\_)$ , and so we get $[S^k\wedge M_+, \operatorname {G/O}]_* \cong [M,\Omega ^k \operatorname {G/O}]$ . Now $\Omega ^{k+1} \operatorname {BG}$ is the homotopy fiber of the map $\Omega ^k\operatorname {G/O}\to \Omega ^k\operatorname {BO}$ , and by obstruction theory (cf. [Reference HatcherHat02, p. 418]), the obstructions to the lifting problem

live in the groups $H^{i+1}(M;\pi _i(\Omega ^{k+1}\operatorname {BG}))\cong H^{i+1}(M;\pi _{k+i+1}(\operatorname {BG}))$ . The homotopy groups $\pi _{k+i+1}(\operatorname {BG})$ are isomorphic to the shifted stable homotopy groups of spheres $\pi _{k+i}^{st}$ by [Reference Lück and MackoLM24, Equation 6.34]. By Serre’s finiteness theorem, these groups are finite for $k+i\ge 1$ , and hence, all of our obstruction groups vanish rationally since we assumed that $k\ge 1$ . Since $\operatorname {maps}_*(M,\Omega ^k \operatorname {BO})$ is an H-space, we see that for every (pointed) map $f\colon M\to \Omega ^k\operatorname {BO}$ , some multiple of f can be lifted to $\Omega ^k \operatorname {G/O}$ . Therefore, it suffices for us to specify an element in

$$\begin{align*}[(D^k, S^{k-1})\times M, (\operatorname{BO},\ast)] = {{\mathrm{KO}}}^0((D^k, S^{k-1})\times M)\end{align*}$$

in order for a multiple of this element to yield a normal invariant $(W,f,\hat f,\xi )$ . Furthermore, by the discussion above, the bundle $\xi $ in this normal invariant is precisely given by the multiple of the element in ${{\mathrm {KO}}}^0((D^k, S^{k-1})\times M)$ , and $\xi $ is trivial when restricted to $S^{k-1}\times M$ . For such a normal invariant, we can hence extend $\xi $ by the trivial bundle to a bundle $\xi '$ over , and the maps f and $\hat f$ can be extended by the identity to a (stable) degree one normal map $(\hat f',f')\colon \nu _{W'}\to \nu _{S^k\times M}\oplus \xi '$ .

Next, we consider the isomorphism given by the Pontryagin character

for $u_k$ the cohomological fundamental class in $H^k(D^k,S^{k-1})\cong H^k({S^k,\ast })$ . The i-th component of the Pontryagin character is given by

$$ \begin{align*} {\mathrm{ph}}_i(\xi) &= {\mathrm{ch}}_{2i}(\xi\otimes\mathbb{C}) = \frac{1}{(2i)!}\Big(\bigl(-2i)c_{2i}(\xi) + f(c_1(\xi),\dots, c_{2i-1}(\xi)\bigr)\Big)\\ &= \frac{(-1)^{i+1}}{(2i-1)!} p_i(\xi), \end{align*} $$

where $f(c_1(\xi ),\dots , c_{2i-1}(\xi ))$ is a polynomial in Chern classes of $\xi $ homogeneous of degree $2i$ which vanishes since all nontrivial products in $H^*((D^k, S^{k-1})\times M;\mathbb {Q})$ are trivial. Hence, for any collection $(x_i)\in H^{4i-k}(M;\mathbb {Q})$ and $(A_i)\in \mathbb {Q}$ there exists a $\lambda \in \mathbb {Z}\setminus \{0\}$ and a normal invariant $ (W,f,\hat f,\xi )\in \mathcal {N}_\partial (D^k\times M)$ such that the bundle $\xi $ has the following Pontryagin classes

$$\begin{align*}p_i(\xi) = (-1)^{i+1}(2i-1)!\lambda A_i\cdot u_k\times x_i,\end{align*}$$

where $u_k$ denotes the cohomological fundamental class of $S^k$ . We observe that $p_i(\xi )\cup p_j(\xi )=0$ for $i,j\ge 1$ since $u_k^2=0$ . Furthermore, $(-1)^{i+1}(2i-1)!\not =0$ for all choices of i, and hence, after replacing $A_i$ by $\frac {(-1)^{i+1} A_i}{(2i-1)!}$ , we may assume that the Pontryagin classes of $\xi $ have the formFootnote ⁵

(2) $$ \begin{align} p_i(\xi) = \lambda A_i\cdot u_k\times x_i. \end{align} $$

This allows us to construct a normal invariant in the kernel of the signature homomorphism and hence an element of such that the underlying stable vector bundle has prescribed Pontryagin classes, which we will do in the succeeding section.

With regard to Pontryagin numbers of the extension $W'$ of W, we remark that $p_i(TW') = p_i(-\nu _{W'})$ and $p_i(-(\nu _{S^k\times M}\oplus \xi '))= p_i(T(S^k\times M)\oplus -\xi ) = p_i(\operatorname {pr}^*TM\oplus -\xi ')$ for $\operatorname {pr}\colon S^k\times M\to M$ . Since $\hat f'\colon \nu _{W'}\to \nu _{S^k\times M}\oplus \xi '$ covers a map of degree one, any Pontryagin number of $W'$ equals the corresponding Pontryagin number of $\operatorname {pr}^*TM\oplus -\xi '$ . Furthermore, as $\xi '$ is trivial on the complement of $W\subset W'$ , the Pontryagin numbers of $\xi '$ are obtained from the ones of $\xi $ by replacing the fundamental class in $H^k(D^k,S^{d-1})$ by the corresponding one in $H^k(S^k,\ast )$ in Equation (2).

3.1 Proof of Theorem A

Part (i) of Theorem A follows from the following ‘block-version’ in combination with Lemma 2.5 (see also Remark 2.6).

Lemma 3.1. Let $4m=d+k$ and let be a monomial in universal Pontryagin classes. Then

Proof. We first show the ‘ $\Leftarrow $ ’ implication. Let $\ell $ be such that $i_\ell $ is maximal with the property above and let $x\in H^*(M;\mathbb {Q})$ be such that

By the discussion in Section 2.2 (in particular, see Equation (2)), there exists a normal invariant $\eta $ such that the underlying (extended) stable vector bundle $\xi \to S^k\times M$ has the following Pontryagin classes:

$$ \begin{align*} p_0(-\xi) &= 1 \qquad p_{i_\ell}(-\xi) = \lambda \cdot u_k\times x\\ p_m(-\xi) &= \lambda A\cdot u_k\times u_M \end{align*} $$

for $u_k$ the cohomological fundamental class of $S^k$ , $A\in \mathbb {Q}$ to be chosen later and $\lambda \in \mathbb {Z}\setminus \{0\}$ determined by A. All other Pontryagin classes of $\xi $ vanish. Note that by assumption, $i_\ell <m$ . ThenFootnote ⁶

(3)

for $\operatorname {pr}\colon S^k\times M\to M$ the projection. Since $p_i(\operatorname {pr}^*TM)=\operatorname {pr}^*p_i(TM)=1\times p_i(TM)$ , we will from now on omit ‘ $\operatorname {pr}^*$ ’ in our computations. Since $i_j<m$ and by our choice of Pontryagin classes of $\xi $ , the expression $p_n(-\xi )\cup p_{i_j-n}(TM)$ is nonzero only if $n=0$ or if $n=i_\ell $ . Furthermore, recall that $p_n(-\xi )\cup p_{n'}(-\xi )=0$ for $n,n'\ge 1$ . We continue the computation

(4) $$ \begin{align} (3)={}& \prod_{i_j\ge i_\ell} \left(p_{i_j}(TM) + p_{i_\ell}(-\xi)\cup p_{i_j-i_\ell}(TM)\right)\cup\prod_{i_j<i_\ell}p_{i_j}(TM)\notag\\={}&\prod_{j=1}^sp_{i_j}(TM)\\& + \prod_{i_j<i_\ell}p_{i_j}(TM) \cup\sum_{i_j\ge i_\ell}\left(p_{i_\ell}(-\xi)\cup p_{i_j-i_\ell}(TM)\cup \prod_{\begin{matrix}q\not=j\\ i_q\ge i_l\end{matrix}}p_{i_q}(TM)\right)\notag, \end{align} $$

where the second equality holds after multiplying out and using the fact that there can only be one factor $p_{i_\ell }(-\xi )$ in each summand. The first summand vanishes for degree reasons. If $i_j>i_\ell $ , then $\prod _{q\not =j}p_{i_q}(TM)=0$ because we chose $i_\ell $ to be maximal such that this product does not vanish. Hence, the latter factor vanishes if $i_j>i_\ell $ and we get

Finally, we need to choose A, such that the surgery obstruction vanishes. We have

$$\begin{align*}p_m(TM\oplus -\xi) = p_m(-\xi) + p_{m-i_\ell}(TM)\cup p_{i_\ell}(-\xi).\end{align*}$$

Consider Hirzebruch’s signature formula:

where $s_m\not =0$ is the leading coefficient of $\mathcal {L}$ . Note that $z=\lambda \cdot z_0$ for some $z_0$ which is independent of A. This is true since there appears precisely one factor $p_{i_\ell }(\xi )$ (which is a multiple of $\lambda $ ) in every monomial summand of $(\mathcal {L}(TM\oplus \xi )-s_mp_m(TM\oplus \xi )+s_m\cdot p_{m-i_\ell }(TM)\cup p_{i_\ell }(-\xi ))$ , while all other factors are coefficients of $\mathcal {L}$ or Pontryagin classes of M which are independent of $\lambda $ . We choose so that $\sigma (\eta )$ vanishes independently of $\lambda $ . We hence obtain a normal invariant with vanishing signature and therefore a block bundle with the desired properties.

For the other implication ‘ $\Rightarrow $ ’, let us assume that for all $i_\ell \ge \frac k4$ .

Since $p_n(\xi )\cup p_{n'}(\xi )=0$ for $n,n'\ge 1$ and $p_{i_1}(TM)\cdots p_{i_s}(TM)=0$ (for degree reasons), every summand in the above expression must contain precisely one factor $p_n(\xi )$ for some $n\ge 1$ . Hence, multiplying out the above delivers

The product $\prod _{r\not = j}p_{i_r}(TM)$ vanishes if $i_j\ge \frac k4$ by assumption. If $i_j<\frac k4$ , then $p_n(-\xi )=0$ for all $1\le n\le i_j$ since every higher Pontryagin class of $\xi $ is of the form $u_k\times *$ and hence is of degree at least $k/4$ . It follows that there is no normal invariant with $p_{i_1}\cup \cdots \cup p_{i_s}(TW)\not =0$ , and hence, there can be no such block bundle.

Next, we turn to the proof of Theorem A(ii) and Theorem B which will again follow from block-analogues thereof combined with Lemma 2.5. Recall the following definitions:

(5)

We assume that M admits at least on nontrivial rational Pontryagin class, so the set $\{i\ge 1 \colon p_i(TM)\not =0\}$ is actually nonempty and $n_{\max }\ge 1$ .

Lemma 3.2. For every $\ell =1,\dots , n_{\max }$ , there exists a normal invariant $\eta _\ell $ with underlying (extended) stable vector bundle $\xi _\ell \to S^k\times M$ with the following property:

$$ \begin{align*} \left\langle {p_{i_{min}}(TM\oplus -\xi_{\ell})^r\cup p_{m-r\cdot i_{\min}}(TM\oplus -\xi_{\ell}),\ [S^k\times M]} \right\rangle \not=0\quad\iff\quad \substack{r=\ell \\\text{ or } r=0} \end{align*} $$

and $\sigma (\eta _\ell )=0$ . For $\ell =1$ , we furthermore have that $\xi _1$ can be chosen such that

$$ \begin{align*} & \left\langle {p_{i_{\min}}(TM\oplus -\xi_1)\cup p_{m-i_{\min}}(TM\oplus -\xi_1),\ [S^k\times M]} \right\rangle \\ &\quad\text{ and }\quad \left\langle {p_{m}(TM\oplus -\xi_1),\ [S^k\times M]} \right\rangle \end{align*} $$

are the only nonvanishing monomial Pontryagin numbers of $TM\oplus -\xi _1$ .

Proof. Let $u_M\in H^{4m-k}(M;\mathbb {Q})$ denote the cohomological fundamental class of M. Since the cup product induces a perfect pairing

$$\begin{align*}H^{4j}(M;\mathbb{Q})\times H^{4(m-j)-k}(M;\mathbb{Q})\to\mathbb{Q},\end{align*}$$

there exists a class such that $x\cup p_{i_{\min }}(TM)^{n_{\max }}=u_M$ . For $r = 0,\dots , n_{\max }$ , we define . Then

$$\begin{align*}x_r\cup p_{i_{\min}}(TM)^r = x \cup p_{i_{\min}}(TM)^{n_{\max}-r}\cup p_{i_{\min}}(TM)^r = u_M.\end{align*}$$

By the discussion in Section 2, we know that for every collection $A_0,\dots , A_{n_{\max }}\in \mathbb {Q}$ , there exists a $\lambda \in \mathbb {Z}\setminus \{0\}$ and a normal invariant $\eta _\ell =(W_\ell ,f_\ell ,\hat f_\ell ,\xi _\ell )$ such that the (extended) stable vector bundle $\xi _\ell '$ has only the following (rational) Pontryagin classes:

$$ \begin{align*} p_0(-\xi_\ell') &= 1\\ p_{m-r\cdot i_{\min}}(-\xi_\ell') &= \lambda A_r \cdot u_k\times x_r \text{ for } r=0,\dots, n_{\max}. \end{align*} $$

Since $r\cdot i_{\min }<m$ and $p_q(TM)=0$ for all $0<q<i_{\min }$ , we have

$$ \begin{align*} p_{i_{min}}(TM\oplus-\xi_\ell') &= \sum_{a=0}^{i_{\min}} p_a(TM)\cup p_{i_{\min}-a}(-\xi_\ell') = p_{i_{\min}} (-\xi_\ell') + p_{i_{\min}} (TM). \end{align*} $$

We will now distinguish two cases: $m=(s+1)\cdot i_{\min }$ for some $1\le s\le n_{\max} $ and $m\not =(r+1)\cdot i_{\min }$ for all r. In the former case, we have $p_{i_{\min }}(-\xi _\ell ')=\lambda A_{s}\cdot u_k\times x_{s}$ and compute

$$ \begin{align*} p_{i_{\min}}(TM\oplus &-\xi_\ell')^{s}\cup p_{\underbrace{m-s\cdot i_{\min}}_{=i_{\min}}}(TM\oplus -\xi_\ell') = p_{i_{\min}}(TM\oplus -\xi_\ell')^{s +1}\\ &= (p_{i_{\min}} (-\xi_\ell') + p_{i_{\min}} (TM) )^{s+1} =(s+1)\cdot p_{i_{\min}}(TM)^{n_{\max}}\cup p_{i_{\min}}(-\xi_\ell')\\ &=(s+1)\lambda A_{s}\cdot \underbrace{p_{i_{\min}}(TM)^{s}\cup u_k\times x_{s}}_{=u_k\cdot u_M} \end{align*} $$

where the third equality follows from multiplying out and the fact that $p_n(\xi )\cup p_{n'}(\xi )=0$ for $n,n'\ge 1$ . For $0\le r\not =s$ we have:

(6) $$ \begin{align} p_{i_{\min}}(TM&\oplus -\xi_\ell')^r \cup p_{m-i_{\min}\cdot r}(TM\oplus -\xi_\ell)\notag\\ ={}& \bigl(p_{i_{\min}} (-\xi_\ell') + p_{i_{\min}} (TM) \bigr)^r\cup \sum_{a=0}^{m-i_{\min}\cdot r}p_a(TM) \cup p_{{m-i_{\min}\cdot r}-a}(-\xi_\ell')\notag\\ ={}& \bigl(p_{i_{\min}} (-\xi_\ell') + p_{i_{\min}} (TM) \bigr)^r\\ &\cup\Big(p_{{m-i_{\min}\cdot r}}(-\xi_\ell') + \sum_{a=i_{\min}}^{m-i_{\min}\cdot r}p_a(TM) \cup p_{{m-i_{\min}\cdot r}-a}(-\xi_\ell')\Big)\notag\\ ={}& \underbrace{ p_{i_{\min}} (TM)^r\cup \lambda A_r u_k\times x_r}_{ = \lambda A_r \cdot u_k\times u_M}\ +\ \ast\cdot u_k\times u_M,\notag \end{align} $$

where $\ast $ is a linear expression in the variables $\lambda A_{r+1},\dots ,\lambda A_{n_{\max }}$ . Now for $b,a_1,\dots , a_{n_{\max }}\in \mathbb {Q}$ , consider the following system of equations:

$$ \begin{align*} b &= \left\langle {p_{m}(TM\oplus-\xi_\ell'), [S^k\times M]} \right\rangle \\ a_1 & = \left\langle {p_{i_{\min}}(TM\oplus-\xi_\ell')\cup p_{m-i_{\min}}(TM\oplus-\xi_\ell'), [S^k\times M]} \right\rangle \\ &\qquad\vdots\\ a_r & = \left\langle {p_{i_{\min}}(TM\oplus-\xi_\ell')^r\cup p_{m-r\cdot i_{\min}}(TM\oplus-\xi_\ell'), [S^k\times M]} \right\rangle \\ &\qquad\vdots\\ a_{n_{\max}} &= \left\langle {p_{i_{\min}}(TM\oplus-\xi_\ell')^{n_{\max}}\cup p_{i_{\min}}(TM\oplus-\xi_\ell'), [S^k\times M]} \right\rangle. \end{align*} $$

By the above computation, this is a linear system of equations in the variables $\lambda A_0,\lambda A_1,\dots , \lambda A_{n_{\max }}$ , and it has the following form:

with $s+1$ in the s-th row. The matrix B is invertible, and hence, we can choose $A_1,\dots A_{n_{\max }}$ such that $a_i=0$ if and only if $i\not =\ell $ . Note that $\lambda $ is not yet determined, as it also depends on $A_0$ , but the condition of $a_i$ being zero or nonzero is independent of $\lambda $ . Furthermore, note that since B is triangular, the values of $a_i$ are independent of $A_0$ . Finally, we need to choose $A_0$ , such that the surgery obstruction vanishes. Consider Hirzebruch’s signature formula:

$$ \begin{align*} \sigma(\eta_\ell) &= \operatorname{sign}(W_\ell') = \left\langle {\mathcal{L}(W_\ell'),\ [W_\ell']} \right\rangle = \left\langle {\mathcal{L}(W_\ell'),\ f_*[S^k\times M]} \right\rangle \\ &= \left\langle {\mathcal{L}(S^k)\mathcal{L}(TM\oplus\xi_\ell),\ [S^k\times M]} \right\rangle = \left\langle {\mathcal{L}(TM\oplus\xi_\ell),\ [S^k\times M]} \right\rangle \\ &= s_m\cdot \lambda\cdot A_0 + \lambda\cdot z_0, \end{align*} $$

where $z_0$ is some number independent of $A_0$ and $s_m$ is the leading coefficient of $\mathcal {L}$ as in the proof of Lemma 3.1. Since $s_m\not =0$ , we can choose so that $\sigma (\eta _\ell )$ vanishes independently of $\lambda $ .

The case $m\not =(r+1)\cdot i_{\min }$ for all r is very similar. By the same computation as (6), we have for all $r\ge 0$ ,

$$ \begin{align*} p_{i_{\min}}(TM&\oplus -\xi_\ell')^r \cup p_{m-i_{\min}\cdot r}(TM\oplus -\xi_\ell)\\ ={}& \underbrace{p_{i_{\min}} (TM)^r\cup \lambda A_r \cup u_k\times x_r}_{ = \lambda A_r \cdot u_k\times u_M}\ +\ \ast\cdot u_k\times u_M. \end{align*} $$

This implies that the respective matrix B has the same form as above with $s+1$ replaced by $1$ . The rest of the argument is verbatim to the first case.

If $\ell =1$ , the above system of linear equations reduces to $a_2=a_3=\dots =a_{n_{\max} }=0$ . From the shape of B, this implies that $A_2=\dots =A_{n_{\max} }$ must be $0$ as well, and hence, bundle $\xi _1'$ only has three nonvanishing Pontryagin classes – namely,

$$ \begin{align*} p_0(\xi_1') &=1 \\ p_{m-i_{\min}}(-\xi_1') &= \lambda A_1 \cdot u_k\times x\\ p_{m}(-\xi_1') &= \lambda A_0 \cdot u_k\times u_M. \end{align*} $$

As noted above, every Pontryagin number of $TM\oplus -\xi _1'$ contains precisely one Pontryagin class of $\xi _1'$ . Since $p_{i_{min}}(TM)$ is the smallest Pontryagin class of M, we deduce that the only possibly nonvanishing Pontryagin numbers of $TM\oplus -\xi _1'$ are $ \left \langle {p_{i_{\min }}(TM\oplus -\xi _1')\cup p_{m-i_{\min }}(TM\oplus -\xi _1'),\ [S^k\times M]} \right \rangle $ and $ \left \langle {p_{m}(TM\oplus -\xi _1'),\ [S^k\times M]} \right \rangle $ . By construction,

$$\begin{align*}\left\langle {p_{i_{\min}}(TM\oplus -\xi_1')\cup p_{m-i_{\min}}(TM\oplus -\xi_1'),\ [S^k\times M]} \right\rangle \not=0.\end{align*}$$

Since $\sigma (\eta _1)=0$ by construction, we deduce that $ \left \langle {p_{m}(TM\oplus -\xi _1'),\ [S^k\times M]} \right \rangle $ is nonzero as well since the leading coefficient in the $\mathcal {L}$ -polynomial is nontrivial by [Reference HirzebruchHir95, p. 12].

3.2 Proof of Corollary C

1. (i) Consider the linear homomorphism $\Omega _{d+k}\longrightarrow \mathbb {Q}^{n_{\max }}$ given by

   $$\begin{align*}[X]\mapsto \Big( \left\langle {p_{i_{\min}}(TX)^\ell\cup p_{\frac{d+k}{4} - \ell\cdot i_{\min}}(TX),[X]} \right\rangle \Big)_{\ell=1,\dots,n_{\max}}.\end{align*}$$

   If $\mathcal {E}$ denotes the vector space generated by $E_1,\dots E_{n_{\max }}$ from Theorem B, the composition

   $$\begin{align*}\mathcal{E}\to\mathrm{Fib}^h(M,k)\to\Omega_{d+k}\otimes\mathbb{Q}\to\mathbb{Q}^{n_{\max}}\end{align*}$$

   is given by a diagonal matrix with nonzero entries on the diagonal by Theorem B. Hence, the first map is forced to be injective.

2. (ii) Let be a monomial Pontryagin number and let $M\overset {\iota }\to E^{4m}\overset {\pi }\to S^k$ be a fiber homotopy trivial bundle. We will show that $p_I[E]$ is a linear combination of $(p_{i_{\min} }^e\cup p_{m-e\cdot i_{\min} }[E])_{e=0..n_{\max} }$ . This implies that $p_I[\_]$ restricted to $\mathrm {Fib}^h_{M,k}$ is a linear combination of $(p_{i_{\min} }^e\cup p_{m-e\cdot i_{\min} }[\_])_{e=0..n_{\max} }$ . Again, since $\Omega _{\ast }$ is classified by Pontryagin numbers, this implies that $\dim \mathrm {Fib}^h_{M,k}\le n_{\max} + 1$ . Furthermore, the signature of any fiber bundle $E\to S^k$ is trivial. Since every coefficient in the $\mathcal {L}$ -polynomial is nontrivial by [Reference Berglund and BergströmBB18], this gives one nontrivial linear relation among the functionals $(p_{i_{\min} }^e\cup p_{m-e\cdot i_{\min} }[\_])_{e=0\dots n_{\max} }$ restricted to $\mathrm {Fib}^h_{M,k}$ , implying our claim:

   $$\begin{align*}\dim\mathrm{Fib}^h_{M,k} = \dim(\mathrm{Fib}^h_{M,k})^*\le n_{\max}.\end{align*}$$

   Now, let us consider the Wang-sequence:

   $$\begin{align*}\dots\longrightarrow H^n(M)\overset{\delta}\longrightarrow H^{k+n}(E)\overset{\iota^*}\longrightarrow H^{k+n}(M)\longrightarrow \dots.\end{align*}$$

   Furthermore,

   $$\begin{align*}\iota^*p_i(TE) = \iota^*p_i(\pi^*TS^k\oplus T_{\pi}E) = \iota^*p_i(T_{\pi}E) = p_i(\iota^*T_{\pi}E) = p_i(TM)\end{align*}$$
   for $T_{\pi }E$ the vertical tangent bundle of E. The fiber homotopy trivialization $h\colon E\to M\times S^k$ yields a retraction of $\iota $ , in particular, $(s\circ \iota )^*TM=TM$ . Therefore, $p_i(TM)=\iota ^*p_i(s^*TM)$ , and by the above computation, it follows for all i that $(p_i(TE)-p_i(s^*TM))=\delta (x_i)$ for some $x_i\in H^{4i-k}(M)$ . Since products of elements in the image of $\delta $ vanish by [Reference WhiteheadWhi78, p. 337, Corollary 3.3], we can multiply out
   $$\begin{align*}p_I(TE) = \prod_{i\in I}(\delta(x_i)+p_i(s^*TM)) = s^*p_I(TM) + \sum_{i\in I}\delta(x_i)\cup s^*p_{I\setminus \{i\}}(TM).\end{align*}$$

   The first summand vanishes for degree reasons, and by our assumption on M, there exist $a_i\in \mathbb {Q}$ such that for $n_i=\frac {m-i}{i_{\min} }$ we have

   $$\begin{align*}p_{I\setminus \{i\}}(s^*TM)=a_i\cdot p_{i_{\min}}(s^*TM)^{n_i} = a_i\cdot (p_{i_{\min}}(TE) - \delta(x_{i_{\min}}))^{n_i}.\end{align*}$$

   Again using that products in the image of $\delta $ vanish, we compute

   (7) $$ \begin{align} p_I(TE)={}&\sum_{i\in I}a_i\cdot\delta(x_i)\cup (p_{i_{\min}}(TE) - \delta(x_{i_{\min}}))^{n_i}\notag\\ ={}&\sum_{i\in I}a_i\cdot\delta(x_i)\cup p_{i_{\min}}(TE)^{n_i} \notag\\ ={}& \sum_{i\in I}a_i\cdot(p_i(TE) - p_i(s^*TM))\cup p_{i_{\min}}(TE)^{n_i}\\ ={}& \sum_{i\in I}a_i\cdot p_i(TE)\cup p_{i_{\min}}(TE)^{n_i} - \sum_{i\in I}a_i\cdot p_i(s^*TM)\cup p_{i_{\min}}(TE)^{n_i}\notag. \end{align} $$

   For the second sum in this formula, we note that there exist $b_i\in \mathbb {Q}$ such that for $m_i=\frac {i}{i_{\min} }$ we have

   $$ \begin{align*} \sum_{i\in I}a_i&\cdot p_i(s^*TM)\cup p_{i_{\min}}(TE)^{n_i} \\ ={}&\sum_{i\in I}a_i b_i p_{i_{\min}}(s^*TM)^{m_i}\cup \bigl(\delta(x_{i_{\min}}) + p_{i_{\min}}(s^*TM)\bigr)^{n_i}\\ ={}&\sum_{i\in I}a_i b_i \binom{n_i}1\delta(x_{i_{\min}})\cup p_{i_{\min}}(s^*TM)^{m_i+n_i-1}\\ &+\sum_{i\in I}a_i b_i \binom{n_i}1 p_{i_{\min}}(s^*TM)^{m_i+n_i}. \end{align*} $$

   Note that is independent of i and the second summand hence vanishes for degree reasons. We compute

   

   Using that $m-n_i\cdot i_{\min } = m-\frac {m-i}{i_{\min} }\cdot i_{\min } = i$ , we can combine this with (7) to obtain

   $$ \begin{align*} p_I(TE) &= \sum_{i\in I}a_i\cdot p_i(TE)\cup p_{i_{\min}}(TE)^{n_i}-\lambda\cdot p_{i_{\min}}(TE)^\ell\\ &=\sum_{i\in I}a_i\cdot p_{m-n_i\cdot i_{\min}}(TE)\cup p_{i_{\min}}(TE)^{n_i}-\lambda\cdot p_{i_{\min}}(TE)^\ell. \end{align*} $$

   Since $a_i$ and $\lambda $ do not depend on E but only on M and I, the restriction of the functional $p_I[\_]$ to $\mathrm {Fib}^h_{M,k}$ is contained in the linear span of the functionals $(p_{i_{\min} }^e\cup p_{m-e\cdot i_{\min} }[\_])_{e=0..n_{\max} }$ , as claimed.

3.3 Upper bounds on $\dim \mathrm {Fib}^h_{M,k}$

Let $\widetilde {\mathrm {Fib}}^h_{M,k}\subset \Omega _{d+k}\otimes \mathbb {Q}$ denote the linear subspace spanned by homotopy trivial M-block bundles. By Lemma 2.5, we have $\widetilde {\mathrm {Fib}}^h_{M,k}\subset \mathrm {Fib}^h_{M,k}$ provided that k is in the unblocking range for M. Furthermore, by Lemma 3.2, $\dim (\widetilde {\mathrm {Fib}}^h_{M,k})\ge n_{\max} $ . In this section, we prove sharpness of the upper bound as claimed in Theorem D or rather its block-analogue.

Recall that $p(n)$ is defined to be the number of partitions and the number of those partitions into natural numbers $\le n'$ is $p(n,n')$ . Proposition 3.3 yields the following upper bound.

Achieving the upper bound

We will now show that this upper bound is sharp (i.e., there exists a manifold for which equality holds).

Definition 3.5. A manifold is said to be P-large if all monomials in rational Pontryagin classes of $TM$ are linearly independent.

If $\tau \colon M\to \operatorname {BO}(d)$ is the classifying map for the tangent bundle $TM$ , then M is P-large if and only if the induced map $\tau ^*\colon H^*(\operatorname {BO}(d);\mathbb {Q})\to H^*(M;\mathbb {Q})$ is injective for $*\le d$ . In the example below, we construct a P-large manifold, which can be made simply connected in dimensions $d\equiv 2,3(4)$ .

Example 3.6. For $n\ge 1$ , let $M^n_{1},\dots ,M^n_{s_n}$ be a basis for $\Omega _{4n}\otimes \mathbb {Q}$ with the property that each of those only has one nontrivial monomial Pontryagin number. Since $\Omega _{*}\otimes \mathbb {Q}$ is generated by $\{\mathbb {CP}^{2n}\}_{n\in \mathbb {N}}$ , we may choose $M^n_i$ to be simply connected. Consider the following d-dimensional manifold:

This manifold has all possible products of Pontryagin classes, and they are all linearly independent since $H^*(M;\mathbb {Q}) = \bigoplus _{n=1}^{d/4}\bigoplus _{j=1}^{s_n} H^*(M^n_j\times S^{d-4n})$ and for every $I=(i_1,\dots , i_s)$ , there is a unique $j\in \{1,\dots , s_{|I|}\}$ such that $p_I(TM^{|I|}_{j})\not =0$ . Note that for $d\not \equiv 1\;(4)$ , every component of M is simply connected. If $d\equiv 2,3\;(4)$ , we can even assume M to be simply connected by performing connected sums. If $d\equiv 0,1\;(4)$ , this is not possible since Pontryagin products of top degree would then live in the $1$ -dimensional space $H^d(M;\mathbb {Q})$ (resp. in the $0$ -dimensional space $H^{d-1}(M;\mathbb {Q})$ ).

For $I=(i_1,\dots , i_s)$ , let $|I|:=\sum i_j$ , and we introduce the short notation

Lemma 3.7. Let M be simply connected and P-large and let $I =\{i_1,\dots ,i_s\}\not =\{m\}$ with $|I|=m$ and $i_j\ge m-\frac {d}4$ for some j. Then there exists a normal invariant $\eta \in \mathcal {N}_\partial (D^{4m-d}\times M)$ with underlying (extended) stable vector bundle $\xi '$ such that

$$ \begin{align*} & \left\langle {p_I(TM\oplus-\xi'), [S^{4m-d}\times M]} \right\rangle \\ &\quad\text{ and }\quad \left\langle {p_m(TM\oplus-\xi'), [S^{4m-d}\times M]} \right\rangle \end{align*} $$

are the only nonvanishing monomial Pontryagin numbers of $TM\oplus -\xi '$ and $\sigma (\eta ) = 0$ .

Proof. Since the cup product induces a perfect pairing and all monomials in Pontryagin classes of M are linearly independent, there exist elements $x_J\in H^{d-4|J|}(M)$ for every collection $J=(j_1,\dots ,j_t)$ with $|J|<\frac d4$ such that $x_J\cup p_{J'}(TM)=u_M\in H^{d}(M)$ is a generator if $J=J'$ and $x_J\cup p_{J'}(TM)=0$ for $J\not =J'$ .

Without loss of generality, let $i_1$ be the biggest element of I and let $a_1$ be the number of elements in I equal to $i_1$ . By assumption, $i_1 \ge m-\frac d4$ and $p_{I\setminus \{i_1\}}(TM)\not =0$ , since M is P-large. By Section 2.2, there exists a normal invariant $\eta $ such that the underlying stable vector bundle $\xi \to S^k\times M$ has the following Pontryagin classes:

$$ \begin{align*} p_0(-\xi') &= 1\\ p_{i_1}(-\xi') &= \lambda \cdot u_{4m-d}\times x_{I\setminus \{i_1\}}\\ p_i(-\xi') &= 0 &\text{ for } i<i_1\\ p_i(-\xi') &= \lambda\cdot u_{4m-d}\times\sum_{J\colon |J| = m-i} A_J x_J &\text{ for } i>i_1 \end{align*} $$

for a generator $u_{4m-d}\in H^{4m-d}(S^{4m-d})$ and $A_J$ to be determined later. Note that for $|J|=m-i$ , the degree of $x_J$ is $d-4|J| = 4i-(4m-d)$ . The same computation as the first one in the proof of Lemma 3.1 that

$$\begin{align*}p_I(TM\oplus -\xi')=\lambda a_1\cdot u_{4m-d}\times u_M \not=0.\end{align*}$$

It remains to show that we can choose $A_J$ such that all other monomial Pontryagin numbers are trivial. Now let $I'=(i_1',\dots ,i_t')$ be different collection with again $|I'|=m$ , $i_1'$ the maximum and $a_1'$ the number of elements in $I'$ equal to $i_1'$ . Then

$$ \begin{align*} p_{I'} (TM\oplus -\xi') &= \prod_{j=1}^t p_{i_j'}(TM\oplus -\xi') = \prod_{j=1}^t \sum_{a=0}^{i_j'} p_a(-\xi')\cup p_{i_j'-a}(TM)\\ &=\prod_{j=1}^t \left(p_{i_j'}(TM) + \sum_{a=i_1}^{i_j'} p_a(-\xi')\cup p_{i_j'-a}(TM) \right). \end{align*} $$

If $i_j'<i_1$ for all j, then the sum on the right vanishes and for degree reasons so does the entire expression. If $i_1' = i_1$ , then we get

$$ \begin{align*} p_{I'} (TM\oplus -\xi') &= \prod_{j\colon i_j'<i_1'} p_{i_j'}(TM)\cup \prod_{j\colon i_j'=i_1'} \left(p_{i_j'}(TM) + p_{i_j'}(-\xi')\right)\\ &=\underbrace{p_{I'}(TM)}_{=0} + \prod_{j\colon i_j'<i_1'} p_{i_j'}(TM)\cup\left(\sum_{j\colon i_j'=i_1'}p_{i_1}(-\xi')\cup p_{i_1}(TM)^{a_1'-1}\right)\\ &=p_{i_1}(-\xi')\cup p_{I'\setminus\{i_1\}}(TM)\\ &=\lambda \cdot u_{4m-d}\times x_{I\setminus\{i_1\}} \cup p_{I'\setminus\{i_1\}}(TM), \end{align*} $$

where the second equality follows from the observation that $p_n(\xi )\cup p_{n'}(\xi )=0$ for $n,n'\ge 1$ . By the choice of $x_J$ , we have that $x_{I\setminus \{i_1\}} \cup p_{I'\setminus \{i_1\}}(TM)=0$ , and therefore, $p_{I'}(TM\oplus -\xi ') = 0$ unless $I=I'$ . It remains to investigate the case $i_1'> i_1$ . The strategy is to choose the coefficients $A_{J}$ by downwards induction with respect to $|J|$ . Note that we have already chosen $A_J$ for $|J|\ge m-{i_1}$ to be either $0$ or $1$ . Let $J = (j_1,\dots ,j_r)$ with $|J|\ge 1$ and let us assume that $A_{J'}$ is already chosen for all $J'$ with $|J'|>|J|$ . By the choice of the Pontryagin classes of $-\xi '$ , this implies that $p_i(-\xi ')$ is already determined for all . If there exists a $j\in J$ such that $j>i_J$ , we set $A_J=0$ . If not, let and note that by assumption, $|I'| = 4m$ and $i_1'$ is the largest entry of $I'$ . We again denote the number of indices agreeing with $i_1'$ by $a_1'$ . We compute

(8) $$ \begin{align} \begin{aligned} p_{I'} (TM\oplus -\xi') ={}& \prod_{\ell=1}^t \left(p_{i_\ell'}(TM) + \sum_{a=i_1}^{i_\ell'} p_a(-\xi')\cup p_{i_\ell'-a}(TM) \right)\\ ={}&\prod_{\ell\colon i_\ell'= i_1'}\Big(p_{i_\ell'}(TM) + p_{i_\ell'}(-\xi') + \sum_{a=i_1}^{i_\ell'-1} p_a(-\xi')\cup p_{i_\ell'-a}(TM)\Big)\\ &\qquad\cup \prod_{\ell\colon i_\ell'< i_1'}\Big(p_{i_\ell'}(TM) + \sum_{a=i_1}^{i_\ell'} p_a(-\xi')\cup p_{i_\ell'-a}(TM)\Big).\\ \end{aligned} \end{align} $$

Extracting all summands containing $p_{i_1'}(-\xi )$ , we obtain

Note that the highest index of a Pontryagin class of $-\xi '$ appearing in $B_{I'}$ is strictly smaller than $i_1' = i_J$ . Hence, it is only dependent on $A_{J'}$ with $|J'|<|J|$ , and by our assumption, this summand is already determined. We get

$$ \begin{align*} p_{I'} (TM\oplus -\xi') &= a_1'\lambda \cdot u_{4m-d}\times\sum_{\tilde J\colon |\tilde J| = m-i_1'} A_{\tilde J} \underbrace{x_{\tilde J}\cup p_{J}(TM)}_{=\begin{cases} 0 &\text{ if } \tilde J \not= J\\ u_M &\text{ if } \tilde J = J\end{cases}} + \lambda\cdot B_{I'}\\[7pt] & = \lambda\cdot (a_1' A_{J}\cdot u_{4m-d}\times u_M + B_{I'}). \end{align*} $$

Therefore, we can choose $A_{J}$ for all J with $|J| = m-i_J$ such that $p_{I'}(TM\oplus -\xi ')=0$ for all $I'$ with $i_1' = i_J$ .Footnote ⁷ It remains to specify $A_{\{0\}}$ and hence $p_m(-\xi )$ . By the same argument as in the proof of Lemma 3.2, we can choose $A_{\{0\}}$ such that $\sigma (\eta )=0$ which finishes the proof.

Corollary 3.8. For a simply connected, P-large manifold M of dimension $d\ge 5$ , we have

$$\begin{align*}\dim\widetilde{\mathrm{Fib}}^h_{M,4m-d} = p(m) - p(m, m- \left\lceil {\frac{d+1}{4}} \right\rceil ) -1.\end{align*}$$

Proof. Since the oriented cobordism ring is classified by Pontryagin numbers, the functionals given by evaluating monomial Pontryagin numbers are all linearly independent. Let . By Lemma 3.7, there exists for every $I\in \mathcal {I}_{m,d}$ an M-block bundle $E_I\to S^k$ such that $ \left \langle {p_J(E_I), [E]} \right \rangle \not =0$ if and only if $I=J$ for all $J\in \mathcal {I}_{m,d}$ . Therefore, $(E_I)_{I\in \mathcal {I}_{m,d}}$ is also linearly independent, and the claim follows from

$$\begin{align*}|\mathcal{I}_{m,d}| = p(m) - p(m, m- \left\lceil {\frac{d+1}{4}} \right\rceil )-1\end{align*}$$

and from the upper bound in Lemma 3.4.

The proof above also shows the following lemma, which states a similar result but does not require the set of all possible Pontryagin classes to be linearly independent.

Lemma 3.9. Let M be a manifold such that the set of all nontrivial Pontryagin classes of M are linearly independent and let I be a partition of $\ell \le d=\dim (M)$ such that $p_I(TM)\not =0$ is the only nontrivial monomial Pontryagin class of degree $\ge \ell $ . Then there exists a normal invariant $\eta \in \mathcal {N}_\partial (D^{4m-d}\times M)$ with underlying stable vector bundle $\xi $ such that

$$ \begin{align*} & \left\langle {p_I(TM\oplus-\xi)\cup p_{4m-|I|}(TM\oplus-\xi), [S^{4m-d}\times M]} \right\rangle \\ &\quad\text{ and }\quad \left\langle {p_m(TM\oplus-\xi), [S^{4m-d}\times M]} \right\rangle \end{align*} $$

are the only nonvanishing monomial Pontryagin numbers of $TM\oplus -\xi $ and $\sigma (\eta )=0$ .

This lemma can be used to prove the following.

Corollary 3.10. There exists a manifold M of dimension $d\ge 8$ divisible by $4$ which is nonconnected but has simply connected components such that

$$\begin{align*}\dim\widetilde{\mathrm{Fib}}^h_{M,4m-d}= p(m) - p(m, m- \left\lceil {\frac{d+1}{4}} \right\rceil )-1.\end{align*}$$

Proof. For $\ell \le m$ , let $P_\ell $ denote the set of partitions of $\ell $ and let . We now construct for each $I\in P$ a manifold $M_I$ satisfying the hypothesis for Lemma 3.9. This will be done by induction on $|I|$ .

For $|I|=1$ , we have $I=(1)$ , and hence, $\mathbb {CP}^{2}\times S^{d-4}$ does the trick.

If $|I|\ge 2$ , let $N_I\times S^{d-|I|}$ be a simply connected manifold of dimension $|I|$ whose only nontrivial monomial Pontryagin number is $p_I$ . Now, we can make all Pontryagin classes of $M_I$ of lower degree linearly independent, and we take connected sums with $M_J$ for $|J|<|I|$ .

We define $M=\amalg _{I\in P} M_I$ . An application of Lemma 3.9 now yields a that for each partition $I\cup \{m-|I|\}$ of m, there exists a manifold $M_I$ and an $M_I$ -block bundle such that $p_{I}\cup p_{m-|I|}$ and $p_m$ are its only nontrivial Pontryagin numbers. Taking the trivial bundle for all other components of M hence yields an M-block bundle with the same property.

Remark 3.11. If $\dim (M)\ge 8$ is divisible by four, then we can even assume that every component of M is $3$ -connected. This is clear for all components arising as products of $4n$ -manifolds and spheres if $n\ge 2$ , as one can choose them to be spin and simply perform surgeries to make them $3$ -connected. Finally, one needs to replace $\mathbb {CP}^{2}\times S^{d-4}$ by the product $X\times S^{d-8}$ for X a $3$ -connected, nullbordant $8$ -manifold with nonvanishing first Pontryagin class. Such a manifold can be constructed, for example, as the linear $S^4$ -bundle over $S^4$ for which the associated vector bundle $V\to S^4$ of rank $5$ has nontrivial first Pontryagin class.

3.4 Pontryagin numbers for topological bundles

Let us now investigate what happens if we leave the smooth world and ask about $(h-)$ sphericity of Pontryagin classes for topological bundles. Obviously, if a class is h-spherical, then it is h-spherical for topological bundles as well.

Now, let $\pi \colon E\to S^k$ be a fiber homotopy trivial, topological M-bundle and let $\rho \colon M\times S^k\to E$ be a homotopy equivalence over $S^k$ . Then we obtain an isomorphism $\rho ^*\colon H^*(E)\overset {\cong }\longrightarrow H^{\ast }(M\times S^k)$ of rational cohomology rings. Therefore, $H^\ast (E)\cong H^\ast (M) \oplus u_k\cdot H^{\ast -k}(M)$ for $u_k$ the cohomological fundamental class of $S^k$ . Therefore, we have $\rho ^*p_i(E) = \operatorname {pr}_M^*p_i(M) + u_k\cdot a_i$ for some $a_i\in H^*(M)$ . Since $u_k^2=0$ , we have

$$ \begin{align*} \rho^*(p_{i_1}\cdots p_{i_r}(E)) = \prod_{\ell=1}^r(\operatorname{pr}^*p_{i_\ell}(M) + u_k\cdot a_{i_\ell}) = \sum_{\ell=1}^r p_{i_1}\cdots\widehat{p_{i_\ell}}\cdots p_{i_r}(M)\cdot u_k\cdot a_{i_\ell}. \end{align*} $$

If $p_{i_1}\cdots \widehat {p_{i_\ell }}\cdots p_{i_r}(M)=0$ for all $\ell \in \{1\dots r\}$ , then this enforces $p_{i_1}\cdots p_{i_r}(E)=0$ , and we obtain the same necessary criterion for sphericity as in Theorem A.

Therefore, if M admits a smooth structure, we have that Theorem A holds for topological M-bundles as well. However, if M is only a topological manifold, one would need a more thorough analysis of topological block bundles and their unblocking in order to prove the analogue of Theorem A.

4.1 The space $\mathcal {R}_C(M)$

Let us now explain the first application of our result to spaces of metrics satisfying positive curvature conditions.

It is well known that the ${\hat {\mathcal {A}}}$ -genus is not multiplicative in fiber bundles: In [Reference Hanke, Schick and SteimleHSS14, Proposition 1.10], Hanke–Schick–Steimle constructed a fiber bundle $E\to S^k$ with ${\hat {\mathcal {A}}}(E)\not =0$ . Their construction however ‘is based on abstract existence results in differential topology [and] does not yield an explicit description of the diffeomorphism type of the […] manifold’ [loc. cit. p. 3]. This has been resolved by Krannich–Kupers–Randal-Williams in [Reference Krannich, Kupers and Randal-WilliamsKKR21], where they constructed a fiber bundle $\mathbb {HP}^{2}\to E\to S^4$ with ${\hat {\mathcal {A}}}(E)\not =0$ . Employing part (ii) of Theorem A together with the computational result [Reference Frenck and ReinholdFR21, Lemma 2.5], we can go far beyond their result.

Next, let us investigate if the bundles we constructed have cross-sections with trivial normal bundle. This is sometimes desirable for applications to positive curvature, as it allows for fiber-wise connected sums. We have the following result.

Proof. Let $\mathrm {triv}\colon S^k\hookrightarrow S^k\times M$ be the trivial section. Since the bundle E from Proposition 4.1 is fiber homotopy trivial via $f\colon S^k\times M\simeq E$ , we get a section . We have

$$ \begin{align*} s^*p_n(TE) & = \mathrm{triv}^*\left(\sum_{i=0}^n p_i(TM)\cup p_{n-i}(-\xi')\right)\\ & = \sum_{i=0}^n \underbrace{\mathrm{triv}^*p_i(TM)}_{=0 \text{ for } i\ge1}\cup \mathrm{triv}^*p_{n-i}(-\xi') = \mathrm{triv}^* p_n(-\xi') \end{align*} $$

with $\xi '$ as in the proof of Lemma 3.2. Recall that the only nonvanishing Pontryagin classes of $\xi '$ are $p_{m-i_{\min }}$ and $p_m$ and let $\nu _s$ denote the normal bundle of s. Since the rank of this bundle is bigger than k, the bundle $\nu _s$ is stable in the sense that it is classified by an element in

$$\begin{align*}\pi_k(\operatorname{BO}) = {\mathrm{KO}}^{-k}({\mathrm{pt}})\cong \begin{cases} \mathbb{Z} & \text{ for } k\equiv 0\;(4)\\ \mathbb{Z}/2 & \text{ for } k\equiv 1,2\;(8)\\ 0 & \text{ otherwise}. \end{cases} \end{align*}$$

Since we are only interested in the problem rationally, it suffices to consider the case $k\equiv 0\;(4)$ . It follows that $\nu _s$ is trivial if $p_{k/4}(\nu _s)=0$ , and as $p(S^k)=1$ , the Pontryagin class $p_{k/4}$ of $\nu _s$ satisfies

$$ \begin{align*} p_{k/4}(\nu_s) = p_{k/4}(s^*TE) = s^*p_{k/4}(TE) = \mathrm{triv}^*p_{k/4}(\xi)=0 \end{align*} $$

since by our assumption, $k/4 < \frac {d+k}{4} - i_{\min } = m-i_{\min }$ and $p_{m-i_{\min }}$ and $p_m$ are the only Pontryagin classes of $\xi $ .

4.1.1 Spin-structures and positive (scalar) curvature

Let M be $\operatorname {Spin}$ and let be the classifying space for M-bundles with a $\operatorname {Spin}$ -structure on the vertical tangent bundle.Footnote ⁸ By [Reference EbertEbe06, Lemma 3.3.6], the homotopy fiber of the forgetful map is a $K(\mathbb {Z}/2,1)$ if M is simply connected. Therefore, the induced map

is an isomorphism, and we may assume without loss of generality that the bundles from Section 3 carry a $\operatorname {Spin}$ -structure on the vertical tangent bundle and hence on the total space, provided that M admits one. Theorem E then follows from Proposition 4.1 by a standard argument that goes back to Hitchin [Reference HitchinHit74] (see [Reference Hanke, Schick and SteimleHSS14, Remark 1.5] or [Reference Frenck and ReinholdFR21, Proposition 3.7]).

Example 4.5.

1. (i) The class of manifolds to which Theorem E is applicable contains $\mathbb {CP}^{2n+1}$ , $\mathbb {HP}^{n}$ , $\mathbb {OP}^{2}$ , as well as iterated products and connected sums of these with arbitrary $\operatorname {Spin}$ -manifolds.

2. (ii) The most interesting examples of spaces $\mathcal {R}_C(M)$ are the ones of positive or nonnegative sectional curvature and positive Ricci curvature metrics. Theorem E implies that for M and $k\ge 2$ as above, we have $\pi _{k-1}(\mathcal {R}_C(M))\otimes \mathbb {Q}\not =0$ for $C=\{{\mathrm {Sec}>0}, {\mathrm {Ric}>0},{\mathrm {Sec}\ge 0}\}$ ,provided the respective spaces are nonempty. The case of nonnegative sectional curvature follows from a Ricci-flow argument; see [Reference Frenck and ReinholdFR21, Proposition 3.3].

According to [Reference ZillerZil14], the only known examples of positively curved manifolds in dimensions $4k+3$ for $k\ge 2$ are spheres. Also, all $7$ -dimensional examples have finite fourth cohomology (cf. [Reference EschenburgEsc92; Reference GoetteGoe14; Reference Goette, Kitchloo and ShankarGKS04]). Therefore, the answer to the following question appears to be unknown. A positive answer would yield the first example of a closed manifold that admits infinitely many pairwise non-isotopic metrics of positive sectional curvature.

Question 4.6. Is there a positively curved manifold of dimension $4k+3$ , $k\ge 1$ with a nonvanishing rational Pontryagin class?

For positive Ricci and nonnegative sectional curvature, lots of examples for such manifolds are known; see Example 4.5.

Applying Theorem E, we get the following classification for the push-forward action on metrics of positive scalar curvature which uses rigidity results from [Reference Ebert and Randal-WilliamsER22] and [Reference FrenckFre21].

Corollary 4.7. Let M be a 2-connected,Footnote ⁹ d-dimensional $\operatorname {Spin}$ -manifold of positive scalar curvature and let k be in the unblocking range for M.

1. (i) Then the orbit map factors through a finite group if and only if $(d+k)$ is not divisible by four or all rational Pontryagin classes of M vanish.

2. (ii) If $k=1$ , then the same holds for instead of .

Proof. The orbit maps

factor through finite groups if all Pontryagin classes of M vanish by [Reference Ebert and Randal-WilliamsER22, Theorem F]. Furthermore, factors through a finite group by [Reference FrenckFre21, Theorem A]. The rest follows from Theorem E.

Theorem E also allows to recover the main result from [Reference Hanke, Schick and SteimleHSS14] (loc.cit. Theorem 1.1 a) and is actually slightly more precise on the dimension restriction.

Corollary 4.8. Let $k\ge 1$ and let N be a $\operatorname {Spin}$ -manifold of positive scalar curvature such that $d+k\equiv 0\;(4)$ and k is in the unblocking range for N. Then the group $\pi _{k-1}(\mathcal {R}_{\mathrm {scal}>0}(N))$ contains an element of infinite order (resp. is infinite if $k=1$ ).

Proof. Let K be a $K3$ -surface. Then for , the manifold $K\times S^n$ satisfies the hypothesis of Theorem E and there is a $K\times S^n$ -bundle $E\to S^k$ that has nonvanishing $\hat {\mathcal {A}}$ -genus and admits a cross-section with trivial normal bundle. Gluing in the trivial $N\setminus D^d$ -bundle along this cross-section yields an $N\#(K\times S^{d-4})$ -bundle over $S^k$ with nonvanishing $\hat {\mathcal {A}}$ -genus. Hence, the group $\pi _{k-1}(\mathcal {R}_{\mathrm {scal}>0}(N\#(K\times S^{d-4})))$ contains an element of infinite order. Since N is cobordant to $N\#(K\times S^{d-4})$ in $\Omega _{\operatorname {Spin}}^d(B\pi _1(N))$ , the corresponding spaces of positive scalar curvature metrics are homotopy equivalent by [Reference Ebert and FrenckEF21, Theorem 1.5].

Remark 4.9. A more general result without any dimension restriction has been proven by Botvinnik–Ebert–Randal-Williams [Reference Botvinnik, Ebert and Randal-WilliamsBER17]. The methods from loc.cit. are, however, not constructive and do not give a way to decide if the obtained elements arise from the orbit of the action . Furthermore, it is unclear if those elements originate from the spaces $\mathcal {R}_{\mathrm {Ric}>0}(M)$ or $\mathcal {R}_{\mathrm {Sec}>0}(M)$ .

4.2 Obstructions to unblocking

One observation from the proofs in Section 3 is that the construction of block bundles works regardless of the dimension k of the base sphere. The same cannot be true for actual fiber bundles over spheres since the tangent bundle of the total space is stably isomorphic to its vertical tangent bundle which is a vector bundle of rank d. Therefore, all Pontryagin classes of degree $*>2d$ vanish. Let $I=(i_1,\dots ,i_s)$ be such that and let M be a manifold such that . If $I=(0)$ , assume instead that M has some nonvanishing rational Pontryagin class. Let

be the map sending a block bundle E classified by f to the Pontryagin number $ \left \langle {p_I(TE)\cup p_{m-|I|}(TE),[E]} \right \rangle $ . The following corollary follows from the proof of Theorem A and Theorem B in Section 3.

4.3 Sphericity of $\kappa $ -classes

Next, consider the group of orientation preserving diffeomorphisms of M equipped with the Whitney $C^\infty $ -topology and the associated classifying space. Since M is assumed to be closed, for any M-bundle $\pi \colon E\to B$ with structure group , there is a map

$$\begin{align*}H^{4m}(\operatorname{BO}(d);\mathbb{Q}) \to H^{{4m}-d}(B;\mathbb{Q})\end{align*}$$

sending a characteristic class $c\in H^{4m}(\operatorname {BO}(d);\mathbb {Q})$ to where $\pi _!\colon H^*(E)\to H^{*-d}(B)$ is the Gysin-homomorphism and $T_\pi E$ is the vertical tangent bundle of $\pi $ . $\kappa _c(E)$ is called the generalized Miller-Morita-Mumford class or simply $\kappa $ -class associated to c. For the universal M-bundle , we hence get universal $\kappa $ -classes

Let be the classifying space for fiber homotopy trivial M-bundles. We define the maps $\Psi _{M,m}$ and $\Psi ^h_{M,m}$ as follows:

where $\mathrm {hur}$ denotes the Hurewicz-homomorphism. Obviously, $\ker (\Psi _{M,m})\subset \ker (\Psi _{M,m}^h)$ . Since the signature of a fiber bundle over $S^k$ vanishes, the Hirzebruch $\mathcal {L}$ -class lies in $\ker (\Psi _{M,m})$ for all M and m. The maps $\Psi ^h_{M,m}$ and $\Psi _{M,m}$ are both given by $c\mapsto (f\mapsto \left \langle {f^*\kappa _c,[S^{4m-d}]} \right \rangle )$ , and we have

$$ \begin{align*} \left\langle {f^*\kappa_{c},[S^{4m-d}]} \right\rangle &= \left\langle {\kappa_c(E), [S^{4m-d}]} \right\rangle = \left\langle {\pi_! c(T_{\pi}^sE),[S^{4m-d}]} \right\rangle \\ &= \left\langle {c(T_{\pi}^sE),[E]} \right\rangle = \left\langle {c(\pi^*TS^{4m-d}\oplus T_{\pi}^sE),[E]} \right\rangle = \left\langle {c(TE),[E]} \right\rangle \end{align*} $$

since $TS^{4m-d}$ is stably parallelizable and hence $c(TS^{4m-d}) = 1$ . With this, Theorem A translates to the following.

Furthermore, we obtain the following bounds on the dimension of $\ker (\Psi _{M,m}^h)$ : For $n_{\max }$ and $p(m,\ell )$ as above, we have

$$\begin{align*}p(m, m- \left\lceil {\frac{d+1}{4}} \right\rceil ) +1\le \dim\ker(\Psi_{M,m}^h)\le p(m)-n_{\max},\end{align*}$$

where the upper bound is attained for manifolds for which all Pontryagin classes are contained in the polynomial algebra generated by $p_{i_{\min} }(TM)$ .