1. Introduction
This paper explores SU(2)-bundles over 8-manifolds, aiming for results akin to those about circle bundles over 4-manifolds [Reference Basu and Basu4, Reference Duan and Liang8]. In the case of simply connected 4-manifolds, the results are established by leveraging the classification of simply connected 5-manifolds achieved by Smale [Reference Smale12] and Barden [Reference Barden3].
A circle bundle 
$S^1 \to X \to M$ over a simply connected 4-manifold M is classified by 
$\alpha\in H^2(M)$, the total space 
$X(\alpha)$ is simply connected if α is primitive, and there are only two possibilities of 
$X(\alpha)$ via the classification of simply connected 5-manifolds. Explicitly, we have [Reference Duan and Liang8, Theorem 2]
(i) For every simply connected 4-manifold M, there is a circle bundle α, such that
$X(\alpha)$ is homotopy equivalent to a connected sum of
$S^2 \times S^3$. If M is spin, among primitive α, this is the only possibility.(ii) For a simply connected 4-manifold M which is not spin and a circle bundle α over it,
$X(\alpha)$ is either homotopy equivalent to a connected sum of
$S^2 \times S^3$ or to a connected sum of
$S^2 \times S^3$ and another manifold B. The manifold B is (up to diffeomorphism unique) a non-spin simply connected 5-manifold whose homology is torsion-free, and
${\text{Rank}}(H_2(B))=1$.
The results of Smale and Barden are geometric in nature and do not generalize easily to higher dimensions. Using homotopy theoretic methods, it was possible to construct sphere fibrations [Reference Basu and Ghosh7] over highly connected Poincaré duality complexes possibly by inverting a few primes or in high enough rank. Among these sphere fibrations, the only case where they could be principal bundles was in dimension 8, and the question whether they may be realized as such was left unresolved.
In this paper, we consider principal SU(2)-bundles which are classified up to homotopy by maps to 
${{\mathbb H}} P^\infty\simeq BSU(2)$, noting that 
$SU(2)=S^3$ is the only case apart from the circle where the sphere is a Lie group. The base space of the SU(2)-bundle which is appropriate for making a similar analysis is a highly connected 8-manifold. More precisely, we consider Poincaré duality complexes M (8-dimensional) that are 3-connected. These are obtained by attaching a single 8-cell to a bouquet of 4-spheres. We denote
\begin{equation*}
{\mathcal{P}} {{\mathcal D}}_3^8 = \text{the collection of } 3\text{-connected } 8 \text{-dimensional Poincar\'{e} duality complexes.}
\end{equation*} The notation 
$M_k\in {\mathcal{P}}{{\mathcal D}}_3^8$ assumes that 
${\text{Rank}}(H_4(M_k))=k$. The attaching map of the 8-cell is denoted by 
$L(M_k)\in \pi_7\big((S^4)^{\vee k}\big) \cong \pi_7(S^4)^{\oplus k} \oplus \pi_7(S^7)^{\oplus \binom{k}{2}}$ and is of the form (once we have chosen a basis 
$\{\alpha_1,\dots,\alpha_k\}$ of 
$\pi_4(M_k)\cong {\mathbb{Z}}^k$)
\begin{equation}
L(M_k)= \sum_{1\leq i \lt j\leq k}g_{i,j}[\alpha_i,\alpha_j] + \sum_{i=1}^{k}g_{i,i}\nu_i + \sum_{i=1}^{k}l_i\nu^{\prime}_i.
\end{equation} The matrix 
$\big((g_{i,j})\big)$ is the matrix of the intersection form and, hence, is invertible. The notation νi stands for 
$\alpha_i\circ \nu$ and 
$\nu^{\prime}_i$ for 
$\alpha_i \circ \nu'$. Here 
$\nu\in \pi_7(S^4)$ is the Hopf map, and 
$\nu'\in \pi_7(S^4)$ is the generator for the 
${\mathbb{Z}}/(12)$ factor satisfying 
$[\iota_4,\iota_4] = 2\nu + \nu'$. For such complexes, we consider
\begin{align*}
{\mathcal{P}}(M_k)=&\,\,\text{the set of principal } SU(2) \text{-bundles } E(\psi)\stackrel{\psi}{\to} M_k \text{ such that } E(\psi) \text{ is }\\ &\,\,3\text{-connected}.
\end{align*} As in the simply connected 4-manifold case, we expect to construct SU(2)-bundles over Mk for which the total space is relatively simple in terms of its homotopy type. In this case, the assumption that 
$E(\psi)$ is 3-connected implies that its homology is torsion-free. The bundle ψ is classified by a primitive element 
$\psi \in H^4(M_k)$, which satisfies a criterion (see proposition 4.4). In this context, we first encounter the question of whether 
${\mathcal{P}}(M_k)$ is non-empty. We prove the following theorem (see proposition 4.1 and proposition 5.1):
Theorem A. For 
$k\geq 3$, the set 
${\mathcal{P}}(M_k)$ is non-empty.
For k = 2, there are examples where 
${\mathcal{P}}(M_k)$ is empty (see the discussion before proposition 4.1 and example 5.1). This means that for every principal SU(2)-bundle over such complexes, the total space has non-trivial π 3. The idea here is that the existence of ψ is given by a certain equation in k variables, and solutions exist once k is large enough.
In the case of simply connected 4-manifolds, the first kind of classification of circle bundles is the result of Giblin [Reference Giblin9], which states that,
\begin{equation*} \text{If } X=S^2\times S^2, \text{then } X(\alpha) \simeq S^2 \times S^3\,\,\text{for any primitive } \alpha.\end{equation*}We also have an analogous result in the 8-dimensional case
\begin{equation*} \text{If } \psi \in {\mathcal{P}}(S^4\times S^4), \text{then } E(\psi) \simeq S^4 \times S^7 .\end{equation*} In fact, this fits into a more general framework. We call a manifold 
$M_k\in {\mathcal{P}}{{\mathcal D}}_3^8$ stably trivial if 
$L(M_k)$ is stably null-homotopic (that is the stable homotopy class of 
$L(M_k) : S^7 \to \big(S^4\big)^{\vee k}$ is 0). In terms of (1.1), this means for every i, 
$g_{i,i}-2 l_i \equiv 0 \pmod{24}$. We have the following theorem (see proposition 4.3):
Theorem B. Suppose Mk is stably trivial. Then, for every 
$\psi \in {\mathcal{P}}(M_k)$, 
$E(\psi)\simeq \#^{k-1} S^4\times S^7$, a connected sum of k − 1 copies of 
$S^4\times S^7$.
This directly generalizes the result for circle bundles over simply connected 4-manifolds that are spin (identifying the spin manifolds as those whose attaching map is stably null).
We proceed towards a more general classification of the homotopy type of the space 
$E(\psi)$ for 
$\psi \in {\mathcal{P}}(M_k)$. Let 
${\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$ be the class of 3-connected 11-dimensional Poincaré duality complexes E such that 
$E\setminus \{pt\}\simeq $ a wedge of equal number of copies of S 4 and S 7. We first observe that 
$E(\psi)\in {\mathcal{P}} {{\mathcal D}}_{4,7}^{11}$ (see proposition 3.1), and we try to address the question of the classification of complexes in 
${\mathcal{P}} {{\mathcal D}}_{4,7}^{11}$ up to homotopy equivalence. The homology of such complexes E is given by
\begin{equation*} H_m(E) \cong \begin{cases} {\mathbb{Z}} & m=0, 11 \\
{\mathbb{Z}}^r & m=4,7 \\
0 & \text{otherwise}.\end{cases} \end{equation*} We denote the number r by 
${\text{Rank}}(E)$. The classification works differently for r = 1, and for 
$r\geq 2$. Table 1 lists the various possibilities for r = 1. For 
$r\geq 2$, E is a connected sum of copies of 
$S^4 \times S^7$, and the complexes 
$E_{\lambda,\epsilon,\delta}$ are defined below. Note that
\begin{align*}
\pi_{10}(S^{4}\vee S^{7})&\cong \pi_{10}(S^{4})\oplus \pi_{10}(S^{7})\oplus \pi_{10}(S^{10}) \\
&\cong \big({\mathbb{Z}}/(24)\{x\}\oplus {\mathbb{Z}}/(3)\{y\}\big) \oplus {\mathbb{Z}}/(24)\{\nu_{(7)}\}\oplus{\mathbb{Z}}\{[\iota_4,\iota_7]\}.
\end{align*} Here, 
$x= \nu\circ \nu_{(7)}$ and 
$y= \nu' \circ \nu_{(7)}$. The notation 
$\nu_{(7)}$ stands for the 3-fold suspension of ν explained in the following convention: for a class 
$\xi \in \pi_r(S^n)$, we write 
$\xi_{(s)} = \Sigma^{s-n}\xi \in \pi_{r+s-n}(S^s)$. Let
\begin{equation*} \phi_{\lambda, \epsilon, \delta}= [\iota_{4},\iota_{7}]+ \lambda (\iota_{7}\circ \nu_{(7)}) + \epsilon (\iota_{4}\circ x ) + \delta (\iota_{4}\circ y),
\end{equation*}Homotopy equivalence classes of 
$E_{\lambda, \epsilon, \delta}.$
and define,
\begin{equation*} E_{\lambda, \epsilon, \delta}= (S^{4}\vee S^{7})\cup_{\phi_{\lambda, \epsilon, \delta}}D^{11}.
\end{equation*}The attaching map of the top cell of E takes the form
\begin{equation*} L(E) : S^{10} \to \big(S^4 \vee S^7\big)^{\vee r}.\end{equation*}The stable homotopy class of L(E) lies in
\begin{equation*}\pi_{10}^s \Big(\big(S^4 \vee S^7\big)^{\vee r}\Big) \cong \big({\mathbb{Z}}/(24)\{\nu\} \oplus {\mathbb{Z}}/(2)\{\nu^2\}\big)^{\oplus r}.\end{equation*} This takes the form 
$\lambda_s \beta\circ \nu + \epsilon_s \alpha \circ \nu^2$ for some 
$\beta\in \pi_7\Big(\big(S^4 \vee S^7\big)^{\vee r}\Big)$ and 
$\alpha \in \pi_4\Big(\big(S^4 \vee S^7\big)^{\vee r}\Big)$. Up to a change of basis, we may assume that 
$\lambda_s \mid 24$, and if λs is even, 
$\epsilon_s\in {\mathbb{Z}}/(2)$. These numbers are invariant over the homotopy equivalence class of E and are denoted by 
$\lambda_s(E)$ and 
$\epsilon_s(E)$ (defined only if 
$\lambda_s(E)$ is even). We use these invariants to classify the homotopy types of elements in 
${\mathcal{P}}{{\mathcal D}}_{4,7}^{11}$ (see theorem 2.6).
Theorem C. Let 
$E \in {\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$. Then the homotopy type of E is determined by the following.
(i) If
$\lambda_s(E)$ is even and
$\epsilon_s(E)=0$, then
\begin{equation*}E \simeq \#^{r-1} E_{0,0,0} \# E_{\lambda_s, \epsilon, \delta} \quad \text{where } \epsilon \equiv 0 \pmod{2}.\end{equation*}(ii) If
$\lambda_s(E)$ is even and
$\epsilon_s(E)=1$, then
\begin{equation*}\begin{aligned}
&E\simeq
\#^{r-1} E_{0,0,0} \# E_{\lambda_s, \epsilon, \delta} \quad & &\text{where } \epsilon \equiv 1 \pmod{2}\\
\text{or } &E\simeq \#^{r-2} E_{0,0,0} \#E_{0,1,0} \# E_{\lambda_s, \epsilon, \delta} \quad & &\text{where } \epsilon \equiv 0 \pmod{2}.
\end{aligned}
\end{equation*}(iii) If
$\lambda_s(E)$ is odd, then
\begin{equation*}E \simeq \#^{r-1} E_{0,0,0} \# E_{\lambda_s, \epsilon, \delta} \quad \text{or} \quad E \simeq \#^{r-2} E_{0,0,0} \#E_{0,1,0} \# E_{\lambda_s, \epsilon, \delta}.\end{equation*}
Further given λs, the choices of ϵ and δ are those that are mentioned in table 1.
We see that in the list given in table 1, for certain cases the homotopy type of E is determined by 
$\lambda_s(E)$ and 
$\epsilon_s(E)$. This happens if 
$\lambda_s(E)=$ 0 or 12. We also observe that the homotopy type of 
$\Omega E$ depends only on the rank r. Now, we look at 
$M_k\in {\mathcal{P}}{{\mathcal D}}_3^8$ and try to determine the set of homotopy equivalence classes of 
$E(\psi)$ for 
$\psi\in {\mathcal{P}}(M_k)$. In this process, we determine a formula for 
$\lambda(\psi) := \lambda_s(E(\psi))$ (proposition 3.5), and using this we determine the set of possible values of 
$\lambda_s(\psi)$ for 
$\psi\in {\mathcal{P}}(M_k)$. The stable homotopy class of 
$L(M_k)$ lies in
\begin{equation*}\pi_7^s \Big((S^4 )^{\vee k}\Big) \cong \big({\mathbb{Z}}/(24)\{\nu\} \big)^{\oplus k}.\end{equation*} This takes the form 
$\sigma_s \alpha\circ \nu$ for some 
$\alpha\in \pi_4\big((S^4 )^{\vee k}\big)$, and up to a change of basis for 
$k\geq 2$, 
$\sigma(M_k):=\gcd(\sigma_s,24)$ is an invariant of the stable homotopy type of Mk. Other than k and 
$\sigma(M_k)$, the explicit stable homotopy class of α above yields a linear map 
$\tau : H^4(M_k)\to {\mathbb{Z}}/(24)$ given by 
$\tau(\psi)= \psi(\sigma_s\alpha)$. We use the invariants k, 
$\sigma(M_k)$, τ, and the intersection form to completely determine the possibilities of 
$\lambda(\psi)$ for 
$\psi\in {\mathcal{P}}(M_k)$ (see theorem 3.6, proposition 4.5, theorem 4.6, theorem 4.9, and theorem 5.2).
Theorem D. For any 
$\psi\in {\mathcal{P}}(M_k)$, 
$\lambda(\psi)$ is a multiple of 
$\sigma(M_k)$ 
$\pmod{24}$. Conversely, the multiples of 
$\sigma(M_k)$ that may be achieved are described as follows:
(i) If the intersection form of Mk is odd and
$k\geq 7$, then
$\{\lambda(\psi) \mid \psi \in {\mathcal{P}}(M_k)\}$ equals the set of multiples of
$\sigma(M_k) \pmod{24}$.(ii) If the intersection form of Mk is even, each
$\psi \in {\mathcal{P}}(M_k)$ satisfies
$\epsilon_s(\psi)\equiv 0 \pmod{2}$.(iii) If
$k\geq 7$, there are
$\psi \in {\mathcal{P}}(M_k)$ such that
$\lambda(\psi)=\sigma(M_k)$, and also there are
$\psi \in {\mathcal{P}}(M_k)$ such that
$\lambda(\psi)=3 \sigma(M_k)$.(iv) If
$\sigma(M_k)\equiv 2$, or
$4 \pmod{8}$ for
$k\geq 5$, there is a
$\psi\in {\mathcal{P}}(M_k)$ such that
$\lambda(\psi)\equiv 0 \pmod{8}$ if and only if the complex satisfies hypothesis
$(H_8)$.(v) If
$\sigma(M_k)\equiv 2 \pmod{8}$ for
$k\geq 5$, there is a
$\psi\in {\mathcal{P}}(M_k)$ such that
$\lambda(\psi)\equiv 4 \pmod{8}$ if and only if the complex satisfies hypothesis
$(H_4)$.
For lower values of k, we do not get systematic results like the above. That is, the set 
$\{\lambda(\psi) \mid \psi \in {\mathcal{P}}(M_k)\}$ is not completely determined by 
$\sigma(M_k)$, k, τ, and the intersection form. Theorem D implies that there are certain Mk whose intersection form is even and there is no 
$\psi \in {\mathcal{P}}(M_k)$ such that 
$E(\psi)\simeq \#^{k-1} S^4\times S^7$, however, if the intersection form is odd, then for 
$k\geq 7$, there is a principal bundle 
$SU(2) \to \#^{k-1}(S^4\times S^7) \to M_k$.
1.1. Organization
In §2, we prove a classification result for certain 3-connected 11-dimensional complexes which proves Theorem C. In §3, we prove formulas relating the stable homotopy type of the total space to that of the base using the Chern character. The results for manifolds with even intersection form are proved in §4, and those with odd intersection form in §5.
2. Homotopy classification of certain 3-connected 11-complexes
We study 3-connected 11-dimensional Poincaré duality complexes E such that 
$E \setminus \{pt\}$ is homotopic to a wedge of copies of S 4 and S 7. We write 
${\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$ for the collection of such complexes. Our target in this section is to analyse them up to homotopy equivalence. We show that these are classified by numbers λ, ϵ, and δ which are explained in detail below.
2.1. The rank one case
Let 
$E \in {\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$ be such that 
$E \setminus \{pt\} \simeq S^4 \vee S^7$, that is, 
${\text{Rank}}(H_4(E))=1$. The homotopy type of E is determined by the attaching map of the top cell, which is an element of
\begin{equation*}\pi_{10}(S^{4}\vee S^{7})\cong \pi_{10}(S^{4})\oplus \pi_{10}(S^{7})\oplus \pi_{10}(S^{10}).\end{equation*}This must be of the form
\begin{equation}
\phi_{\lambda, \epsilon, \delta}= [\iota_{4},\iota_{7}]+ \lambda (\iota_{7}\circ \nu_{(7)}) + \epsilon (\iota_{4}\circ x ) + \delta (\iota_{4}\circ y).
\end{equation} The total space associated with 
$\phi_{\lambda, \epsilon, \delta}$ is denoted by
\begin{equation*} E_{\lambda, \epsilon, \delta}= (S^{4}\vee S^{7})\cup_{\phi_{\lambda, \epsilon, \delta}}D^{11}.
\end{equation*} Note that as 
$(-\iota_4)\circ \nu= \nu + \nu'$, we have 
$(-\iota_4)\circ x= x+ y.$ For given any 
$\lambda,\epsilon$, and δ, we observe the effect of the self-homotopy equivalences on 
$E_{\lambda, \epsilon, \delta}$ as follows:
\begin{equation}
\begin{aligned}
& \iota_4\mapsto -\iota_4,\quad \iota_7\mapsto -\iota_7 & &\implies E_{\lambda,\epsilon, \delta}\xrightarrow[]{\simeq} E_{-\lambda, \epsilon,\epsilon -\delta},\\
&\iota_{4} \mapsto \iota_{4}, \quad \iota_{7}\mapsto \iota_{7} + a\iota_{4}\circ \nu & &\implies E_{\lambda,\epsilon, \delta}\xrightarrow[]{\simeq} E_{\lambda, \epsilon + (\lambda +2)a,\delta}, \nonumber \\
&\iota_{4} \mapsto \iota_{4}, \quad \iota_{7}\mapsto \iota_{7} + b\iota_{4}\circ \nu' & &\implies E_{\lambda,\epsilon, \delta}\xrightarrow[]{\simeq} E_{\lambda, \epsilon-4b ,\delta + (\lambda+ 1)b}. \nonumber
\end{aligned}
\end{equation} This leads us to homotopy equivalences between 
$E_{\lambda, \epsilon, \delta}$’s depending on the choice of 
$\lambda \in \pi_{10}(S^7) \cong {\mathbb{Z}}/{24}$. Table 1 lists the different homotopy types in 
${\mathcal{P}} {{\mathcal D}}^{11}_{4,7}$ of rank 1.
2.2. A simplification of the ataching map
We simplify and reduce the attaching map of the top cell of E.
Proposition 2.1. Let 
$E \in {\mathcal{P}}{{\mathcal D}}_{4,7}^{11}$ with 
${\text{Rank}}(E)=k-1$. The attaching map ϕ of the top cell of E as in (2.1) can be reduced, up to homotopy, to the following form
\begin{equation*}\phi= \sum_{i=1}^{k-1}[\iota_4^i,\iota_7^i]+ \sum_{i=1}^{k-1}\lambda_i \iota_7^i\circ \nu_{(7)}+ \sum_{i=1}^{k-1}s_i\nu_i\circ \nu_{(7)}+ \sum_{i=1}^{k-1}r_i\nu_i'\circ \nu_{(7)}.\end{equation*}Proof. By Hilton–Milnor decomposition, we have the following equivalence
\begin{align*}
\pi_{10}((S^4\vee S^7)^{\vee k-1})\cong \pi_{10}(S^4) ^{\oplus (k-1)} & \oplus \pi_{10}(S^7)^{\oplus (k-1)} \oplus \pi_{10}(S^7) ^{\oplus {k-1\choose 2}} \oplus \\
& \pi_{10}(S^{10})^{\oplus {(k-1)\times (k-1)}} \oplus \pi_{10}(S^{10}) ^{\oplus {k-1 \choose 3}}.
\end{align*} We choose 
$\eta_1,\dots, \eta_{k-1}\in \pi_4(E)$ and 
$\gamma_1,\dots, \gamma_{k-1}\in \pi_7(E)$ such that they correspond to the homology generators, say 
$\tilde{\eta}_1,\dots, \tilde{\eta}_{k-1}\in H_4(E)$ and 
$\tilde{\gamma}_1,\dots, \tilde{\gamma}_{k-1}\in H_7(E)$ such that
\begin{equation}
\tilde{\eta}_i^* \cup \tilde{\gamma}_j^*= \begin{cases}
1 \quad \text{if } i=j,\\
0 \quad \text{if } i \neq j.
\end{cases}
\end{equation} Let 
$\tilde{f} \colon (S^4\vee S^7)^{\vee k-1} \to E$ be the inclusion that sends 
$\iota_4^i \mapsto \eta_i$ and 
$\iota_7^i \mapsto \gamma_i$ for 
$1 \leq i \leq k-1$. Then 
$\tilde{f}\circ \phi\in\pi_{10}(E) $ whose image under the map 
$\rho\colon \pi_{10}(E)\rightarrow \pi_9(\Omega E)\rightarrow H_9(\Omega E)$ is 0 in the tensor algebra 
$T(\tilde{\eta}_1,\dots, \tilde{\eta}_{k-1}, \tilde{\gamma}_1,\dots, \tilde{\gamma}_{k-1})/(\sum_{i=1}^{k-1}[\tilde{\eta}_i,\tilde{\gamma}_i])$.
The attaching map ϕ may contain triple Whitehead products as 
$[\iota^4_i,[\iota^4_j,\iota^4_{\ell}]]$, the Whitehead products of the form 
$[\iota_i^4,\iota_j^7]$ for i ≠ j, the terms involving 
$[\iota_i^4,\iota_j^4]\circ \nu_{(7)}$ and 
$[\iota_i^4,\iota_j^4]\circ \nu_{(7)}'$. The triple Whitehead product maps injectively to the loop homology and hence they cannot occur in the attaching map. The cup product formula in (2.2) implies that there is no Whitehead product of the form 
$[\iota_i^4,\iota_j^7]$ for i ≠ j. If 
$[\iota_i^4,\iota_j^4]\circ \nu_{(7)}$ and 
$[\iota_i^4,\iota_j^4]\circ \nu_{(7)}'$ appear in the attaching map, we update the map 
$\tilde{f}$ by appropriately sending 
$\iota_i^7 \mapsto \gamma_i-\eta_j\circ\nu'$, 
$\iota_i^7 \mapsto \gamma_i-\eta_j\circ\nu$ and 
$\iota_j^7 \mapsto \gamma_j- [\eta_i,\eta_j]$ to get the desired form of the attaching map.
We note that the composition is given by
\begin{equation*}S^{10} \to (S^4 \vee S^7)^{\vee k-1} \to (S^7)^{\vee k-1}\end{equation*} which is an element of 
$(\pi_{10} S^7)^{\oplus k-1} \cong ({\mathbb{Z}}/{24} \{\nu\})^{\oplus k-1}$. This can be calculated using the real e-invariant, see [Reference Adams2]. We use this to reduce proposition 2.1 to the case 
$\lambda_i=0$ for 
$i \leq k-2$.
Proposition 2.2. Let 
$E \in {\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$ and 
${\text{Rank}}(E)=k-1$. Then the attaching map ϕ of the top cell of E can be reduced to the following form
\begin{equation}
\phi= \sum_{i=1}^{k-1}[\iota_4^i,\iota_7^i]+ \lambda \iota_7^{k-1}\circ \nu_{(7)}+ \sum_{i=1}^{k-1}\epsilon_i\nu_i\circ \nu_{(7)}+ \sum_{i=1}^{k-1}\delta_i\nu_i'\circ \nu_{(7)}.
\end{equation} As a consequence, 
$E \simeq \#_{i=1}^{k-2} E_{0, \epsilon_i, \delta_i} \# E_{\lambda, \epsilon_{k-1} , \delta_{k-1}}$.
Proof. Let 
$\tau \colon H^{7}(E)\cong\mathbb{Z}\{\tilde{\gamma}_1^*,\tilde{\gamma}_2^*,\dots,\tilde{\gamma}_{k-1}^*\} \rightarrow \mathbb{Z}/24 $ be the linear map defined by
\begin{equation*}\tau(\tilde{\gamma}_i^*)= e(r_i\circ \phi), \quad \quad \text{for } 1\leq i\leq k-1,\end{equation*} where e denotes for the real e-invariant and 
$r_i:(S^4 \vee S^7)^{\vee k-1} \to S^7$ is the retraction onto the i-th factor. Let 
$\tilde{\tau}\colon H^{7}(E)\rightarrow \mathbb{Z}$ be the lift of τ and 
$\lambda= \gcd \big((\tilde{\tau}(\tilde{\gamma}_1^*),\dots,\tilde{\tau}(\tilde{\gamma}_{k-1}^*)\big)$. Then we can change the generators 
$\tilde{\gamma}_1,\dots, \tilde{\gamma}_{k-1}$ such that 
$\tilde{\tau}(\tilde{\gamma}_i)=0$ for 
$1 \leq i \lt k-1$ and 
$\tilde{\tau}(\tilde{\gamma}_{k-1})= \lambda.$ So, for a suitable choice of dual bases, we can have ϕ as in (2.3).
2.3. A general classification up to homotopy
We now proceed to the classification of elements in 
${\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$. In the connected sum 
$E_{\lambda_1, \epsilon_1, \delta_1} \# E_{\lambda_2, \epsilon_2, \delta_2}$, we transform 
$\alpha_1= \alpha_1^{\prime} + \alpha_2^{\prime}$ and 
$\alpha_2= \alpha_2^{\prime}$ by 
$A=\begin{pmatrix}
1 & 1\\ 0& 1
\end{pmatrix}$. If 
$2|\lambda_2$, then we transform
\begin{equation*}\beta_1= \beta_1^{\prime}-\epsilon_1\nu_2- \frac{\lambda_2\epsilon_1}{2}\nu_2^{\prime} , \quad \beta_2 = -\beta_1^{\prime}+\beta_2^{\prime}-\epsilon_1[\alpha_1^{\prime},\alpha_2^{\prime}].\end{equation*}Hence by the following expression
\begin{align*}
&[\alpha_1,\beta_1]+[\alpha_2,\beta_2]+ \lambda_1\beta_1\circ\nu_{(7)}+ \lambda_2\beta_2\circ \nu_{(7)}+ \epsilon_1 x_1 + \epsilon_2 x_2+\delta_1 y_1+ \delta_2 y_2 \\
=&[\alpha_1^{\prime},\beta_1^{\prime}]+ [\alpha_2^{\prime},\beta_2^{\prime}] +(\lambda_1-\lambda_2)\beta_1^{\prime}\circ\nu_{(7)}+ \lambda_2\beta_2^{\prime}\circ\nu_{(7)} +\epsilon_1x_1\\ &\quad +(-\epsilon_1+\epsilon_2 + 2\lambda_2\epsilon_1-\lambda_1\epsilon_1)x_2 \\
&\qquad\qquad\quad\qquad \delta_1 y_1+ (\delta_1+\delta_2+\lambda_2\epsilon_1(1+\lambda_1))y_2,
\end{align*}we conclude
\begin{align}
E_{\lambda_1,\epsilon_1,\delta_1} \# E_{\lambda_2,\epsilon_2,\delta_2} \simeq E_{\lambda_1-\lambda_2, \epsilon_1, \delta_1}&\# E_{\lambda_2,\epsilon_2 + \epsilon_1(2 \lambda_2 - \lambda_1-1), \delta_1+\delta_2+ (1 + \lambda_1)\epsilon_1 \lambda_2}\nonumber\\ &\qquad \qquad\qquad\qquad\qquad \text{when } 2|\lambda_2.
\end{align}Proposition 2.3. For any unit 
$a \in {\mathbb{Z}}/24$, we have homotopy equivalence
\begin{equation*}E_{\lambda, \epsilon, \delta} \# E_{0,0,0} \simeq \begin{cases} E_{a\lambda, \epsilon, \delta} \# E_{0,0,0} \quad &\text{if } a \equiv 1 \pmod{3} \\ E_{-a\lambda, \epsilon, \delta} \# E_{0,0,0} \quad &\text{if } a \equiv 2 \pmod{3}.
\end{cases}\end{equation*}Proof. We transform
\begin{align*}
&\alpha_1= a \alpha_1^{\prime} + b \alpha_2^{\prime} & & \beta_1= a\beta_1^{\prime}-24\beta_2^{\prime}-b\epsilon \nu_2- 24b\epsilon\nu_1\\
&\alpha_2= 24 \alpha_1^{\prime} + a \alpha_2^{\prime} & & \beta_2= -b\beta_1^{\prime}+a\beta_2^{\prime}-\epsilon b[\alpha_1^{\prime},\alpha_2^{\prime}] \nonumber
\end{align*} with 
$a^2-24b=1$ and calculate 
$[\alpha_1,\beta_1]+[\alpha_2,\beta_2]+ \lambda \beta \circ\nu_{(7)} + \epsilon x_1 + \delta y_2$. This gives the homotopy equivalence
\begin{equation}
E_{\lambda,\epsilon,\delta}\# E_{0,0,0} \\
\simeq
E_{a\lambda, a^2\epsilon, a \delta +{{a}\choose 2}\epsilon} \# E_{0, -b^2\epsilon -b\lambda \epsilon, b \delta + {{b}\choose 2}\epsilon}.
\end{equation} This proves the proposition except for when 
$\lambda \equiv 0 \pmod{2}$ and 
$\epsilon, b \equiv 1 \pmod{2}$. In that case, we have 
$E_{0,1,0}$ instead of 
$E_{0,0,0}$ in the second component of the connected sum. From (2.4), we get
\begin{equation}
E_{\lambda,\epsilon,\delta} \# E_{0,1,0} \simeq E_{\lambda, \epsilon, \delta}\# E_{0,1 + \epsilon(- \lambda-1), \delta}
\end{equation}which we apply following the equivalence in (2.5) for λ even and ϵ, b odd. This concludes the proof.
The transformations above further simplify the possibilities of 
$E \in {\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$ listed in proposition 2.2.
Proposition 2.4. Let 
$E \in {\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$ and 
${\text{Rank}}(E)=k-1$. Then
\begin{equation*}
E\simeq \#^{k-3}E_{0,0,0}\#E_{0,\hat{\epsilon},0}\#E_{\lambda,\epsilon,\delta}
\end{equation*} for some 
$\lambda,\epsilon \in \mathbb{Z}/24$, 
$\delta\in {\mathbb{Z}}/3$, 
$\hat{\epsilon} \in {\mathbb{Z}}/2$.
Proof. From proposition 2.2, we have the attaching map ϕ as in (2.3). The repeated use of the homotopy equivalences in (2.2) and (2.6) gives the reduced form (2.3) as follows
\begin{equation*}\phi= \sum_{i=1}^{k-1}[\iota_4^i,\iota_7^i]+ \lambda \iota_7^{k-1}\circ \nu_{(7)}+ \hat{\epsilon}x_{k-2}+ \epsilon x_{k-1} + \delta y_{k-1},\end{equation*} where 
$\lambda,\epsilon \in \mathbb{Z}/24$, 
$\delta\in {\mathbb{Z}}/3$, 
$\hat{\epsilon} \in {\mathbb{Z}}/2$.
Corollary 2.5. Let 
$E \in {\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$ and 
${\text{Rank}}(E)=k-1$. Then stably we have
\begin{equation}
\Sigma^{\infty}E\simeq \Sigma^{\infty}(S^4\vee S^7)^{\vee {k-2}}\vee \Sigma^{\infty}Cone(\lambda_s(E) \nu_{(11)} +\epsilon_s(E) x ),
\end{equation} where x is the generator of the stable homotopy group 
$\pi_{14}(S^8)\cong \mathbb{Z}/2$ and 
$\lambda_s(E)\in \mathbb{Z}/24$, 
$\epsilon_s(E)\in \mathbb{Z}/2.$ Moreover, for 
$k-1\geq 2$ if 
$\lambda_s(E) \equiv 1 \pmod{2}$ in (2.7), then 
$\epsilon_s(E)=0\in \mathbb{Z}/2$.
Proof. From (2.3) and proposition 2.4,
\begin{equation*}
\Sigma^4\phi= \lambda\iota_{11}^{k-1}\circ \nu_{(11)}+ \hat{\epsilon}\iota_{8}^{k-2}\circ \nu_{(11)}+\epsilon\iota_8^{k-1}\circ \nu_{(11)}.\end{equation*} Note that 
$\nu_{(11)}\in \pi^{s}_{3}(S^0)\simeq \mathbb{Z}/24$ and 
$\Sigma^4 (\nu\circ \nu_{(7)})\in \pi^{s}_{6}(S^0)\simeq \mathbb{Z}/2$ are the generators. If 
$\hat{\epsilon}=0\in \mathbb{Z}/2$, the result readily follows. Otherwise, if 
$\epsilon=1,\hat{\epsilon}=1\in \mathbb{Z}/2$, we apply the transformation 
$\iota_8^{k-2}+\iota^{k-1}_{8}\mapsto \iota^{k-1}_8.$ If 
$\epsilon=0,\hat{\epsilon}=1\in \mathbb{Z}/2$, we interchange 
$\iota_8^{k-2}$ and 
$\iota_8^{k-1}$ to deduce the result.
We see that the stable homotopy type of 
$E \in {\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$ is determined by 
$\lambda_s(E)$ which is a divisor of 24 and 
$\epsilon_s(E) \in {\mathbb{Z}}/2$. The following theorem classifies the different homotopy types of E given the values of λs and ϵs.
Theorem 2.6. Let 
$E \in {\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$ and 
${\text{Rank}}(E)= k-1$. Then depending on 
$\lambda_s=\lambda_s(E)$ and 
$\epsilon_s=\epsilon_s(E)$, the homotopy type of E is determined by the following.
(i) If λs is even and
$\epsilon_s=0$, then
\begin{equation*}E \simeq \#^{k-2} E_{0,0,0} \# E_{\lambda_s, \epsilon, \delta} \quad \text{where } \epsilon \equiv \epsilon_s \pmod{2}.\end{equation*}(ii) If λs is even and
$\epsilon_s=1$, then
\begin{equation*}
\begin{aligned}
&E\simeq
\#^{k-2} E_{0,0,0} \# E_{\lambda_s, \epsilon, \delta} \quad & &\text{where } \epsilon \equiv 1 \pmod{2}\\
\text{or } &E\simeq \#^{k-3} E_{0,0,0} \#E_{0,1,0} \# E_{\lambda_s, \epsilon, \delta} \quad & &\text{where } \epsilon \equiv 0 \pmod{2}.
\end{aligned}
\end{equation*}(iii) If λs is odd, then
\begin{equation*}E \simeq \#^{k-2} E_{0,0,0} \# E_{\lambda_s, \epsilon, \delta} \quad \text{or} \quad E \simeq \#^{k-3} E_{0,0,0} \#E_{0,1,0} \# E_{\lambda_s, \epsilon, \delta}.\end{equation*}
Further given λs, the choices of ϵ and δ are those that are mentioned in table 1.
Proof. We write 
$Y_1=\#^{k-2} E_{0,0,0} \# E_{\lambda, \epsilon, \delta}$ and 
$Y_2=\#^{k-2} E_{0,0,0} \# E_{\lambda, \epsilon', \delta'}$, and let 
$Y_1 \stackrel{f}{\to} Y_2$ be a homotopy equivalence with homotopy inverse g. We show that in this case, the pair 
$(\epsilon,\delta)$ is related to 
$(\epsilon',\delta')$ by the transformations (2.2). There exists a unique (up to homotopy) factorization 
$(S^7)^{\vee k-1} \to (S^7 \vee S^4)^{\vee k-1} \to Y_2$ through cellular approximation 
$\tilde{f}$ as in the following diagram.
Now we consider the composition 
$f_7\colon =\rho \circ \tilde{f}$, where ρ is the projection map.
From the stable homotopy type, we get an isomorphism
where 
$f_7(\beta_{k-1})\equiv\gamma_{k-1}\pmod{24}$. Hence, the corresponding matrix of f 7 is
\begin{equation*}\equiv \begin{pmatrix}
& & & 0\\ & A & & \vdots\\ & & & 0 \\ * & \dots & * & 1\\
\end{pmatrix} \pmod{24}\end{equation*} for some 
$A \in GL_{k-2} ({\mathbb{Z}})$. From the inverse homotopy equivalence 
$g \colon Y_2 \to Y_1$, we can construct a corresponding block matrix of g 7 similar to that of f 7 for some 
$B \in GL_{k-2}({\mathbb{Z}})$ 
$\pmod{24}$. Through suitable pre-composition of f 7 and post-composition of g 7, we may assume that 
$A=B=I$ where I is the identity matrix in 
$GL_{k-2} ({\mathbb{Z}})$. Thus both f 7 and g 7 are composition of shearing maps with 
$\beta_{k-1} \mapsto \gamma_{k-1}$, that is, they are composition of maps associated with 
$\beta_i \mapsto \gamma_i + c_i \gamma_{k-1}$ for some 
$c_i \in {\mathbb{Z}}$.
We now consider the map f on the 7-skeleton
which takes the form
\begin{align*}
&\beta_{i}\mapsto \gamma_i + c_i \gamma_{k-1} + \sum_{j=1}^{k-1} a_{i,j} \nu_j + \sum_{j=1}^{k-1} a_{i,j}'\nu_j'+ \sum_{\substack{j=1\\ \ell=1 \\ j \neq \ell }}^{\substack{j=k-1\\ \ell=k-1}} a_{i,j, \ell}[\eta_j,\eta_{\ell}], \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad \alpha_i \mapsto \eta_i, \quad \text{for } 1 \leq i \leq k-2,\\
&\beta_{k-1} \mapsto \gamma_{k-1} +\sum_{j=1}^{k-1} b_{j} \nu_j + \sum_{j=1}^{k-1} b_{j}'\nu_j'+ \sum_{\substack{j=1\\ \ell=1 \\ j \neq \ell }}^{\substack{j=k-1\\ \ell=k-1}} b_{j ,\ell}[\eta_j,\eta_{\ell}], \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad \alpha_{k-1} \mapsto \eta_{k-1}-\sum_{j=1}^{k-2} c_j \eta_j.
\end{align*} As f is a homotopy equivalence, we must have that the attaching map of the 11-cell of Y 1 must be carried by 
$f^{(7)}$ to the attaching map of Y 2, that is, 
$f^{(7)} \circ L(Y_1) \simeq L(Y_2)$. We now look at the coefficients of 
$\eta_{k-1}\circ x$ and 
$\eta_{k-1}\circ y$ that arises in 
$f^{(7)}\circ L(Y_1)$ and note that the only terms which contribute to these coefficients are
\begin{equation*}f^{(7)}( [\alpha_{k-1},\beta_{k-1}]+\lambda_s \beta_{k-1}\circ \nu_{(7)}+ \epsilon \alpha_{k-1} \circ x+ \delta \alpha_{k-1} \circ y).\end{equation*}We now deduce
\begin{equation*}\epsilon'=(\lambda_s+2)b_{k-1}-4b^{\prime}_{k-1} \quad \quad \text{and} \quad \quad \delta'= (\lambda_s+1)b_{k-1}',\end{equation*} which verifies the result for complexes of the type 
$ \#^{k-2} E_{0,0,0} \# E_{\lambda, \epsilon, \delta}$.
For the remaining cases, we follow the same argument with 
$Y_1 = \#^{k-3} E_{0,0,0} \#E_{0,1,0} \# E_{\lambda_s, \epsilon, \delta}$ with ϵ even if λs is, and 
$Y_2=\#^{k-3} E_{0,0,0}$ 
$\#E_{0,\hat{\epsilon},0} \# E_{\lambda_s, \epsilon', \delta'}$, where 
$\hat{\epsilon}=0$ or 1. Note that the only terms which contribute to 
$\eta_{k-2}\circ x$ are
\begin{equation*}f^{(7)}( [\alpha_{k-1},\beta_{k-1}]+[\alpha_{k-2},\beta_{k-2}]+\alpha_{k-2}\circ x + \lambda_s \beta_{k-1}\circ \nu_{(7)}+ \epsilon \alpha_{k-1} \circ x+ \delta \alpha_{k-1} \circ y).\end{equation*}A direct computation implies
\begin{equation}
\lambda_s b_{k-2}+\epsilon c_{k-2}^2+1\equiv \hat{\epsilon} \pmod{2}.
\end{equation} First, let λs be even and 
$\epsilon\equiv 0\pmod{2}$. This implies 
$\hat{\epsilon}=1$. Finally, let λs is odd. We look at the coefficients of 
$[\eta_{k-1},[\eta_{k-2}, \eta_{k-1}]]$ and 
$[\eta_{k-2}, \eta_{k-1}] \circ \nu_{(7)}$ in 
$f^{(7)}\circ L(Y_1)$ 
$\pmod{2}$, which are 
$b_{k-2,k-1} +c_{k-2} b_{k-1} -a_{k-2, k-1}$ and 
$\lambda_s b_{k-2, k-1} + a_{k-2, k-1} + b_{k-2} -c_{k-2}b_{k-1}-\epsilon c_{k-2}$. Since both these coefficients are zero, we have 
$b_{k-2}\equiv\epsilon c_{k-2}\pmod{2}$. Using the relation (2.8), we observe that 
$\hat{\epsilon}=1$. The conditions on 
$\epsilon'$ and 
$\delta'$ are verified analogously as in the previous case. This completes the proof of the various implications in the theorem.
2.4. The loop space homotopy type
We study the loop space homotopy type of 
$E \in {\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$ with 
$E^{(7)} \simeq (S^4 \vee S^7)^{\vee k-1}$ and show that the loop space homotopy is independent of the 
$\lambda, \epsilon$, and δ occurring in the attaching map ϕ of E.
Theorem 2.7. The homotopy type of the loop space of E is a weak product of loop spaces on spheres and depends only on 
$k-1={\text{Rank}} (H_4(E))$. In particular,
\begin{equation*}\Omega E \simeq \Omega (\#^{k-1} (S^4 \times S^7) ).\end{equation*}Proof. This follows from the arguments in [Reference Basu and Basu5] and [Reference Basu and Basu6]. More explicitly, we first compute the homology of 
$\Omega E$ via cobar construction, see [Reference Adams1]. In this case, 
$H_*(E)$ is a coalgebra which is quasi-isomorphic to 
$C_*(E)$, and hence we may compute the cobar construction of 
$H_*(E)$ and deduce as in [Reference Basu and Basu6, Proposition 2.2]
\begin{equation*}H_*(\Omega E) \cong T(a_1, b_1, \dots, a_{k-1}, b_{k-1})/ (\sum [a_i, b_i])\end{equation*} where 
$\rho(\alpha_i)=a_i$ and 
$\rho(\beta_i)=b_i$ with ρ defined as
\begin{equation*}\rho \colon \pi_r (E) \cong \pi_{r-1} (\Omega E) \xrightarrow{Hur} H_{r-1}(\Omega E).\end{equation*} We then note that 
$H_*(\Omega E)$ is the universal enveloping algebra of the graded Lie algebra 
$L(a_1, b_1, \dots, a_{k-1}, b_{k-1})/ (\sum [a_i, b_i])$, where L is the free Lie algebra functor. Now we apply the Poincaré–Birkhoff–Witt theorem as in [Reference Basu and Basu6, Proposition 3.6] to deduce the result.
3. Stable homotopy type of the total space
In this section, we examine the possible stable homotopy types of the total space E for a principal SU(2)-fibration over 
$M_k \in {\mathcal{P}}{{\mathcal D}}^8_3$. We relate this to the stable homotopy type of Mk.
Let 
$f \colon M_k \to {\mathbb H} P^{\infty}$ be a map such that 
$\pi_4(f)\colon \pi_{4}(M_k)\rightarrow \pi_{4}(\mathbb{H}P^{\infty})\cong \mathbb{Z}$ is surjective. This ensures that the homotopy fibre E(f) is 3-connected and is a Poincaré duality complex of dimension 11. One easily deduces
\begin{equation*} H_i(E(f)) = \begin{cases}
\mathbb{Z} &\quad i= 0,11 \\
\mathbb{Z}^{\oplus{k-1}}&\quad i= 4,7\\
0&\quad\text{otherwise}
\end{cases} \end{equation*} from the Serre spectral sequence associated with the fibration 
$S^3 \to E(f) \to M_k$. We may now consider a minimal CW-complex structure on 
$E:=E(f)$ with 
$(k-1)$ 4-cells, 
$(k-1)$ 7-cells, and one 11-cell, see [Reference Hatcher10, Section 2.2]. The 7-th skeleton 
$E^{(7)}$ is, therefore, a pushout
We now observe that the ϕi are all 0.
Proposition 3.1. The maps 
$\phi_i \simeq 0$ for 
$1 \leq i \leq k-1$.
Proof. Note that the homotopy class of each of the attaching maps is in 
$\pi_6(S^4)^{\oplus {k-1}}$ which lies in the stable range. Applying the 
$\Sigma^{\infty}$ functor on the diagram (3.1), we get the cofibre sequence
\begin{equation*}\Sigma^{\infty}(S^6)^{\vee{(k-1)}}\xrightarrow{\Sigma^{\infty}(\vee_{i=1}^{k-1}\phi_i)} \Sigma^{\infty}(S^4)^{\vee{(k-1)}}\rightarrow \Sigma^{\infty} E^{(7)}\end{equation*}which in turn induces a long exact sequence (on the stable homotopy groups)
\begin{equation}
\dots\rightarrow \pi_{7}^{s}(E^{(7)}) \rightarrow \pi_6^{s}(S^6)^{\oplus k-1} \xrightarrow{\Phi} \pi_6^{s}(S^4)^{\oplus k-1} \rightarrow \pi_{6}^{s}(E^{(7)})\rightarrow \dots
\end{equation} where Φ is the induced map of 
$\vee_{i=1}^{k-1} \phi_i$. We have the following commutative diagram
The second row is a part of a long exact sequence and so, the map 
$\pi_6^s(S^4)^{\oplus k}\to \pi^s_6(M_k)$ is injective. Hence, the map 
$ \pi_6^{s}(S^4)^{\oplus{(k-1)}}\rightarrow \pi_6^{s}(E)$ is injective, in (3.2) Φ is forced to be 0. The result follows.
Proposition 3.1 implies that E fits into the pushout
for some 
$[L(E)]\in \pi_{10}((S^4\vee S^7)^{\vee(k-1)})$. Hence, E belongs to 
${\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$. We consider
which is of the form
\begin{equation*}
\sum_{i=1}^{k-1}\lambda_i \iota^{i}_{7}\circ \nu_{(7)}\in \pi_{10}(S^7) ^{\oplus{k-1}} \cong \bigoplus_{i=1}^{k-1}\mathbb{Z}/24\{\nu_{(7)}\}.\end{equation*} The coefficients λi can be computed via the e-invariant, see [Reference Adams2]. Recall that the e-invariant of a map 
$g \colon S^{11} \to S^8$ can be computed using Chern character. The complex K-theoretic e-invariant 
$e_{{\mathbb C}}$ is computed via the diagram
We obtain
\begin{equation*}ch(b_{12})= a_{12}, \quad ch(b_{8})= a_{8} + r{a}_{12}.\end{equation*} If 
$g=\lambda \nu_{(7)}$, 
$e_{{\mathbb C}}(g)=r=\frac{\lambda}{12} \in {\mathbb Q}/{\mathbb{Z}}$, see [Reference Adams2]. We also have 
$e_{{\mathbb C}}=2e$, where e is computed using the Chern character of the complexification 
$c \colon KO \to K$. Therefore, from [Reference Adams2, Proposition 7.14]
\begin{equation}
b_8 \in Im(c) \implies e(g)=\frac{r}{2} \in {\mathbb Q} / {\mathbb{Z}}.
\end{equation}3.1. K-theory of Mk
Consider the Atiyah–Hirzebruch spectral sequence
\begin{equation*}E_2^{**}= H^{*}(M_k; \pi_*K) \implies K^{*}(M_k).\end{equation*} As M has only even dimensional cells, this has no non-trivial differential for degree reasons. This gives the additive structure of 
$K^{0}(M_k)$. Let 
$H^4(M_k) \cong {\mathbb{Z}} \{\psi_1,\dots, \psi_{k}\}$ and 
$H^8(M_k) \cong {\mathbb{Z}}\{z\}$. Note that if 
$\alpha_1,\dots, \alpha_k\in \pi_4(M_k) \cong H_4(M_k)\cong {\mathbb{Z}}^k$ is dual to the basis 
$\psi_1, \dots, \psi_k$, in the expression (1.1), the matrix 
$\big((g_{i,j})\big)$ of the intersection form is related to the cup product via the equation 
$\psi_i\cup \psi_j = g_{i,j} z$. Let 
$\tilde{\psi_1},\dots, \tilde{\psi_{k}}, and \tilde{z}$ be classes in 
$K(M_k)$ corresponding to 
$\psi_1,\dots, \psi_{k}, and z$ respectively, in the 
$E^{\infty}$-page. We have the diagram
where
\begin{equation}
ch(\tilde{z})= z,\quad ch (i^*(\tilde{\psi_j}))= {\psi_j} (\implies ch(\tilde{\psi_j})= {\psi_j}+ \tau_j z), 1\leq j\leq k.
\end{equation} Note that in terms of the formula (1.1), 
$\psi_i \psi_j=g_{i,j}z$. We now use the fact that ch is a ring map to get
\begin{equation*}ch(\tilde{\psi_i}\tilde{\psi_j})= (\psi_i+\tau_i z)\cup (\psi_j+\tau_j z)= \psi_i\cup \psi_j= g_{i,j}z.\end{equation*} As 
$ch \colon K(M_k) \to H^{ev}(M_k;{\mathbb Q})$ is injective, we deduce 
$\tilde{\psi_i} \tilde{\psi_j}=g_{i,j} \tilde{z}$.
Further let 
$q_i \colon (S^4)^{\vee k} \to S^4$ be the retraction onto the i-th factor, and note that 
$q_i \circ L(M)$ is stably equivalent to 
$(g_{i,i} -2l_i) \alpha_i \circ \nu_{(7)}$. Thus the e-invariant
\begin{equation*}
e_{{\mathbb C}}(q_i\circ L(M))= (g_{i,i}-2l_i)e_{{\mathbb C}}(\Sigma \nu_{(7)})= \frac{g_{i,i}-2l_i}{12}\in \mathbb{Q}/\mathbb{Z}.\end{equation*}We summarize these observations in the following proposition.
Proposition 3.2. Let 
$M_k \in {\mathcal{P}}{{\mathcal D}}^8_3$ with 
$L(M_k)$ as in (1.1). Then, 
${K}^{0}(M_k) \cong {\mathbb{Z}}\{1, \tilde{\psi}_1,\tilde{\psi}_{2},\dots, \tilde{\psi_{k}},\tilde{z} \}$. The ring structure is given by
\begin{equation*}
\tilde{\psi}_i \tilde{\psi}_j= g_{i,j} \tilde{z} \quad \text{for } 1\leq i, j\leq k. \end{equation*}and
\begin{equation*}e_{{\mathbb C}}(q_i\circ L(M))= \frac{g_{i,i}-2l_i}{12}\in \mathbb{Q}/\mathbb{Z} \quad \text{for } 1\leq i \leq k.\end{equation*}3.2. K-theory of E(f)
The space 
$E:=E(f)$ is the total space of the sphere bundle associated with the quaternionic line bundle classified by f. We note that the quaternionic line bundle has a complex structure, and therefore, has a K-orientation.
Let 
$\gamma_{\mathbb{H}}$ be the canonical 
$\mathbb{H}$-bundle over 
$\mathbb{H}P^{\infty}$. The K-theoretic Thom class of 
$\gamma_{\mathbb{H}}$ is given by 
$\gamma_{\mathbb{H}}-2\in \tilde{K}^{0}(\mathbb{H}P^{\infty})$. As the total space of the sphere bundle is contractible, the Thom space 
$Th(\gamma_{{\mathbb H}}) \simeq {\mathbb H} P^{\infty}$. Consider the map 
$\pi \colon \mathbb{C}P^{\infty} \to \mathbb{H}P^{\infty}$. The pullback bundle 
$\pi^*(\gamma_{\mathbb{H}})=\gamma_{{\mathbb C}} \oplus \bar{\gamma}_{{\mathbb C}}$, where 
$\gamma_{{\mathbb C}}$ is the canonical line bundle over 
${{\mathbb C}}P^{\infty}$. Therefore,
\begin{equation*}\pi^*ch(\gamma_{\mathbb{H}}-2) =ch( \lt pi_ \gt \lt /pi_ \gt ^{*}(\gamma_{\mathbb{H}}-2)) = ch(\gamma_{\mathbb{C}})+ ch(\bar{\gamma}_{\mathbb{C}})-2=e^x + e^{-x} -2,\end{equation*} where 
$H^\ast({\mathbb{C}} P^\infty;{\mathbb{Z}}) \cong {\mathbb{Z}}[x]$ and 
$x=c_1(\gamma_{\mathbb{C}})$. Since 
$\pi^*$ is injective on 
$H^*$, we may use this formula to deduce
\begin{equation}
ch(\Phi_{K}(\gamma_{\mathbb{H}}))= \Phi_H(\gamma_{\mathbb{H}})(1+ \frac{y}{12}+ \frac{y2}{360}+\dots)
\end{equation} where 
$H^*({\mathbb H} P^{\infty}) \cong {\mathbb{Z}}[y]$ and 
$\pi^*(y)=x^2$. We use the Thom isomorphism associated to 
$f^*(\gamma_{{\mathbb H}})$ to deduce the following.
Proposition 3.3. Assume that 
$f^*(y)=\psi_k$. Then,
\begin{equation*}\tilde{K}^{0}(Th(f^{*}(\gamma_{\mathbb{H}})))\cong \tilde{K}^{0}(M)\{\Phi_{K}(f^{*}\gamma_{\mathbb{H}})\} \end{equation*} as a 
$\tilde{K}^{0}(M)$-module, and
\begin{equation*}ch(\Phi_{K}(f^{*}\gamma_{\mathbb{H}}))= \Phi_H(f^*\gamma_{\mathbb{H}}) (1 + \frac{\psi_k^{}}{12}+ \frac{\psi_k^{2}}{360}+ \dots).\end{equation*}Proof. From the naturality of the Chern character as well as the Thom class, we have
\begin{equation*}
ch\Big(\Phi_{K}(f^{*}(\gamma_{\mathbb{H}}))\Big)= ch\Big(f^{*}(\Phi_{K}((\gamma_{\mathbb{H}}))\Big)= f^{*}ch(\Phi_{K}(\gamma_{\mathbb{H}})).\end{equation*} The result follows from (3.5) and the fact 
$f^*(y)=\psi_k$.
Notation 3.1.
Suppose we are in the situation of proposition 3.3, that is, 
${f^*(y)=\psi_k}$. We now assume that the basis 
$\{\psi_1, \dots, \psi_k \}$ is such that one of the following cases occur
Case 1 If
$\psi_k\cup \psi_k= \pm z$, then
$\psi_j \psi_k=0$ for
$1\leq j \leq k-1$.Case 2 If
$\psi_k\cup \psi_k= g_{k,k}z$ for some integer
$g_{k,k}\neq \pm 1,$ then assume
$\psi_{k-1} \psi_k=1$ and
$\psi_j \psi_k=0$ for
$1 \leq j \leq k-2$.
In terms of these notations, we prove the following calculation. Here we consider the cofibre sequence
\begin{equation*}E\rightarrow M_k \rightarrow Th(f^*(\gamma_{\mathbb{H}}))\rightarrow \Sigma E\rightarrow \Sigma M\rightarrow \dots, \end{equation*} which demonstrates 
$K(\Sigma E)$ as a submodule of 
$K(Th(f^* \gamma_{{\mathbb H}}))$ because 
$K(\Sigma M_k)=0$. The following proposition identifies this submodule.
Proposition 3.4. (i) Suppose we are in case 1 of notation 3.1, then we have
\begin{equation*}\tilde{K}^{0}(\Sigma E)\cong\mathbb{Z}\{ \Phi_K(f^*(\gamma_{\mathbb{H}}))\tilde{\psi}_1,\dots,\Phi_K(f^*(\gamma_{\mathbb{H}}))\tilde{\psi}_{k-1} , \Phi_K(f^*(\gamma_{\mathbb{H}}))\tilde{z}\}.\end{equation*}(ii) Suppose we are in case 2 of notation 3.1, then
\begin{align*}\tilde{K}^{0}(\Sigma E)&\cong\mathbb{Z}\left\{
\Phi_K(f^*(\gamma_{\mathbb{H}}))\tilde{\psi}_1,\dots,\Phi_K(f^*(\gamma_{\mathbb{H}}))\tilde{\psi}_{k-2} , \Phi_K(f^*(\gamma_{\mathbb{H}}))(\tilde{\psi}_k-g_{k,k}\tilde{\psi}_{k-1}),\right.\\
&\qquad \left.\Phi_K(f^*(\gamma_{\mathbb{H}}))\tilde{z}\right\}.\end{align*}Proof. We have the following short exact sequence
which implies that
\begin{equation*} \tilde{K}^{0}(\Sigma E)= \text{Ker} (s_0\colon \tilde{K}^{0}\Big(Th(f^*(\gamma_{\mathbb{H}}))\Big)\rightarrow \tilde{K}^0(M))\end{equation*} where s 0 is the restriction along the zero section. Note that s 0 is a 
$\tilde{K}^{0}(M)$-module map. Hence,
\begin{align*}
s_0(\Phi_{K}(f^*\gamma_{\mathbb{H}})\tilde{z})&= e_{K}(f^*\gamma_{\mathbb{H}})\tilde{z}=0 \nonumber\\
s_0(\Phi_{K}(f^*\gamma_{\mathbb{H}})\tilde{\psi}_i)&= e_{K}(f^*\gamma_{\mathbb{H}})\tilde{\psi}_i= g_{i k}\tilde{z} , 1\leq i \leq k
\end{align*} since 
$e_{K}(f^*\gamma_{\mathbb{H}})= \tilde{\psi}_k + m\tilde{z} $ for some m. The result follows from a direct calculation of the kernel and the assumptions in the respective cases.
We now choose the various maps 
$\chi_j \colon S^7 \to E$ for 
$1 \leq j \leq k-1$ such that on K-theory they precisely represent the choice of the first 
$(k-1)$-elements in the basis of proposition 3.4. In these terms, we calculate the 
$e_{{\mathbb C}}$-value of the composite
\begin{equation*}r_j \circ L(E) \colon S^{10} \stackrel{L(E)}{\to} (S^4 \vee S^7)^{\vee k-1}\stackrel{r_j}{\to} S^7\end{equation*}where rj is the restriction onto the j-th factor, and L(E) is the attaching map of the top cell of E.
Proposition 3.5. (i) If we are in case 1, then
\begin{equation*}
e_{{\mathbb C}}(r_j\circ L(E))=\tau_j= \frac{g_{j,j}-2l_j}{12}\in \mathbb{Q}/\mathbb{Z}, 1\leq j\leq k-1 .\end{equation*}(ii) If we are in case 2, then
\begin{align*}
e_{{\mathbb C}}(r_j\circ L(E))&= \tau_j= \frac{g_{j,j}-2l_j}{12}\in \mathbb{Q}/\mathbb{Z}, 1\leq j\leq k-2; \quad \\ e_{{\mathbb C}}(r_{k-1}\circ L(E) )&= \tau_k- g_{k,k}\tau_{k-1} \in \mathbb{Q}/\mathbb{Z}.\end{align*}Proof. For the e-invariant, we calculate the Chern character 
$ch\colon \tilde{K}^{0}(Th(f^*\gamma_{\mathbb{H}}))\rightarrow H^{ev}(Th(f^*\gamma_{\mathbb{H}});\mathbb{Q})$ in terms of (3.4) as follows
\begin{align*}
ch(\Phi_{K}(f^*\gamma_{\mathbb{H}})\tilde{z})&= \Phi_H(f^*\gamma_{\mathbb{H}})(1+ \frac{\psi_k}{12} +\dots )z= \Phi_H(f^*\gamma_{\mathbb{H}})z,
\\
ch(\Phi_{K}(f^*\gamma_{\mathbb{H}})\tilde{\psi}_j)&= \Phi_H(f^*\gamma_{\mathbb{H}})(1+ \frac{\psi_k}{12} +\dots )(\psi_j + \tau_j z)\\ &= \Phi_H(f^*\gamma_{\mathbb{H}})\psi_j + \tau_j \Phi_H(f^*\gamma_{\mathbb{H}})z,
\\
ch\Big(\Phi_{K}(f^*\gamma_{\mathbb{H}})(\tilde{\psi}_k-g_{k,k}\tilde{\psi}_{k-1})\Big)&= \Phi_H(f^*\gamma_{\mathbb{H}})(1+ \frac{\psi_k}{12} +\dots )\left(\psi_k +\tau_k z- g_{k,k}\psi_{k-1}\right.\\ &\quad \left. -g_{k,k}\tau_{k-1}z\right)\nonumber\\ &= \Phi_H(f^*\gamma_{\mathbb{H}})(\psi_k -g_{k,k}\psi_{k-1})+ \Phi_H(f^*\gamma_{\mathbb{H}})\\ &\quad\times(\tau_k - g_{k,k}\tau_{k-1})z.
\end{align*}We now turn our attention to the attaching map
\begin{equation*}S^{10} \stackrel{L(E)}{\to} (S^4 \vee S^7)^{\vee k-1}\to{(S^7)}^{\vee k-1}.\end{equation*}In order to identify the composite, we are required to compute the KO-theoretic e-invariant. We know that
\begin{align*}
KO^*= \mathbb{Z}[\eta, u][\mu^{\pm 1}]/(2\eta, \eta^3,\eta u, u^2-4\mu) \quad \text{and} \quad
K^*= \mathbb{Z}[\beta^{\pm 1}],
\end{align*} with 
$|\eta| = -1,~ | u|= -4,~ |\mu| =-8$ and 
$|\beta| = -2$. The complexification map 
$c\colon KO\rightarrow K$ induces a graded ring homomorphism 
$c\colon KO^*(X) \rightarrow K^*(X)$ with
\begin{equation*}
c(\eta)=0, \quad c(u)= 2\beta^2, \quad c(\mu)= \beta^4.\end{equation*}Theorem 3.6. Let 
$S^3\rightarrow E \rightarrow M_k$ be the principal SU(2)-fibration classified by a given map 
$f\colon M_k\rightarrow {\mathbb{HP}}^{\infty}$. Suppose that 
$\Sigma L(M_k) \equiv 0\pmod{\lambda}$. Then 
$\Sigma (r_j\circ L(E)) \equiv 0\pmod{\lambda}, 1\leq j\leq k-1$ where 
$r_j:(S^4 \vee S^7)^{\vee k-1} \to S^7$ is the retraction onto the j-th factor.
Proof. We consider the Atiyah–Hirzebruch spectral sequences for 
${\mathbb H} P^{\infty}$ and Mk
\begin{align*}
E^{*,*}_2&= H^{*}(\mathbb{H}P^{\infty};\mathbb{Z})\otimes KO^*(pt)\implies KO^*({\mathbb{HP}}^{\infty}),\\
E^{*,*}_2&= H^{*}(M_k;\mathbb{Z})\otimes KO^*(pt)\implies KO^*(M_k).
\end{align*} The spectral sequences have no non-trivial differentials for degree reasons. Thus, 
$KO^*({\mathbb{HP}}^{\infty})\cong KO^*[\hat{y}],$ where 
$\hat{y} \in KO^4({\mathbb H} P^\infty)$. The class 
$\hat{y}$ serves as KO-theoretic Thom class for 
$\gamma_{{\mathbb H}}$. We have
\begin{equation*}c (\Phi_{KO}(\gamma_{\mathbb{H}}))= c(\hat{y})= \beta^{-2}\Phi_{K}(\gamma_{\mathbb{H}}).\end{equation*} Let 
$\hat{\psi}_{j}\in \tilde{KO}^{4}(M_k)$, 
$\hat{z} \in \tilde{KO}^{8}(M_k)$ be the class in the 
$E^{\infty}$-page represented by 
$\psi_j \in H^4(M_k)$ and 
$z \in H^8(M_k)$. Then we get
\begin{equation*}c(\hat{\psi}_{i})= \beta^{-2}\tilde{\psi}_{i}, \quad c(\hat{z})= \beta^{-4}\tilde{z} \quad \text{and} \quad \hat{\psi}_{i}\hat{\psi}_{j}=g_{i,j}\hat{z}. \end{equation*} It follows that the K-theoretic generators 
$\Phi_K(f^* \gamma_{{\mathbb H}})\tilde{\psi}_j$ and 
$\Phi_K(f^* \gamma_{{\mathbb H}})(\tilde{\psi}_k-g_{k,k}\tilde{\psi}_{k-1}) $ lie in the image of the map c. Therefore by (3.3), we get in case 2 that
\begin{equation*} e(r_{j}\circ L(E))\equiv
\begin{cases}
\frac{g_{j,j}-2l_j}{24}
&\pmod{\mathbb{Z}} , \quad\text{for } j \lt k-1,\\
\frac{(g_{kk}-2l_k)-g_{k,k}(g_{k-1,k-1}-2l_{k-1})}{24} &\pmod{\mathbb{Z}} ,\quad\text{for } j=k-1;
\end{cases}\end{equation*}and in case 1 that
\begin{equation*}e(r_{j}\circ L(E))\equiv
\frac{g_{j,j}-2l_j}{24}
\pmod{\mathbb{Z}} , \quad\text{for } j\leq k-1.\end{equation*}The result now follows from proposition 3.2.
4. SU(2)-bundles over even complexes
We study the homotopy type of 
$E \in {\mathcal{P}}{{\mathcal D}}_{4,7}^{11}$, the total space of a principal SU(2)-bundle over 
$M_k \in {\mathcal{P}}{{\mathcal D}}^8_3$. In this section, we consider the case where the intersection pairing 
$\langle-,-\rangle \colon H_{4}(M_k)\times H_{4}(M_k) \rightarrow \mathbb{Z}$ is even i.e. 
$\langle x,x \rangle \in 2\mathbb{Z}$ for all 
$x\in H_{4}(M_k)$. Note that in this case, k must be even. We observe the following:
(i) If
$k \geq 4$, Mk supports a principal SU(2)-bundle whose total space E is 3-connected.(ii) The possible stable homotopy types of E can be determined directly from the stable homotopy type of Mk and the intersection form. In this regard, the formulas in theorem 3.6 are used to demonstrate this connection.
4.1. Existence of SU(2)-bundles
We discuss the existence of principal SU(2)-bundle over 
$M_k \in {\mathcal{P}}{{\mathcal D}}^8_3$ with an attaching map as in (1.1) whose intersection form is even. If 
${\text{Rank}}(H_{4}(M))= 2$, then up to isomorphism 
$\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}$ is the only possible matrix for the intersection form. The attaching map 
$L(M)\in \pi_{7}(S^{4}\vee S^{4})$ of M is of the form
\begin{equation}
L(M)= [\alpha_{1},\alpha_{2}]+ l_{1}\nu^{\prime}_{1}+ l_{2}\nu^{\prime}_{2},
\end{equation} where 
$l_{1}, l_{2}\in \mathbb{Z}/12$. For a principal bundle 
$SU(2) \to E \to M_k$ where E is 3-connected, the classifying map 
$f_E \colon M_k\rightarrow \mathbb{H}P^{\infty}$ is such that 
$\pi_{4}(f)$ is surjective. Suppose 
$\alpha_{i}\mapsto n_{i}\iota_{4}, i=1,2$ such that 
$\gcd(n_{1},n_{2})=1$ and 
$f\circ L(M)\simeq *$. If both l 1 and l 2 are odd, no such n 1 and n 2 exists. For, the equation 
$f \circ L(M) \simeq \ast$ translates to 
$n_1n_2+n_1+n_2 \equiv 0 \pmod{12}$ [Reference Basu and Ghosh7, Example 4.13]. However, for 
$k \geq 4$, one may always construct a suitable map.
Proposition 4.1. Suppose 
$M_k\in \mathcal{PD}_{3}^{8}$ such that 
${\text{Rank}}(H_{4}(M_k))=k\geq 4$ and the intersection form is even. Then there exists a map 
$\psi \colon M_k\rightarrow \mathbb{H}P^{\infty}$ such that 
$\text{hofib}(\psi)$ is 3-connected.
Proof. From [Reference Basu and Ghosh7, Theorem 4.20], if 
$k\geq 6$, the attaching map of Mk can be expressed as
\begin{equation*}L(M_k)= \sum_{1\leq i \lt j\leq k}g_{i,j}[\alpha_{i},\alpha_{j}]+ \sum_{1=1}^{k}\frac{g_{i,i}}{2}[\alpha_{i},\alpha_{i}]+ \sum_{i=1}^{k}s_{i}\nu_{i}'\end{equation*} such that 
$s_{i}=0$ for 
$i\geq 2$ for a choice of basis 
$\{\alpha_{1}, \dots, \alpha_{k}\}$ of 
$H_{4}(M_k)$. Then the map 
$\tilde{\psi}\colon (S^{4})^{\vee k}\xrightarrow{(0,0,\dots,0, 1)} \mathbb{H}P^{\infty}$ extends to a map 
$\psi\colon M_k\rightarrow \mathbb{H}P^{\infty}$ such that 
$\pi_{4}(\psi)$ is surjective.
Now if 
$k=4,$ by the classification of inner product spaces in low dimensions [Reference Milnor and Husemoller11, Ch. II, Theorem (2.2)], the attaching map of Mk can be expressed as
\begin{equation*}L(M_k)= [\alpha_{1},\alpha_{2}]+ [\alpha_{3},\alpha_{4}]+ \sum_{i=1}^{4}l_{i}\nu_{i}'\end{equation*} for a choice of basis of 
$H_{4}(M_k).$ Choose two positive integers 
$m, n$ such that 
$\gcd(m,n)=1$ and 
$ml_{1}+ nl_{3} \equiv 0 \pmod{12}$. Then the map 
$\tilde{\psi}\colon (S^{4})^{\vee 4}\xrightarrow{(m,0,n,0)} \mathbb{H}P^{\infty}$ extends to a map 
$\psi\colon M_k\rightarrow \mathbb{H}P^{\infty}$ such that 
$\pi_{4}(\psi)$ is surjective.
We now focus on the stable homotopy type of the total space E(f) for 
$f \colon M_k \to {\mathbb H} P^{\infty}$ such that 
$\pi_4(f)$ is injective. From the attaching map of Mk as in (1.1) and even intersection form, we have
\begin{equation}
\Sigma^{\infty}M_k\simeq \Sigma^{\infty} (S^4)^{\vee k-1}\vee\Sigma^{\infty}(Cone({\sigma(M_k)}\nu_{(7)})
\end{equation} for some even 
$\sigma(M_k)$. Hence the stable homotopy type of Mk is determined by 
$\sigma(M_k)$.
Proposition 4.2. Let 
$E(f_{\psi})$ be the total space of a principal SU(2)-bundle over 
$M_{k}\in \mathcal{PD}^{8}_{3}$, classified by a map 
$f_\psi\colon M_k\rightarrow \mathbb{H}P^{\infty}$ for 
$k\geq 4.$ Then
\begin{equation*}
\Sigma^{\infty}E(f_{\psi}) \simeq \Sigma^{\infty}(S^4\vee S^7)^{\vee {k-2}}\vee \Sigma^{\infty}Cone({{\lambda (\psi) }}\Sigma^4 \nu_{(7)} ),\end{equation*} where 
$\lambda(\psi) := \lambda_s(E(f_{\psi}))$, is even and a multiple of 
$\sigma(M_k)$.
Proof. Note that 
$E(f_{\psi}) \in \mathcal{PD}_{4,7}^{11}$ and its stable homotopy type is given in the Corollary 2.5 where 
$\epsilon= \epsilon_s(E(f_\psi))\in \mathbb{Z}/2$ and 
$\lambda(\psi)\in\mathbb{Z}/24$. So, it suffices to show that 
$2|\lambda(\psi)$ and ϵ = 0. The fact 
$2|\lambda(\psi)$ follows from (4.2) and theorem 3.6.
Now, the cofibre sequence obtained from cell structure of E and Mk induces the following commutative diagram
of stable homotopy groups, where ϕ is the attaching map of top cell in E.
Since 
$\lambda(\psi) \beta_{k-1} \circ \nu_{(7)} + \epsilon \alpha_{k-1} \circ x=0$ in 
$\pi^s_{10} (E)$, its image in 
$\pi_{10}^s(M)$ is 0. Note that bottom left map 
$\pi_{10}^s(\sum^{\infty} L(M))=0$ because 
$2|\sigma(M_k)$ in (4.2) and hence the bottom right map is injective, where 
$\pi_{10}^s(S^4)^{\oplus k}\cong \mathbb{Z}/2\{\alpha_1\circ \nu^2,\dots,\alpha_{k}\circ\nu^2\}$. Since 
$2|\lambda(\psi)$, the middle vertical arrow sends 
$\Sigma^{\infty}\phi= \lambda(\psi) \beta_{k-1}\circ\nu_{(7)}+\epsilon \alpha_{k-1}\circ\nu^2$ to 
$\epsilon\alpha_{k-1}\circ\nu^2$ which is in turn mapped to 0 via the bottom right map as 
$\Sigma^{\infty}\phi=0\in\pi_{10}^s(E).$ Hence 
$\epsilon=0\in\mathbb{Z}/2$.
4.2. Stably trivial manifolds
The following result states that the total space of a principal SU(2)-bundle over stably trivial 
$M_{k}\in \mathcal{PD}^{8}_{3}$ ( i.e., 
$\sigma(M_k)\equiv 0\pmod{24}$) is itself a connected sum of copies of 
$S^4\times S^7$.
Proposition 4.3. Let 
$E\in {\mathcal{P}}{{\mathcal D}}^{11}_{4,7}$ be the total space of a principal SU(2)-bundle over stably trivial 
$M_{k}\in \mathcal{PD}^{8}_{3}$. Then 
$E\simeq \#^{k-1}(S^4\times S^7)$.
Proof. From theorem 2.6, we have 
$E\simeq \#^{k-2}E_{0,0,0}\#E_{0,\epsilon,\delta}$. Then the result follows from proposition 4.2 for 
$k \geq 4$. For k = 2, the attaching map of M 2 is of the form (4.1). Stably trivial condition implies 
$M_2=S^4 \times S^4$ which further implies 
$E=S^4 \times S^7$.
4.3. Possible stable homotopy types of the total space
Let 
$\psi\in H^{4}(M_k;\mathbb{Z})$ be a cohomology class represented by a map 
$\overline{\psi}\colon M_{k}\rightarrow K(\mathbb{Z},4)$, which has a unique lift 
$\tilde{\psi}\colon M_k\rightarrow S^4$ up to homotopy if 
${\overline{\psi}} \big{|}_{{(S^4)}^{\vee {k-1}}}\circ L(M_k) \in \pi_7(S^4)$ is 0. As the inclusion 
$(S^4)^{\vee{k}}\rightarrow M_{k}$ induces an isomorphism on H 4 and H 4, a cohomology class 
$\psi\in H^4(M_k)$ always induces a map 
$\tilde{\psi}\colon (S^4)^{\vee{k}}\rightarrow S^4.$ We consider the following diagram
and formulate when the map 
$M_{k}\rightarrow \mathbb{H}P^2$ exists. Note that if the map exists then its homotopy fibre will be 3-connected.
For 
$\psi\in H^4(M_k),$ consider the composite
\begin{equation*} S^7\xrightarrow{L(M_k)}(S^4)^{\vee{k}}\xrightarrow{\tilde{\psi}}S^4\end{equation*} and define 
$\tau(\psi)=[\tilde{\psi}\circ L(M)]\in \pi_{7}^s(S^4)$. Thus
\begin{equation*}
\tau\colon H^4(M_k)\rightarrow \mathbb{Z}/24\end{equation*}and one can check that it is a linear map.
Proposition 4.4. Suppose 
$\psi \in H^4(M_k)$ is primitive. The map 
$\tilde{\psi}\colon (S^4)^{\vee{k}}\rightarrow S^4$ extends to a map 
$f_\psi: M_k\rightarrow \mathbb{H}P^2$ if and only if 
$\psi\cup\psi\equiv \tau(\psi)z\pmod{24}$ for some 
$z \in H^8(M_k)$.
Proof. Consider a primitive element 
$\psi \in H^4(M_k)$. We extend ψ to a basis of 
$H^{4}(M_k)$ and use the dual basis of 
$\pi_4(M_k)$ to write down the attaching map of Mk is as in (1.1). In this notation, we have 
$\psi^2=g_{k,k} z$, where 
$z \in H^8(M_k)$ is the chosen generator and 
$\tau (\psi)=g_{k,k}-2l_k$. Thus, 
$\tilde{\psi}\circ L(M)$ maps to 
$0\in \pi_{7}(\mathbb{H}P^{2})$ if and only if 
$l_k=0$, that is 
$\psi\cup \psi\equiv \tau{(\psi)}z\pmod{24}$. Hence, the result follows.
Proposition 4.4 gives us criteria for constructing the maps fψ out of cohomology classes ψ. Using this we determine which multiples of 
$\sigma(M_k)$ may occur as 
$\lambda_s(E)$ for 
$E\to M_k$ a principal SU(2)-bundle where E is 3-connected. We first show that there exists fψ such that 
$\lambda(\psi)=\sigma(M_k)$ if k is large enough.
Proposition 4.5. Suppose the stable homotopy type of Mk is determined by 
$\sigma(M_k).$ Then there exists 
$f_\psi\colon M_{k}\rightarrow \mathbb{H}P^{\infty}$ such that
(1)
$\lambda(\psi)\equiv \sigma(M_k) \pmod{3}$ for
$k \geq 5$,(2)
$\lambda(\psi) \equiv \sigma(M_k) \pmod{8}$ for
$k \geq 7$.
Proof. We begin the proof with the first case. If 
$\tau \equiv 0 \pmod{3}$, the proof follows from theorem 3.6. Let 
$\tau \not\equiv 0 \pmod{3}$ and ψ 0 a primitive cohomology class such that 
$\tau(\psi_{0}) \equiv \sigma(M_k) \not \equiv 0 \pmod{3}$. We need to choose a 
$\psi \in \ker (-\cup\psi_{0})\cap \ker(\tau)$ such that 
$\psi^2 \equiv \tau(\psi)\equiv 0\pmod{3}$. This we can do for 
$k-2 \geq 3$, see [Reference Milnor and Husemoller11, Chapter II, (3.2)–(3.4)]. By Poincaré duality, we get 
$\psi'$ such that 
$\psi\cup \psi'=z$. We write 
$\psi'= \psi^{\prime}_{\ker (\tau)}+t\psi_{0}$ for some t where 
$\psi^{\prime}_{\ker(\tau)}\in \ker(\tau)$. Since 
$z=\psi'\cup \psi= \psi^{\prime}_{\ker(\tau)} \cup \psi$, we may assume 
$\psi' =\psi^{\prime}_{\ker\tau}$. Thus by assigning 
$\tau_1=\tau(\psi_{0}) \equiv \sigma(M_k) \pmod{3}$, 
$\tau_{k-1}=\tau(\psi') \equiv 0 \pmod{3}$ and 
$\tau_{k}=\tau(\psi)\equiv 0 \pmod{3}$; and choosing other ψi’s such that 
$\tau_i=0$ for 
$i=2, \dots, k-2$, we have
\begin{equation*}\lambda(\psi)\equiv \sigma(M_k) \pmod{3} \quad \quad \text{for } k \geq 5.\end{equation*} Now we look into the second case. The proof goes similarly to that of the above, except when we choose 
$\psi \in \ker (-\cup\psi_{0})\cap \ker(\tau)$ such that 
$\psi^2 \equiv \tau(\psi)\equiv 0\pmod{8}$. We can choose such ψ for 
$k-2 \geq 5$, see [Reference Milnor and Husemoller11, Chapter II, (3.2)–(3.4)]. Then following similar arguments, one can deduce
\begin{equation*}\lambda(\psi)\equiv \sigma(M_k) \pmod{8} \quad \quad \text{for } k \geq 7.\end{equation*} The following theorem constructs fψ with 
$\lambda(\psi)=3 \sigma(M_k)$ if 
$\sigma(M_k)$ is not divisible by 3.
Theorem 4.6. Suppose 
$3 \nmid \sigma(M_k)$. Then for 
$k\geq 7$, there exists 
$f_\psi\colon M_k\rightarrow \mathbb{H}P^{2}$ such that 
$\lambda(\psi)=3 \sigma(M_k)$.
Proof. Let
\begin{equation*}\tau^{(3)}\colon H^{4}(M_k,\mathbb{F}_{3})\rightarrow \mathbb{F}_3\end{equation*} be the restriction of τ in modulo 3. As 
$\tau^{(3)}$ is surjective, there exists 
$\psi_{0}\in H^{4}(M_k,\mathbb{F}_3)$ such that for all cohomology class ψ, 
$\tau^{(3)}(\psi)z\equiv\psi\cup \psi_0\pmod{3}$. In particular, 
$\tau^{(3)}(\psi_0)z\equiv \psi_0\cup\psi_0\pmod{3}$. We consider two cases, 
$\psi_0^2=0$ or ψ 02 is unit.
First let ψ 02 is unit. Then we can choose 
$\psi_1, \dots, \psi_{k-1}$ such that 
$\psi_0 \cup \psi_i=0$. We take the dual basis αi corresponding to ψi and αk corresponding to ψ 0. Thus 
$\psi_0(\alpha_i)=0$ for 
$i=1, \dots, k-1$ and 
$\psi_0(\alpha_{k})=1$. Hence
\begin{equation*}\tau_1 \equiv \dots \equiv \tau_{k-1} \equiv 0 \pmod{3}, \quad \text{and} \quad \tau_k \equiv g_{k,k} \equiv 1 \pmod{3},\end{equation*} which implies 
$\lambda(\psi)= \gcd(\tau_1,\dots,\tau_{k-2},\tau_{k-1}) \equiv 0 \pmod{3}$.
Now let 
$\psi_0^2=0$. Then 
$\tau^{(3)}(\psi_0)z=0$. We choose 
$\psi_1, \dots, \psi_{k-1}$ such that 
$\psi_{k-1} \cup \psi_0=1$ and 
$\psi_i \cup \psi_0=0$ for 
$i=1 , \dots, k-2$. After taking the dual basis, with similar argument, we have
\begin{align*}\tau_1 &\equiv \dots \equiv \tau_{k-2}\equiv 0 \pmod{3}, \quad \tau_{k-1}\equiv \sigma(M_k) \pmod{3}, \quad \text{and} \quad\\ \tau_k &\equiv g_{k,k} \equiv 0 \pmod{3}.\end{align*} Hence 
$\lambda(\psi)= \gcd(\tau_1,\dots,\tau_{k-2},\tau_k-g_{k,k}\tau_{k-1})\equiv0\pmod{3}$.
Remark 4.7. We note that proposition 4.4, proposition 4.5, and theorem 4.6 do not use the fact that the intersection form is even and also holds in the case where the intersection form is odd.
Now we look to prove similar results modulo 8, which in turn provide us desired construction ψ as in proposition 4.4 using the Chinese reminder theorem. However, in this case, certain conditions are required for obtaining analogous fψ.
• A complex Mk with
$\sigma(M_k)=2 $ or 4 is said to satisfy hypothesis
$(H_8)$ if
$(\ker \tau)^{\perp}=(\sigma(M_k) \psi) \pmod{8}$, where
$\psi \in H^4(M_k)$ (which is unique
$\pmod{\frac{8}{\sigma(M_k)}}$) satisfies
\begin{equation*}\begin{cases} \psi ^2 \equiv 0 \pmod{8} & \text{if } \sigma(M_k)=2 \\
\psi ^2 \equiv 0 \pmod{4} & \text{if } \sigma(M_k)=4.\end{cases}\end{equation*}• A complex Mk with
$\sigma(M_k)=2 $ is said to satisfy hypothesis
$(H_4)$ if
$(\ker \tau)^{\perp}=(2\psi) \pmod{4} $, where
$\psi \in H^4(M_k)$ (which is unique
$\pmod{2}$) satisfies
\begin{equation*}\psi ^2 \equiv \tau (\psi) \equiv 0 \text{or } 4 \pmod{8}.\end{equation*}
Note that the hypotheses 
$(H_8)$ and 
$(H_4)$ depend only on the intersection form and τ and not on the choice of ψ. We now prove the existence of fψ under the hypothesis defined above.
(i) Suppose
$8 \nmid \sigma(M_k)$. For
$k\geq 5$, there exists
$f_\psi \colon M_k\rightarrow \mathbb{H}P^{2}$ such that
$\lambda(\psi)\equiv 0 \pmod{8}$ if and only if the complex satisfies hypothesis
$(H_8)$.(ii) Suppose
$ \sigma(M_k)=2$. Then for
$k\geq 5$, there exists
$f_\psi\colon M_k\rightarrow \mathbb{H}P^{2}$ such that
$\lambda(\psi)\equiv 4 \pmod{8} $ if and only if the complex satisfies hypothesis
$(H_4)$.
Proof. The condition 
$k\geq 5$ comes from the fact that we are required to make certain choices modulo 3 using proposition 4.5. First suppose in case 1, a fψ exists such that 
$\lambda(\psi)=8 \sigma(M_k)$. Then there exists a basis 
$\{\psi_1, \dots, \psi_{k-2}, \psi', \psi\}$ satisfying (working 
$\pmod{8}$)
\begin{equation}
\begin{aligned} &\psi \cup \psi_i=0 \text{for }1 \leq i \leq k-2, \quad &&\psi \cup \psi'=1, \\ &\tau(\psi_i)=0 \text{for }1 \leq i \leq k-2, \quad &&\tau (\psi') = \sigma(M_k) \quad \text{and} \quad \tau(\psi)=\psi^2=0.\end{aligned}
\end{equation} Note that 
$\langle \psi \rangle^{\perp}= \langle \psi_1, \dots, \psi_{k-2}, \psi \rangle \subset \ker (\tau)= \langle \psi_1, \dots, \psi_{k-2}, \psi, \frac{8}{\sigma(M_k)} \psi' \rangle$. This implies 
$(\ker (\tau))^{\perp} = (\sigma(M_k) \psi)$, and thus the hypothesis 
$(H_8)$ is satisfied.
For the converse part if the complex satisfies hypothesis 
$(H_8)$, one can check that there is a choice of ψ such that (4.3) is satisfied. We look into the cases 
$\sigma(M_k)=2, 4$.
First, let 
$\sigma(M_k)=2$. Then 
$(\ker (\tau))^{\perp} = (2\psi)$ and ψ is well-defined modulo 4. We note that
\begin{equation}
\chi \cup (2\psi) \equiv \tau(\chi)z \pmod{8} ~\forall \chi \in H^4(M_k).
\end{equation}This implies
\begin{equation*}\tau (\psi )\equiv 2\psi^2 \equiv 0 \pmod{8}.\end{equation*} For any 
$\chi \in \ker (\tau)$, we have 
$(2 \psi) \chi =\tau (\chi) \equiv 0 \pmod{8}$. In particular, 
$2 \psi^2 = \tau(\psi) \equiv 0 \pmod{8}$. Together these two imply 
$\psi^2 \equiv 0 \pmod{8}$. Eq. (4.4) implies that if 
$\chi\cup \psi =0$, 
$\tau(\chi)=0$. Now choosing a basis as in case 2 of proposition 3.5, we obtain the conditions in (4.3).
Now let 
$\sigma(M_k)=4$. Then 
$(\ker (\tau))^{\perp} = (4\psi)$ and ψ is determined modulo 2. We proceed analogously observing that
\begin{equation*}
\chi \cup (4\psi) \equiv \tau(\chi)z \pmod{8} ~\forall \chi \in H^4(M_k),
\end{equation*}which implies
\begin{equation*}\tau (\psi )\equiv 4\psi^2 \equiv 0 \pmod{8}.\end{equation*} Let 
$\psi'$ be such that 
$\tau(\psi')=4$, and so we have that 
$\psi \cup \psi'$ is an odd multiple of z. If ψ 2 is 
$4 \pmod{8}$, we change ψ to 
$\psi + 2 \psi'$ to ensure 
$\psi^2 = \tau(\psi) \equiv 0 \pmod{8}$. Now choosing a basis as in case 2 of proposition 3.5, we obtain the conditions in (4.3).
Case 2 also proceeds along analogous lines. For the existence (of fψ for some ψ) question, we need a basis satisfying
\begin{align}
\begin{aligned} &\psi \cup \psi_i=0 \text{ for }1 \leq i \leq k-2, \quad \psi \cup \psi'=1, \\
&\tau(\psi_i)\equiv 0 \pmod{4} \text{ for }1 \leq i \leq k-2, \quad
\tau (\psi') = \sigma(M_k), \quad \tau(\psi)=\psi^2 \equiv 0 \text{ or } 4 \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\pmod{8}, \\
&\text{such that at least one of } \tau(\psi_i) \text{ for } 1\leq i \leq k-2, \text{ or } \tau(\psi) \equiv 4 \pmod{8} .
\end{aligned}\nonumber\\
\end{align} Note that 
$\langle \psi \rangle^{\perp}= \langle \psi_1, \dots, \psi_{k-2},\psi \rangle \subset \ker (\tau)= \langle \psi_1, \dots, \psi_{k-2},\psi, 2 \psi' \rangle \pmod{4}$ and 
$(\ker (\tau))^{\perp} = (2 \psi)\pmod{4}$. Hence, the hypothesis 
$(H_4)$ is satisfied.
Conversely, if 
$(H_4)$ is satisfied, we obtain a ψ such that 
$\psi^2 \equiv \tau(\psi) \equiv 0 \text{or } 4 \pmod{8}$. This ψ also satisfies
\begin{equation*}
\chi \cup (2\psi) \equiv \tau(\chi)z \pmod{4} ~\forall \chi \in H^4(M_k).
\end{equation*} Let 
$\psi'$ be such that 
$\tau(\psi')=2$, which implies that 
$\psi \cup \psi'$ is an odd multiple of z. Now replacing ψ by 
$\psi + 2 \psi'$ if required we may assume that 
$\psi^2 \equiv \tau(\psi) \equiv 4 \pmod{8}$. Now choosing a basis as in case 2 of proposition 3.5, we obtain the conditions in (4.5).
If 
${\text{Rank}}(H_4(M_k))=k \geq 5$, the above results indicate a systematic computation of possible stable homotopy types of the total space depending on k, 
$\sigma(M_k)$, and the intersection form. In lower rank cases, the results depend on the explicit formula for the attachment 
$L(M_k)$, and not just on these variables. Hence the systematic description turns out to be cumbersome. We demonstrate some observations on the 
${\text{Rank}}(H_4(M)) =2$ case.
Example 4.1. Recall that the attaching map 
$L(M)\in \pi_{7}(S^{4}\vee S^{4})$ of M 2 is of the form
\begin{equation*}L(M)= [\alpha_{1},\alpha_{2}]+ l_{1}v^{\prime}_{1}+ l_{2}v^{\prime}_{2}\end{equation*} where 
$l_{1}, l_{2}\in \mathbb{Z}/12$. We already have
• If both l 1 and l 2 are odd, there does not exist
$f \colon M\rightarrow \mathbb{H}P^{\infty}$ such that
$E=\text{hofib}(f)$ is 3-connected.
If one of l 1 and l 2 is even, or both are even, there exists 
$f \colon M\rightarrow \mathbb{H}P^{\infty}$ such that 
$E=\text{hofib}(f)$ is 3-connected. Via an explicit calculation using proposition 3.5, we observe the following:
(i) If none of l 1 and l 2 are divisible by 3, then we obtain
$\lambda(E) \equiv 0 \pmod 3$.(ii) If either of l 1 or l 2 but not both are divisible by 3, then
$\lambda(E) \not \equiv 0 \pmod 3$.(iii) If
$\sigma(M) \equiv 4 \pmod{8}$ and
$ l_1 l_2 \equiv 0 \pmod{8}$, where none of l 1 and l 2 are divisible by 3, then
$\lambda(E) \equiv 0 \pmod{8}$.(iv) If
$\sigma(M) \equiv 2 \pmod{8}$, we can never obtain
$\lambda(E) \equiv 0, 4 \pmod{8}$.
5. SU(2)-bundles over odd complexes
We now work out the case of 
$M_k \in {\mathcal{P}}{{\mathcal D}}_3^8$ for which the intersection form is odd. Recall that the notation Mk means that 
${\text{Rank}}(H_4(M_k))=k$. The intersection form being odd implies that there are two possibilities of 
$\sigma(M_k)$, namely, 1 and 3 among the divisors of 24. Here, we prove
(i) For 
$k\geq 3$, it is possible to obtain a SU(2)-bundle whose total space is 3-connected.
(ii) Further if 
$k\geq 7$, it is possible to obtain maps 
$\psi(j) : M_k \to {\mathbb H} P^\infty$ with 
$\lambda(\psi(j))=j$ for every multiple 
$j \pmod{24} $ of 
$\sigma(M_k)$ which is also a divisor of 24.
5.1. Existence of SU(2)-bundles
Through an explicit computation, we demonstrate the existence of principal SU(2)-bundles over 
$M_k \in {\mathcal{P}} {{\mathcal D}} ^8_3$ if 
$k\geq 3$ when the intersection form is odd.
Proposition 5.1. Suppose 
$M_k\in \mathcal{PD}_{3}^{8}$ such that 
${\text{Rank}}(H_{4}(M_k))=k\geq 3$ and the intersection form is odd. Then there exists a map 
$\psi \colon M_k\rightarrow \mathbb{H}P^{\infty}$ such that 
$\text{hofib}(\psi)$ is 3-connected.
Proof. Recall that the attaching map of Mk can be expressed as (1.1). Using proposition 4.4, we are required to find a primitive element ψ such that 
$\tau(\psi)z \equiv \psi^2 \pmod{24}$, which is equivalent to checking that the coefficient of 
$\nu'$ in 
$\tilde{\psi}\circ L(M_k)$ is 
$0 \pmod{12}$. It suffices to find 
$\psi \pmod{8}$ and 
$\psi \pmod{3}$ separately.
We first work out the 
$\pmod{3}$ case, where the base ring is a field of characteristic ≠ 2 so that the form is diagonalizable. Considering the map 
$(S^4)^{\vee k}\xrightarrow[]{(0,\dots,0, n_1, n_2, n_3)} {\mathbb H} P^{\infty}$ which sends 
$L(M_k)$ to
\begin{equation*}
\Big(\pm{n_1 \choose 2} \pm{n_2 \choose 2} \pm{n_3 \choose 2} + n_1 l_{k-2}+n_2 l_{k-1}+ n_3 l_k\Big) \cdot \nu' + \text{multiple of } \nu,
\end{equation*} where the ± correspond to the diagonal entries 
$\pmod{3}$. We observe through a direct calculation that for every fixed choice 
$\epsilon_1, \epsilon_2,\epsilon_3$ of ±1s, there is a 
$n_1, n_2, n_3$ with 
$\gcd(n_1, n_2, n_3)=1$ such that
\begin{equation*}
\Big(\epsilon_1{n_1 \choose 2} +\epsilon_2{n_2 \choose 2} +\epsilon_3{n_3 \choose 2} + n_1 l_{k-2}+n_2 l_{k-1}+ n_3 l_k\Big) \equiv 0 \pmod{3}.
\end{equation*} This completes the argument 
$\pmod{3}$.
Working 
$\pmod{8}$, the fact that the intersection form is odd implies that we may choose a basis such that 
$g_{k,k} \equiv \pm 1 \pmod{8}$ and 
$g_{k, k-1} \equiv 0 \pmod{8}$, see [Reference Milnor and Husemoller11, Chapter II, (4.3)]. Also the intersection form can be written as the block matrix 
$\begin{pmatrix}
A & 0 \\0 & \pm 1
\end{pmatrix}.$
If A is an even intersection form, then the result follows from proposition 4.1 for 
$k \geq 5$. Now let A is not even, then the intersection form is 
$\begin{pmatrix}
A' & 0 \\ 0 & B'
\end{pmatrix}$ where B ʹ is a diagonal matrix of order 2 with diagonal entries ±1. If A ʹ is an even intersection form, then for 
$k \geq 6$ the result follows from proposition 4.1. If A ʹ is not even, the intersection form updates to
\begin{equation*}
\begin{pmatrix}
A^{\prime\prime} & 0 \\0 & B^{\prime\prime}
\end{pmatrix}
\end{equation*} where B ʹʹ is a diagonal matrix of order 3 with diagonal entries ±1. For 
$B^{\prime\prime}=I_3$, the map 
$(S^4)^{\vee k}\xrightarrow[]{(0,\dots,0, n_1, n_2, n_3)} {\mathbb H} P^{\infty}$ sends 
$L(M_k)$ to
\begin{equation*}
\Big({n_1 \choose 2} +{n_2 \choose 2} +{n_3 \choose 2} + n_1 l_{k-2}+n_2 l_{k-1}+ n_3 l_k\Big) \cdot \nu' + \text{multiple of } \nu,
\end{equation*} where 
$\gcd (n_1, n_2, n_3)=1$ ensures that the corresponding ψ is primitive. We may directly compute and observe that the equations
\begin{equation*} \Big({n_1 \choose 2} +{n_2 \choose 2} +{n_3 \choose 2} + n_1 l_{k-2}+n_2 l_{k-1}+ n_3 l_k\Big) \equiv 0 \pmod{8}, \gcd (n_1, n_2, n_3)=1,\end{equation*} have a common solution. A similar argument works for the other diagonal ±1 matrix choices for 
$B^{\prime\prime\prime}$. This proves the result for 
$k\geq 6$.
If 
$3 \leq k \leq 5$, we know from [Reference Milnor and Husemoller11, Chapter II, (3.2)–(3.4)] that the form is a direct sum of ±1 and the hyperbolic form. The argument in the even case in proposition 4.1 implies the result for a sum of two hyperbolic forms, and the above argument implies the result for a sum of 3 ±1s. The remaining cases are taken care of if we show the result for the intersection form
\begin{equation*}
\begin{pmatrix}
H & 0 \\ 0 & \pm 1
\end{pmatrix}
\end{equation*} where 
$H= \begin{pmatrix}
0 &1 \\1 &0
\end{pmatrix}$ is the hyperbolic matrix. We consider the map 
$(S^4)^{\vee 3} \xrightarrow[]{(n_1, n_2, n_3)} {\mathbb H} P^{\infty}$, and as mentioned above we need to find a common solution of
\begin{equation*}{n_3 \choose 2}+ n_1 n_2+\sum_{i=1}^3 n_i l_i \equiv 0 \pmod{8}, \gcd (n_1, n_2, n_3)=1.\end{equation*}Through a direct calculation, we check that such solutions always exist. This completes the proof.
The following example shows that proposition 5.1 does not extend to the k = 2 case.
Example 5.1. Consider M 2 such that
\begin{equation*}
L(M_2) = \nu_1 + \nu_2 + l_1 \nu_1^{\prime} + l_2 \nu_2^{\prime}, \quad l_1\equiv l_2 \equiv 2 \pmod{3}.
\end{equation*} A map 
$M_2 \to {\mathbb H} P^\infty$ which restricts to 
$(n_1,n_2)$ on the 4-skeleton sends 
$L(M_2)$ to
\begin{equation*}
\Big({n_1 \choose 2} +{n_2 \choose 2} + n_1 l_1+n_2 l_2\Big) \cdot \nu' + \text{multiple of } \nu.
\end{equation*}We may check directly that
\begin{equation*}
\Big({n_1 \choose 2} +{n_2 \choose 2} + n_1 l_1+n_2 l_2\Big) \equiv 0 \pmod{3} \implies n_1 \equiv n_2 \equiv 0 \pmod{3}.
\end{equation*} Therefore, there is no map 
$M_2 \to {\mathbb H} P^\infty$ whose homotopy fibre is 3-connected.
5.2. Possible stable homotopy type of the total space
We note from §4 that propositions 4.4, 4.5, and theorem 4.6 are also valid when the intersection form is odd. We check that all stable homotopy types are achievable in the odd case if the rank k of 
$H_4(M)$ is 
$\geq 7$. Applying the results from §4, it only remains to check that the different possibilities 
$\pmod{8}$ are achievable. We do this in the theorem below.
Theorem 5.2. Suppose 
$\sigma(M_k)$ is odd. Then for 
$k \geq 5$ and 
$j\in \{0,2,4\}$, there exists 
$\psi \colon M_k \to {\mathbb H} P^2$ such that 
$\lambda(\psi) \equiv j \pmod{8}$.
Proof. We work 
$\pmod{8}$, knowing that if 
$k\geq 5$, proposition 4.5 allows us to make a choice of ψ so that 
$\lambda(\psi)$ is as required 
$\pmod{3}$. The proof is very similar to the proof of theorem 4.6. If 
$\sigma(M_k)$ is odd, the linear map 
$\tau \colon ({\mathbb{Z}}/8)^k \to {\mathbb{Z}}/8$ is represented by some primitive class ψ (that is, 
$\tau (\chi)= \langle \chi, \psi \rangle \pmod{8}$, and 
${\mathbb{Z}}\{\psi\}$ is a summand of 
$H^4(M_k)$). In particular, 
$\psi^2=\tau (\psi)$. We have two cases.
First let 
$\tau(\psi)$ be odd, i.e. ψ 2 is unit in modulo 8. Then we can extend ψ to a basis 
$\psi, \psi_1, \dots, \psi_{k-1}$ such that 
$\psi \cup \psi_i=0$ for 
$1\leq i \leq k-1$. We take the dual basis αi corresponding to ψi and αk corresponding to ψ. Thus 
$\psi(\alpha_i)=0$ for 
$i=1, \dots, k-1$ and 
$\psi(\alpha_{k})=1$. Hence
\begin{equation*}\tau_1 \equiv \dots \equiv \tau_{k-1} \equiv 0 \pmod{8}, \quad \text{and} \quad \tau_k \equiv g_{k,k} \pmod{8},\end{equation*} which implies 
$\lambda(\psi)= \gcd(\tau_1,\dots,\tau_{k-2},\tau_{k-1}) \equiv 0 \pmod{8}$ by case 1 of proposition 3.5.
Now let ψ 2 be even. Extend ψ to a basis 
$\psi, \psi_1, \dots, \psi_{k-1}$ such that 
$\psi_{k-1} \cup \psi=1$ and 
$\psi_i \cup \psi=0$ for 
$i=1 , \dots, k-2$. After taking the dual basis, with similar argument we have
\begin{equation*}\tau_1 \equiv \dots \equiv \tau_{k-2}\equiv 0 \pmod{8}, \quad \tau_{k-1}\equiv 1 \pmod{8}, \quad \text{and} \quad \tau_k \equiv g_{k,k} \pmod{8}.\end{equation*} Hence 
$\lambda(\psi)= \gcd(\tau_1,\dots,\tau_{k-2},\tau_k-g_{k,k}\tau_{k-1})\equiv0\pmod{8}$ by case 2 of proposition 3.5. This proves the result for j = 0.
For 
$j=2, $ or 4, we can use 
$\psi(j)= \psi+j \psi_1 $, and note that
\begin{equation*} \tau(\psi(j))= \tau(\psi), ~ \psi(j)^2 = \psi^2 + j^2 \psi_1^2 \equiv \psi^2 \pmod{8}.\end{equation*} The last equivalence comes from the fact that 
$\tau(\psi_1)z = \psi\cup \psi_1 \equiv 0 \pmod{8}$, and so ψ 12 is forced to be an even multiple of z. We may now compute using the formulas of proposition 3.5 to conclude that 
$\lambda(\psi(j))\equiv j \pmod{8}$.
As in the even case, when the rank is high enough we have a systematic idea of the possibilities of the total space. However, in the low rank cases (
$k\leq 6$), the results are not systematic and may depend on individual cases rather than only on 
$\sigma(M_k), k$, and the intersection form.
