# PL cobordism and classification of PL manifolds

Wen Shen  Department of Mathematics, Capital Normal University, Beijing,
P.R.China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we present more conclusions for $C^{\ast}(\mathrm{PL})$ and
oriented PL cobordism. Furthermore, we compute the string PL bordism groups of
certain dimensions. Finally, we apply these results to classify certain PL
manifolds of dimensions $10$ and $13$.

###### 2020 Mathematics Subject Classification:

Primary 57Q20, 57Q25, 55T15

###### Contents

  1. 1 Introduction
  2. 2 $C^{\ast}(\mathrm{PL})$ and $\Omega_{\ast}^{\mathrm{SPL}}$ cobordism
  3. 3 $\Omega_{\ast}^{\mathrm{PL}\langle 8\rangle}$ cobordism
  4. 4 Kervaire invariant
  5. 5 Classification for certain 13-manifolds

##  1\. Introduction

Since Kervaire [17] gave an invariant of $(4k+2)$-manifolds for $k\neq 0,1,3$,
the invariant, called Kervaire invariant, has been of interest. In particular,
the existence of framed manifolds of Kervaire invariant one is one of the
oldest issues in differential and algebraic topology, and has been solved
completely by [3] [5] [13] [15] [25] [26] [27], and the recent work of Lin-
Wang-Xu.

From [17], the Kervaire invariant can be defined for any $4$-connected closed
manifold of dimension $10$, and we see that the Kervaire invariant of any
smooth $4$-connected closed manifold of dimension $10$ is $0$. Naturally, we
will ask whether a $4$-connected closed manifold of dimension $10$ admits a
smooth structure if its Kervaire invariant equals to $0$. Kervaire reduced
this question to the following conjecture.

######  Conjecture 1.1.

The $5$-th Betti number $B_{5}(M)$ and Kervaire invariant
$\Phi(M)\in\mathbb{Z}_{2}$ are a complete set of invariants of the homotopy
type of the $4$-connected closed manifold $M$ of dimension $10$.

In this paper, we prove the conjecture of Kervaire as follows.

######  Theorem 1.2.

The $5$-th Betti number $B_{5}(M)$ and Kervaire invariant
$\Phi(M)\in\mathbb{Z}_{2}$ are a complete set of invariants of the
homeomorphism type of the $4$-connected closed manifold $M$ of dimension $10$.

Indeed, any $4$-connected closed manifold of dimension $10$ admits a unique
piecewise linear (PL) structure. So these invariants in Theorem 1.2 also
determine the PL homeomorphism types of the $4$-connected closed manifolds of
dimension $10$.

Consider all $4$-connected closed manifolds of dimension $10$ with the fixed
$B_{5}=2s>0$. By observing that $\Phi$ is additive with respect to the
connected sum of manifolds, one of the homeomorphism types is represented by
the connected sum of $s$ copies of $S^{5}\times S^{5}$, the other type is the
connected sum of $M_{0}$ and $s-1$ copies of $S^{5}\times S^{5}$ where the
manifold $M_{0}$ was constructed by Kervaire in [17]. Then, we have

######  Corollary 1.3.

The Kervaire invariant of a $4$-connected closed manifold of dimension $10$ is
$0$ iff it admits a smooth structure.

Classification of the manifolds with a given cohomology ring (up to
diffeomorphism, homeomorphism, or homotopy) is one of the central problems in
geometric topology. The Poincar$\acute{e}$ theorem is a special case. For a
more complicated cohomology ring, there also exist many investigations, such
as the classifications of certain 5-manifolds [2, 23], simply connected
$6$-manifolds [16, 40], the $7$-manifolds modeled on Aloff-Wallach spaces [21,
22, 24], the Bazaikin spaces [10]. Different from the above classifications
that focus on smooth manifolds, Theorem 1.2 and the following classification
are for certain PL manifolds.

Let $\mathcal{M}$ denote certain PL $13$-manifolds, which have the same
cohomology ring as $\mathrm{CP}^{3}\times S^{7}$. Partial these manifolds have
been considered in the smooth category [35]. Recall a definition in [35]

######  Definition 1.4.

Let $x:\mathcal{M}\to\mathrm{CP}^{\infty}$ denote a generator of
$H^{2}(\mathcal{M};\mathbb{Z})$. We call that $\mathcal{M}$ admits a
restriction lift if there exists a lift $f:\mathcal{M}\to\mathrm{CP}^{4}$ such
that the following diagram is homotopy commutative

$\textstyle{\mathrm{CP}^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\scriptstyle{f}$$\textstyle{\mathrm{CP}^{\infty}}$

Now, we are ready to present the classifying theorem.

######  Theorem 1.5.

Assume that $\mathcal{M}$, $\mathcal{M}^{\prime}$ admit restriction lifts.
Then

$\mathcal{M}$ is $\mathrm{PL}$-homeomorphic to $\mathcal{M}^{\prime}$ if and
only if $p_{1}(\mathcal{M})=p_{1}(\mathcal{M}^{\prime})$.

$\mathcal{M}$ is homotopy equivalent to $\mathcal{M}^{\prime}$ if and only if
$p_{1}(\mathcal{M})=p_{1}(\mathcal{M}^{\prime})\mod 24$.

######  Remark 1.6.

Although $\mathcal{M}$, $\mathcal{M}^{\prime}$ are $\mathrm{PL}$ manifolds, we
can also define the same Stiefel-Whitney classes $w_{i}$ for $i<7$, and
Pontryagin class $p_{1}$ since $H_{i}(\mathrm{BPL},\mathrm{BO};\mathbb{Z})=0$
for $i\leq 7$.

######  Remark 1.7.

For any $s\in\mathbb{Z}$, there exists an $8$-dimensional vector bundle $\eta$
over $\mathrm{CP}^{3}$ so that $p_{1}(\eta)=sx^{2}$. The total space
$\overline{\mathcal{M}}$ of the sphere bundle of $\eta$ has the first
Pontrjagin class $(4+s)x^{2}$. By the (1) of Theorem 1.5, any $\mathcal{M}$
with restriction lift is PL-homeomorphic to such the total space
$\overline{\mathcal{M}}$.

The $h$-cobordism theorem is a general method to classify simply connected
manifolds. We mainly use the PL cobordism and surgery to construct the
$h$-cobordism, thus proving Theorem 1.2 and 1.5. Next, we introduce some
conclusions of PL cobordism.

Throughout the rest of the paper, let $H^{i}(-)$ (or $H_{i}(-)$) denote the
$\mathbb{Z}_{2}$-cohomology (or homology) group $H^{i}(-;\mathbb{Z}_{2})$ (or
$H_{i}(-;\mathbb{Z}_{2})$).

The methods developed by Thom to study the unoriented differentiable cobordism
ring $\Omega_{\ast}^{\mathrm{O}}$ have been applied successfully to various
classes of smooth manifolds. Through the theory of microbundles developed by
Milnor [29], it is possible to apply Thom’s ideas to study PL manifolds.

Given a vector (or PL) bundle $\pi:E\to X$, one can define a right
$\mathscr{A}$ action on $H^{\ast}(X)$ (see [6]) via

$x\cdot\alpha=\Psi^{-1}(\chi(\alpha)\Psi(x))$ (1.1)

for $x\in H^{\ast}(X)$, $\alpha\in\mathscr{A}$ where $\chi$ is the conjugation
of $\mathscr{A}$, $T(E)$ is the Thom space of $E$, and $\Psi$ is the Thom
isomorphism $\Psi:H^{\ast}(X)\to H^{\ast}(T(E))$.

Analogous to vector bundles, there exists corresponding Thom isomorphism and
classifying space for PL bundles [34]. Let BPL be the universal classifying
space for stable PL bundles. $H^{\ast}(\mathrm{BPL})$ is a Hopf algebra, as to
$H^{\ast}(\mathrm{BO})$. From [4], there is a Hopf algebra
$C^{\ast}(\mathrm{PL})$ over the mod $2$ Steenrod algebra $\mathscr{A}$ such
that

$H^{\ast}(\mathrm{BPL})\cong H^{\ast}(\mathrm{BO})\otimes
C^{\ast}(\mathrm{PL})$ (1.2)

as Hopf algebras over $\mathscr{A}$, and
$\Omega_{\ast}^{\mathrm{PL}}\cong\Omega_{\ast}^{\mathrm{O}}\otimes
C^{\ast}(\mathrm{PL})$ as algebras. In particular, the isomorphism (1.2) is an
isomorphism of right $\mathscr{A}$-modules, and the right $\mathscr{A}$ action
on $H^{\ast}(\mathrm{BO})\otimes C^{\ast}(\mathrm{PL})$ is given by the
formula

$(b\otimes c)\cdot\alpha=\Sigma
b\cdot\alpha^{\prime}\otimes\chi(\alpha^{\prime\prime})c$ (1.3)

where $\Delta(\alpha)=\Sigma\alpha^{\prime}\otimes\alpha^{\prime\prime}$ is
the diagonal map in $\mathscr{A}$.

It is well known that the homotopy fiber $\mathrm{PL/O}$ of the natural map
$\mathscr{D}:\mathrm{BO}\to\mathrm{BPL}$ is 6-connected. Then we take the
$(n-1)$-connected covers of BPL for $n=2,4,8$, analogous to BO. Moreover, we
have the natural map $\mathscr{D}\langle n\rangle:\mathrm{BO}\langle
n\rangle\to\mathrm{BPL}\langle n\rangle$.

######  Theorem 1.8.

Let the right $\mathscr{A}$ action on $H^{\ast}(\mathrm{BO}\langle
n\rangle)\otimes C^{\ast}(\mathrm{PL})$ be given by the formula $($1.3$)$.
Then $H^{\ast}(\mathrm{BPL}\langle n\rangle)\cong H^{\ast}(\mathrm{BO}\langle
n\rangle)\otimes C^{\ast}(\mathrm{PL})$ as right $\mathscr{A}$-modules for
$n=2,4,8$.

The cases for $n=2,4$ have been proved in [4]. We only prove the case $n=8$ in
this paper.

For the Hopf algebra $C^{\ast}(\mathrm{PL})$, we have

######  Theorem 1.9.

$(1)$ $C^{i}(\mathrm{PL})=0$ for $1\leq i\leq 7$.

$(2)$ $C^{8}(\mathrm{PL})=\mathbb{Z}_{2}$,
$C^{9}(\mathrm{PL})=\mathbb{Z}_{2}^{2}$.

$(3)$ $C^{10}(\mathrm{PL})=\mathbb{Z}^{3}_{2}$,
$C^{11}(\mathrm{PL})=\mathbb{Z}_{2}^{2}$.

$(4)$ $C^{i}(\mathrm{PL})\neq 0$ for $i\geq 12$.

The notation $\mathbb{Z}^{j}_{2}$ denotes the direct sum of $j$ groups with
order $2$.

The (1) and (2) in Theorem 1.9 are known (see [39]), and the (3) and (4) are
proved in this paper.

Let $\Omega_{n}^{\mathrm{SPL}}$ be the oriented cobordism group of PL
manifolds. In [42], Williamson presented partial structures of the groups
$\Omega_{n}^{\mathrm{SPL}}$ for $0\leq n\leq 18$. For example,
$\Omega_{10}^{\mathrm{SPL}}=\mathbb{Z}_{2}\oplus\mathrm{2gp}$,
$\Omega_{11}^{\mathrm{SPL}}=\mathbb{Z}_{2}\oplus\mathbb{Z}_{3}\oplus\mathrm{2gp}$
where $\mathrm{2gp}$ denotes a group whose order is a power of $2$. In this
paper, we see more precise conclusions for the dimensions $n=10,11$.

######  Proposition 1.10.

$\Omega_{10}^{\mathrm{SPL}}=\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}^{2}$,
$\Omega_{11}^{\mathrm{SPL}}=\mathbb{Z}_{2}\oplus\mathbb{Z}_{3}$.

As to smooth string cobordism $\Omega_{\ast}^{\mathrm{O}\langle 8\rangle}$,
one can define the string cobordism $\Omega_{\ast}^{\mathrm{PL}\langle
8\rangle}$ in the PL category. For certain dimensions, we have

######  Proposition 1.11.

$\Omega_{n}^{\mathrm{O}\langle 8\rangle}=\Omega_{n}^{\mathrm{PL}\langle
8\rangle}$ for $n\leq 7$, $\Omega_{8}^{\mathrm{PL}\langle
8\rangle}=\mathbb{Z}\oplus\mathbb{Z}_{4}$,

$\Omega_{9}^{\mathrm{PL}\langle 8\rangle}=0$, $\Omega_{10}^{\mathrm{PL}\langle
8\rangle}=\mathbb{Z}_{2}$, $\Omega_{11}^{\mathrm{PL}\langle 8\rangle}=0$,
$\Omega_{13}^{\mathrm{PL}\langle 8\rangle}=0$.

By considering the map $\Omega_{n}^{\mathrm{O}\langle
8\rangle}\to\Omega_{n}^{\mathrm{PL}\langle 8\rangle}$, we see

######  Corollary 1.12.

There exists an $(n-1)$-connected closed $\mathrm{PL}$ manifold of dimension
$2n$ for $n=4,5$, which does not admit any smooth structure.

The above corollary implies that the generators with finite order in
$\Omega_{n}^{\mathrm{PL}\langle 8\rangle}$ for $n=8,10$ can not be represented
by any smooth manifolds as in [38]. In particular, the nontrivial element of
$\Omega_{10}^{\mathrm{PL}\langle 8\rangle}$ can be represented by the manifold
$M_{0}$. Moreover, the corollary gives an answer for the Milnor’s question in
the theorem $4$ of [28].

Note that every $4$-connected closed manifold of dimension $10$ admits a
unique $\mathrm{BPL}\langle 8\rangle$-structure, i.e. the classifying map of
its PL normal bundle has a lift to $\mathrm{BPL}\langle 8\rangle$, unique up
to homotopy. By $\Omega_{10}^{\mathrm{PL}\langle 8\rangle}=\mathbb{Z}_{2}$,
all $4$-connected closed manifolds of dimension $10$ are divided into two
$\mathrm{PL}\langle 8\rangle$ bordism classes, which are distinguished by the
Kervaire invariant. Then, Theorem 1.2 follows by the classifying theorem 4.1.

The plan of this paper is then as follows. In Section 2, we do the preparatory
work for $C^{\ast}(\mathrm{PL})$ and $\Omega_{\ast}^{\mathrm{SPL}}$; in
Section 3 we compute $\Omega_{\ast}^{\mathrm{PL}\langle 8\rangle}$ in certain
dimensions; in Section 4 we apply the PL cobordism to prove the conjecture of
Kervaire; another application in Section 5 is to classify certain PL
$13$-manifolds.

##  2\. $C^{\ast}(\mathrm{PL})$ and $\Omega_{\ast}^{\mathrm{SPL}}$ cobordism

We first recall the isomorphism of Hopf algebras [9]

$H_{\ast}(\mathrm{BSPL})\cong H_{\ast}(\mathrm{BSO})\otimes V^{(1)}(K[3]\oplus
R\langle 4\rangle)\otimes\Gamma(T)\otimes H_{\ast}(\mathscr{K})$ (2.1)

where $\mathscr{K}$ is the product of Eilenberg-MacLane spaces

$\Pi_{n\neq
2^{i}}\mathrm{K}(\mathbb{Z}_{2},4n-2)\times\Pi_{n>1}\mathrm{K}(\mathbb{Z},4n).$

Other notations will be explained briefly as follows.

$R[k]$ is a coalgebra over $\mathscr{A}$, its dual $R[k]^{\ast}$ is a
polynomial algebra,

$R[k]^{\ast}=P[\xi_{1k},\cdots,\xi_{kk}]\quad\mathrm{deg}(\xi_{ik})=2^{k-i}(2^{i}-1)$

and is an unstable algebra over $\mathscr{A}$. Let $R\langle
n\rangle=\oplus_{k\geq n}R[k]$.

Let $\mathrm{PH}^{\ast}(\mathrm{K}(\mathbb{Z},4))$ be the primitive elements
of $H^{\ast}(\mathrm{K}(\mathbb{Z},4))$, and let
$\alpha:\mathrm{PH}^{\ast}(\mathrm{K}(\mathbb{Z},4))\to R[3]^{\ast}$ be the
$\mathscr{A}$-module homomorphism with $\alpha(\ell_{4})=\xi_{13}$. Indeed,
the homomorphism $\alpha$ is a monomorphism [9]. Let $K[3]$ be the kernel of
the dual epimorphism $\alpha^{\ast}:R[3]\to
Q(H_{\ast}(\mathrm{K}(\mathbb{Z},4)))$ where $Q(-)$ is the vector space of
indecomposable elements.

For a graded module $N$, let $sN$ be $N$ with degrees shifted up by $1$. We
define $V^{(1)}(N)=T(s\bar{N})/\mathscr{F}$ where $T(s\bar{N})$ is the tensor
algebra on $s\bar{N}$, $\bar{N}\subset N$ the subspace of elements of positive
dimension, and $\mathscr{F}$ is a certain ideal.

There is a monomorphism $i_{1}:E(R[2])\to H_{\ast}(\mathscr{K}_{1})$ where
$E(R[2])$ is the exterior algebra, $\mathscr{K}_{1}=\Pi_{n\geq
2}\mathrm{K}(\mathbb{Z}_{2},2^{n}-2).$ Let
$\Gamma(T)=H_{\ast}(\mathscr{K}_{1})//i_{1}$.

One can see more details about the above notations in [9]. Here we pay more
attention to the structure of $\mathbb{Z}_{2}$-vector space.

Let $\mathscr{V}=V^{(1)}(K[3]\oplus R\langle 4\rangle)\otimes\Gamma(T)$. Since
the isomorphism (1.2), we see that $C^{\ast}(\mathrm{PL})$ is isomorphic to
$\mathscr{V}\otimes H_{\ast}(\mathscr{K})$ as vector spaces.

######  Lemma 2.1.

Let $\mathscr{V}_{i}$ be the set of all elements of degree $i$.
$\mathscr{V}_{8}=0$, $\mathscr{V}_{9}=\mathbb{Z}_{2}^{2}$,
$\mathscr{V}_{10}=\mathbb{Z}_{2}$, $\mathscr{V}_{11}=0$. Furthermore, one of
generators of $\mathscr{V}_{9}$ is provided by $\Gamma(T)$,
$\mathscr{V}_{10}\subset\Gamma(T)$.

###### Proof.

Note that $C^{8}(\mathrm{PL})=\mathbb{Z}_{2}$ and
$C^{9}(\mathrm{PL})=\mathbb{Z}^{2}_{2}$ [39]. By
$H_{8}(\mathscr{K})=\mathbb{Z}_{2}$ and $H_{9}(\mathscr{K})=0$, we have
$\mathscr{V}_{8}=0$ and $\mathscr{V}_{9}=\mathbb{Z}_{2}^{2}$. Since
$H_{9}(\mathscr{K}_{1})=\mathbb{Z}_{2}^{8}$ and
$E(R[2])_{9}=\mathbb{Z}_{2}^{7}$, we have $\Gamma(T)_{9}=\mathbb{Z}_{2}$.

Since that $K[3]\oplus R\langle 4\rangle$ does not contain any element of
degree $9$, $V^{(1)}({K[3]\oplus R\langle 4\rangle})$ does not contain any
element of degree $10$. Since $H_{10}(\mathscr{K}_{1})=\mathbb{Z}_{2}^{9}$ and
$E(R[2])=\mathbb{Z}_{2}^{8}$,
$\mathscr{V}_{10}=\Gamma(T)_{10}=\mathbb{Z}_{2}$.

Note that ${R[3]}$ only has an element of degree $10$, which is the dual of
$\xi_{13}\xi_{23}$ in $R[3]^{\ast}$. Note that
$\alpha(\mathrm{Sq}^{4}\mathrm{Sq}^{2}\ell_{4})=\xi_{13}\xi_{23}$. From the
definition, $K[3]$ does not have any element of degree $10$. Since that the
smallest degree of the elements of $R\langle 4\rangle$ is $8$, and the
smallest degree of $R[3]$ is $4$, we see that $V^{(1)}({K[3]\oplus R\langle
4\rangle})$ does not have any element of degree $11$. By
$H_{11}(\mathscr{K}_{1})=\mathbb{Z}_{2}^{11}$ and
$E(R[2])_{11}=\mathbb{Z}_{2}^{11}$, $\mathscr{V}_{11}=\Gamma(T)_{11}=0$. ∎

Recall the work of Sullivan, i.e. there is a map

$G/\mathrm{PL}\to\mathrm{K}(\mathbb{Z}_{2},2)\times_{\delta\mathrm{Sq}^{2}}\mathrm{K}(\mathbb{Z},4)\times\Pi_{n\geq
1}(\mathrm{K}(\mathbb{Z}_{2},4n+2)\times\mathrm{K}(\mathbb{Z},4n+4))$

inducing an isomorphism in $\mathbb{Z}_{2}$ cohomology. Let $k_{i}\in
H^{i}(\mathrm{G/PL})$ correspond to the fundamental class $\ell_{i}\in
H^{i}(\mathrm{K}(\mathbb{Z}_{2},i))$ or $H^{i}(\mathrm{K}(\mathbb{Z},i))$.
From [8] and [9], we have the following lemma.

######  Lemma 2.2.

$\Gamma(T)\otimes H_{\ast}(\mathscr{K})$ as the subspace of
$H_{\ast}(\mathrm{G/PL})$ is mapped injectively into $H_{\ast}(\mathrm{BSPL})$
through the natural homomorphism

$H_{\ast}(\mathrm{G/PL})\to H_{\ast}(\mathrm{BSPL}).$

Conversely, for any $i\neq 2^{j}-1$, there exists an element $x\in
C^{2i}(\mathrm{PL})$ such that its image is $k_{2i}+\cdots$ under the natural
homomorphism

$H^{\ast}(\mathrm{BSPL})\to H^{\ast}(\mathrm{G/PL}).$

######  Corollary 2.3.

$C^{n}(\mathrm{PL})\neq 0$ for $n\geq 12$.

###### Proof.

Note that $4i\neq 2^{j+1}-2$. Let $i\geq 2$. By Lemma 2.2, there exists an
element $x_{4i}\in C^{4i}(\mathrm{PL})$ such that its image is
$k_{2i}+\cdots$. Applying the Steenrod operations, we have
$\mathrm{Sq}^{k}x_{4i}\neq 0$, thus $C^{4i+k}(\mathrm{PL})\neq 0$ for any
$i\geq 2$, $2\leq k\leq 5$. ∎

From [39], $C^{8}(\mathrm{PL})=\mathbb{Z}_{2}$,
$C^{9}(\mathrm{PL})=\mathbb{Z}_{2}^{2}$. Let $p_{8}$ be the generator of
$C^{8}(\mathrm{PL})$. By Lemma 2.2, we have

######  Lemma 2.4.

$\mathrm{Sq}^{1}p_{8}=0,$ $\mathrm{Sq}^{i}p_{8}\neq 0$ for $2\leq i\leq 8$.

For some elements of $C^{\ast}(\mathrm{PL})$, we can not directly get the
$\mathscr{A}$ actions by Lemma 2.2. Then, we will use the Adams spectral
sequence (ASS) for ${{}_{2}}\pi_{\ast}(\mathrm{MSPL})$ to detect these
$\mathscr{A}$ actions on $C^{\ast}(\mathrm{PL})$.

######  Lemma 2.5.

Let $p_{9,1}$, $p_{9,2}$ denote two generators of $C^{9}(\mathrm{PL})$. Then,
$\mathrm{Sq}^{1}p_{9,1}=0$, $\mathrm{Sq}^{1}p_{9,2}\neq 0$.

###### Proof.

It is easy to see that

$h_{0}^{s}p_{8}\in\mathrm{Ext}_{\mathscr{A}}^{s,s+8}(H^{\ast}(\mathrm{MSPL};\mathbb{Z}_{2}),\mathbb{Z}_{2})\quad
s\geq 0$
$p_{9,1},p_{9,2}\in\mathrm{Ext}_{\mathscr{A}}^{0,9}(H^{\ast}(\mathrm{MSPL};\mathbb{Z}_{2}),\mathbb{Z}_{2}).$

Let $N^{n}=H^{\ast}(\mathrm{MSO})\otimes C^{\geq n}(\mathrm{PL})$. Applying
the functor $\mathrm{Ext}_{\mathscr{A}}^{\ast,\ast}(-,\mathbb{Z}_{2})$ on the
short exact sequence

$0\to N^{n+1}\to N^{n}\to H^{\ast}(\mathrm{MSO})\otimes C^{n}(\mathrm{PL})\to
0$

we get a long exact sequence for the $\mathrm{Ext}$-groups, thus an exact
couple that establishes the spectral sequence (AAHSS)

$E_{1}^{s,m,n}(\mathrm{SPL})=\mathrm{Ext}_{\mathscr{A}}^{s,s+m}(H^{\ast}(\mathrm{MSO};\mathbb{Z}_{2}),\mathbb{Z}_{2})\otimes
C^{n}(\mathrm{PL})$
$E_{1}^{s,m,n}(\mathrm{SPL})\Longrightarrow\mathrm{Ext}_{\mathscr{A}}^{s,s+m+n}(H^{\ast}(\mathrm{MSPL};\mathbb{Z}_{2}),\mathbb{Z}_{2})$
(2.2)

Combining the results for
$\mathrm{Ext}_{\mathscr{A}}^{s,t}(H^{\ast}(\mathrm{MSO}),\mathbb{Z}_{2})$ [30]
and the above spectral sequence, we have
$\mathrm{Ext}_{\mathscr{A}}^{s,s+7}(H^{\ast}(\mathrm{MSPL};\mathbb{Z}_{2}),\mathbb{Z}_{2})=0$
for $s>0$.

In the Adams spectral sequence

$\mathrm{Ext}_{\mathscr{A}}^{s,t}(H^{\ast}(\mathrm{MSPL};\mathbb{Z}_{2}),\mathbb{Z}_{2})\Longrightarrow{{}_{2}\pi}_{\ast}(\mathrm{MSPL}),$
(2.3)

$d_{r}(p_{8})=0$ for any $r\geq 2$. By
$\Omega_{8}^{\mathrm{SPL}}=\mathbb{Z}^{2}\oplus\mathbb{Z}_{4}$ [42] and
$\Omega_{8}^{\mathrm{SO}}=\mathbb{Z}^{2}$, we have
$d_{2}(p_{9,1})=h_{0}^{2}p_{8}$ and $\mathrm{Sq}^{1}p_{9,1}=0$. By
$\Omega_{9}^{\mathrm{SPL}}=\mathbb{Z}_{2}^{3}$ [42] and
$\Omega_{9}^{\mathrm{SO}}=\mathbb{Z}_{2}^{2}$, we have that
$\mathrm{Sq}^{1}p_{9,2}\neq 0$, and $p_{9,2}$ survives to the
$E_{\infty}$-term. ∎

######  Proposition 2.6.

$C^{10}(\mathrm{PL})=\mathbb{Z}_{2}\langle
p_{10,1}\rangle\oplus\mathbb{Z}_{2}\langle
p_{10,2}\rangle\oplus\mathbb{Z}_{2}\langle p_{10,3}\rangle$ where
$\mathrm{Sq}^{2}p_{8}=p_{10,1}$, $\mathrm{Sq}^{1}p_{9,2}=p_{10,2}$, and
$\mathrm{Sq^{1}}p_{10,3}\neq 0$.

###### Proof.

Since $\mathrm{Sq}^{2}p_{8}\neq 0$ and $\mathrm{Sq}^{1}p_{9,2}\neq 0$,
$C^{10}(\mathrm{PL})$ has generators so that $p_{10,1}=\mathrm{Sq}^{2}p_{8}$
and $p_{10,2}=\mathrm{Sq}^{1}p_{9,2}$. By Lemma 2.2, we see that $p_{10,1}$
corresponds to $\mathrm{Sq}^{2}\ell_{8}\in
H^{10}(\mathrm{K}(\mathbb{Z},8))\subset H^{10}(\mathscr{K})$. By Lemma 2.2,
$C^{10}(\mathrm{PL})$ has another generator $c_{10}$ that corresponds to
$\ell_{10}\in H^{10}(\mathrm{K}(\mathbb{Z}_{2},10))\subset
H^{10}(\mathscr{K})$. Thus, $\mathrm{Sq}^{1}c_{10}\neq 0$. Since
$\mathrm{Sq}^{1}p_{10,2}=\mathrm{Sq}^{1}\mathrm{Sq}^{1}p_{9,2}=0$, $c_{10}\neq
p_{10,2}$. Moreover, the dual of $p_{10,2}$ is in
$\mathscr{V}_{10}=\mathbb{Z}_{2}$ by Lemma 2.1. ∎

######  Proposition 2.7.

$\Omega_{10}^{\mathrm{SPL}}=\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}^{2}$.

###### Proof.

By the AAHSS (2.2),
$\mathrm{Ext}_{\mathscr{A}}^{s,s+9}(H^{\ast}(\mathrm{MSPL}),\mathbb{Z}_{2})=0$
for any $s>0$. By Proposition 2.6, $p_{10,1}$,
$p_{10,3}\in\mathrm{Ext}_{\mathscr{A}}^{0,10}(H^{\ast}(\mathrm{MSPL}),\mathbb{Z}_{2})$,
thus survive to the $E_{\infty}^{0,10}$ in the ASS (2.3). Since
$\mathrm{Sq}^{1}p_{10,1}\neq 0$ and $\mathrm{Sq}^{1}p_{10,3}\neq 0$, we get
the proposition. ∎

######  Proposition 2.8.

$C^{11}(\mathrm{PL})=\mathbb{Z}_{2}\langle
p_{11,1}\rangle\oplus\mathbb{Z}_{2}\langle p_{11,3}\rangle$ where
$\mathrm{Sq}^{1}p_{10,1}=p_{11,1}$, and $\mathrm{Sq^{1}}p_{10,3}=p_{11,3}$.

###### Proof.

Since $V^{11}=0$ (see Lemma 2.1), $C^{11}(\mathrm{PL})$ corresponds to

$H^{11}(\mathscr{K})=H^{11}(\mathrm{K}(\mathbb{Z},8)\times\mathrm{K}(\mathbb{Z}_{2},10))=\mathbb{Z}_{2}^{2}.$

So the desired results follow from Proposition 2.6. ∎

By Proposition 2.8, the homomorphism

$\mathrm{Ext}_{\mathscr{A}}^{s,s+11}(H^{\ast}(\mathrm{MSO}),\mathbb{Z}_{2})\to\mathrm{Ext}_{\mathscr{A}}^{s,s+11}(H^{\ast}(\mathrm{MSPL}),\mathbb{Z}_{2})$

is isomorphic. By the AAHSS (2.2),
$\mathrm{Ext}_{\mathscr{A}}^{s,s+10}(H^{\ast}(\mathrm{MSPL}),\mathbb{Z}_{2})=0$
for any $s>0$. Since
$\Omega_{11}^{\mathrm{SPL}}=\mathbb{Z}_{2}\oplus\mathrm{2gp}\oplus\mathbb{Z}_{3}$
[42], we have

######  Corollary 2.9.

$\Omega_{11}^{\mathrm{SPL}}=\mathbb{Z}_{2}\oplus\mathbb{Z}_{3}$.

Based on Lemma 2.2 and the proof of Proposition 2.8, we have

######  Corollary 2.10.

$\mathrm{Sq}^{2}p_{9,1}=0$, $\mathrm{Sq}^{2}p_{9,2}=0$.

Note that the Thom isomorphism $H_{\ast}(\mathrm{BSPL};\mathbb{Z})\cong
H_{\ast}(\mathrm{MSPL};\mathbb{Z})$. Finally, we consider
$H_{\ast}(\mathrm{BSPL};\mathbb{Z})$ by the spectral sequence (AHSS)

$E_{2}^{p,q}(\mathrm{SPL})=H_{p}(\mathrm{MSPL};\pi_{q}(S^{0}))\Longrightarrow\pi_{p+q}(\mathrm{MSPL})$
(2.4)

where $S^{0}$ is the sphere spectrum.

######  Lemma 2.11.

$H_{n}(\mathrm{BSPL};\mathbb{Z})$ does not have odd torsion for $n<11$.

###### Proof.

It is well known that $H_{n}(\mathrm{BSO};\mathbb{Z})\cong
H_{n}(\mathrm{BSPL};\mathbb{Z})$ for $n\leq 7$, and
$H_{\ast}(\mathrm{BSO};\mathbb{Z})$ does not have odd torsion. Recall that, in
dimensions $i\leq 11$ [37], $\pi_{i}(S^{0})$ is as follows.

$i$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
---|---|---|---|---|---|---|---|---|---|---|---|---
$\pi_{i}(S^{0})$ | $\mathbb{Z}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{24}$ | 0 | 0 | $\mathbb{Z}_{2}$ | $\mathrm{Z}_{240}$ | $\mathbb{Z}_{2}^{2}$ | $\mathbb{Z}^{3}_{2}$ | $\mathbb{Z}_{6}$ | $\mathbb{Z}_{504}$

Note that $\mathrm{PL/O}$ is $6$-connected,
$\pi_{7}(\mathrm{PL/O})=H_{7}(\mathrm{PL/O};\mathbb{Z})=\mathbb{Z}_{28}$. By
the Serre spectral sequence for the fibration,

$\mathrm{PL/O}\to\mathrm{BSO}\to\mathrm{BSPL}$

$H_{8}(\mathrm{BSPL};\mathbb{Z})$ does not have any element with finite order
$i$ so that $i$ is prime to $2$, $7$; and
$H_{8}(\mathrm{BSO};\mathbb{Z}_{3})\to H_{8}(\mathrm{BSPL};\mathbb{Z}_{3})$ is
isomorphic.

Note that for $p<7$, $E_{2}^{p,7-p}(\mathrm{SPL})$ is a finite group that does
not have any element with order $i$ so that $(i,7)=7$. By
$\pi_{8}(\mathrm{MSPL})=\mathbb{Z}^{2}\oplus\mathbb{Z}_{4}$ [42] and the
differentials of the spectral sequence (2.4),
$H_{8}(\mathrm{BSPL};\mathbb{Z})$ does not have odd torsion.

By the same arguments, the desired results for $n=9,10$ follow. ∎

##  3\. $\Omega_{\ast}^{\mathrm{PL}\langle 8\rangle}$ cobordism

We first consider the integral homology group for $\mathrm{BPL}\langle
n\rangle$.

######  Lemma 3.1.

Let $n=4$, $8$. $H_{i}(\mathrm{BPL}\langle n\rangle;\mathbb{Z})$ does not have
odd torsion for $i<11$.

###### Proof.

Recall the fibrations

$\mathrm{K}(\mathbb{Z}_{2},1)\to\mathrm{BPL}\langle 4\rangle\to\mathrm{BSPL}$
$\mathrm{K}(\mathbb{Z},3)\to\mathrm{BPL}\langle 8\rangle\to\mathrm{BPL}\langle
4\rangle$

Since $H^{i}(\mathrm{K}(\mathbb{Z}_{2},1);\mathbb{Z})=\mathbb{Z}_{2}$ or $0$
for any $i>0$, the homomorphism

$H^{\ast}(\mathrm{BSPL};\mathbb{Z}_{p^{r}})\to H^{\ast}(\mathrm{BPL}\langle
4\rangle;\mathbb{Z}_{p^{r}})$

is isomorphic for any $r>0$ and any odd prime number $p$. The desired results
for $n=4$ follow by Lemmma 2.11.

By the conclusions for $n=4$, we compute the case for $n=8$. Recall that, in
dimensions $\leq 11$, the nontrivial integral cohomology groups of
$\mathrm{K}(\mathbb{Z},3)$ are as follows

i | 3 | 6 | 8 | 9 | 10 | 11
---|---|---|---|---|---|---
$H^{i}$ | $\mathbb{Z}\langle l_{3}\rangle$ | $\mathbb{Z}_{2}\langle l_{3}^{2}\rangle$ | $\mathbb{Z}_{3}\langle a_{8}\rangle$ | $\mathbb{Z}_{2}\langle l^{3}_{3}\rangle$ | $\mathbb{Z}_{2}\langle a_{10}\rangle$ | $\mathbb{Z}_{3}\langle a_{8}l_{3}\rangle$

For the $7$-connected covers of $\mathrm{BO}$ and $\mathrm{BPL}$, we have

$\textstyle{\mathrm{K}(\mathbb{Z},3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{K}(\mathbb{Z},3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{K}(\mathbb{Z},3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{BO}\langle
8\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathscr{D}\langle
8\rangle}$$\scriptstyle{p_{1}}$$\textstyle{\mathrm{BPL}\langle
8\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\textstyle{\mathrm{PK}(\mathbb{Z},4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{BO}\langle
4\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathscr{D}\langle
4\rangle}$$\textstyle{\mathrm{BPL}\langle
4\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{l}}$$\textstyle{\mathrm{K}(\mathbb{Z},4)}$

As the proof of Lemma 2.11, $H_{8}(\mathrm{BO}\langle
4\rangle;\mathbb{Z}_{3})\to H_{8}(\mathrm{BPL}\langle
4\rangle;\mathbb{Z}_{3})$ is isomorphic. Note $H_{10}(\mathrm{BO}\langle
8\rangle;\mathbb{Z}_{3})=0$ [12]. Considering these
$E_{2}^{p,q}(\mathrm{PL}8)$-terms for $8\leq p+q\leq 10$ and the following
morphism

$\textstyle{{E}_{2}^{p,q}(\mathrm{O}8)=H_{p}(\mathrm{BO}\langle
4\rangle;H_{q}(\mathrm{K}(\mathbb{Z},3);\mathbb{Z}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{p+q}(\mathrm{BO}\langle
8\rangle;\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{E}_{2}^{p,q}(\mathrm{PL}8)=H_{p}(\mathrm{BPL}\langle
4\rangle;H_{q}(\mathrm{K}(\mathbb{Z},3);\mathbb{Z}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{p+q}(\mathrm{BPL}\langle
8\rangle;\mathbb{Z})}$

we have the desired results for the case $n=8$. ∎

Then, we compute the $\mathbb{Z}_{2}$-cohomology of $\mathrm{BPL}\langle
8\rangle$.

######  Lemma 3.2.

$H^{\ast}(\mathrm{BPL}\langle 8\rangle)\cong H^{\ast}(\mathrm{BO}\langle
8\rangle)\otimes C^{\ast}(\mathrm{PL})$ as right $\mathscr{A}$-modules. The
formula $($1.3$)$ induces the right $\mathscr{A}$ action on
$H^{\ast}(\mathrm{BO}\langle 8\rangle)\otimes C^{\ast}(\mathrm{PL}).$

###### Proof.

Let $g_{o}=\mathscr{D}\langle 4\rangle\circ g_{l}$. Then
$g_{o}^{\ast}(\ell_{4})=w_{4}$, where $\ell_{4}$ and $w_{4}$ are the
generators of $H^{4}(\mathrm{K}(\mathbb{Z},4))$ and $H^{4}(\mathrm{BO}\langle
4\rangle)$. Recall that $H^{\ast}(\mathrm{K}(\mathbb{Z},4))$ is a polynomial
ring with generators $\ell_{4}$ and $\mathrm{Sq}^{\mathrm{I}}\ell_{4}$, where
$\mathrm{I}=(i_{1},i_{2},\cdots,i_{r})$ is the admissible sequence satisfying
$e(\mathrm{I})=2i_{1}-(i_{1}+\cdots+i_{r})<4$ and $i_{r}>1.$ By [36],
$g_{o}^{\ast}:H^{\ast}(\mathrm{K}(\mathbb{Z},4))\to
H^{\ast}(\mathrm{BO}\langle 4\rangle)$ is a monomorphism. Thus,
$g_{l}^{\ast}:H^{\ast}(\mathrm{K}(\mathbb{Z},4))\to
H^{\ast}(\mathrm{BPL}\langle 4\rangle)$ is also a monomorphism. By the case
$n=4$ of Theorem 1.8, all differentials of the spectral sequence

${E}_{2}^{m+p,q}=H^{m+p}(\mathrm{BPL}\langle
4\rangle;H^{q}(\mathrm{K}(\mathbb{Z},3)))\Longrightarrow
H^{m+p+q}(\mathrm{BPL}\langle 8\rangle)$

are determined by the differentials of the following spectral sequences

$E_{2}^{p,q}=H^{p}(\mathrm{K}(\mathbb{Z},4);H^{q}(\mathrm{K}(\mathbb{Z},3)))\Longrightarrow
H^{p+q}(\mathrm{PK}(\mathbb{Z},4))$ $E_{2}^{p,q}=H^{p}(\mathrm{BO}\langle
4\rangle;H^{q}(\mathrm{K}(\mathbb{Z},3)))\Longrightarrow
H^{p+q}(\mathrm{BO}\langle 8\rangle).$

Thus $H^{\ast}(\mathrm{BPL}\langle 8\rangle)\cong H^{\ast}(\mathrm{BO}\langle
8\rangle)\otimes C^{\ast}(\mathrm{PL})$ as vector spaces. The desired
$\mathscr{A}$-module isomorphism follows by the commutative diagram

$\textstyle{H^{\ast}(\mathrm{BO}\langle 4\rangle)\otimes
C^{\ast}(\mathrm{PL})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{right-\mathscr{A}-module}$$\scriptstyle{p_{1}^{\ast}\otimes
1}$$\scriptstyle{right-\mathscr{A}-module}$$\textstyle{H^{\ast}(\mathrm{BPL}\langle
4\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}^{\ast}}$$\scriptstyle{right-\mathscr{A}-module}$$\textstyle{H^{\ast}(\mathrm{BO}\langle
8\rangle)\otimes
C^{\ast}(\mathrm{PL})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{H^{\ast}(\mathrm{BPL}\langle
8\rangle)}$

where $p_{1}^{\ast}$ is an epimorphism [12]. ∎

Indeed, by formula (1.1), the right $\mathscr{A}$ action on
$H^{\ast}(\mathrm{BPL}\langle n\rangle)$ is induced by the canonical left
$\mathscr{A}$ action on $H^{\ast}(\mathrm{MPL}\langle n\rangle)$, where
$\mathrm{MPL}\langle n\rangle$ is the Thom spectrum associated with the
universal bundle over $\mathrm{BPL}\langle n\rangle$.

######  Corollary 3.3.

$H^{\ast}(\mathrm{MPL}\langle n\rangle)\cong H^{\ast}(\mathrm{MO}\langle
n\rangle)\otimes C^{\ast}(\mathrm{PL})$ as canonical left
$\mathscr{A}$-modules for $n=2,4,8$, and the formula

$\alpha(b\otimes
c)=\Sigma\alpha^{\prime}b\otimes\alpha^{\prime\prime}c\quad\Delta(\alpha)=\Sigma\alpha^{\prime}\otimes\alpha^{\prime\prime}$

presents the left $\mathscr{A}$ action on $H^{\ast}(\mathrm{MO}\langle
n\rangle)\otimes C^{\ast}(\mathrm{PL})$.

Now, we discuss some representatives of $\Omega^{\mathrm{O}\langle
8\rangle}_{\ast}=\pi_{\ast}(\mathrm{MO}\langle 8\rangle)$. Recall that, in
dimension $i\leq 14$ [12, 14], $\pi_{i}(\mathrm{MO}\langle 8\rangle)$ is as
follows.

$i$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
$\pi_{i}(\mathrm{MO}\langle 8\rangle)$ | $\mathbb{Z}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{24}$ | 0 | 0 | $\mathbb{Z}_{2}$ | 0 | $\mathbb{Z}\oplus\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$

$i$ | 10 | 11 | 12 | 13 | 14
---|---|---|---|---|---
$\pi_{i}(\mathrm{MO}\langle 8\rangle)$ | $\mathbb{Z}_{6}$ | 0 | $\mathbb{Z}$ | $\mathbb{Z}_{3}$ | $\mathbb{Z}_{2}$

######  Lemma 3.4.

Any torsion element $\sigma_{n}\in\pi_{n}(\mathrm{MO}\langle 8\rangle)$ for
$n=8,9,10$ is represented by an exotic sphere $\Sigma^{n}$.

###### Proof.

By [17] and [18], we have the exact sequence

$0\to\mathrm{bP}_{n+1}\to\Theta_{n}\to coker({J}_{n})\to 0$

for $n=8,9,10$ where $J_{n}:\pi_{n}(\mathrm{SO})\to\pi_{n}(S^{0})$. By [1] and
[31], $coker(J_{n})$ has the following generators

$\phi\in\mathbb{Z}_{2}\subset\pi_{8}(S^{0}),\quad\eta\phi\in\mathbb{Z}_{2}\subset\pi_{9}(S^{0}),$
$\kappa\in\mathbb{Z}_{2}\subset\pi_{9}(S^{0}),\quad\eta\kappa\in\mathbb{Z}_{2}\subset\pi_{10}(S^{0}),\quad\gamma\in\mathbb{Z}_{3}\subset\pi_{10}(S^{0})$

where $\eta\in\pi_{1}(S^{0})$ is represented by $h_{1}$ in the ASS for
$S^{0}$. Note that exotic sphere $\Sigma^{n}$ has the trivial
$\mathrm{BO}\langle 8\rangle$-structure. Considering the unit
$\iota:S^{0}\to\mathrm{MO}\langle 8\rangle$, we have that
$\iota_{\ast}(\phi)=c_{0}$, $\iota_{\ast}(\kappa)=h_{1}\omega$,
$\pi_{10}(S^{0})\cong\pi_{10}(\mathrm{MO}\langle 8\rangle)$ by [12, 14] and
[32]. Thus, the lemma follows. ∎

Next, we consider the ASS for ${{}_{2}}\pi_{\ast}(\mathrm{MPL}\langle
8\rangle)$. Recall the $E_{2}$-terms of the Adams spectral sequence (ASS) for
$\mathrm{MO}\langle 8\rangle$ as in Fig.1. From [12], we have
$H^{\ast}(\mathrm{MO}\langle 8\rangle)\cong\mathscr{A}//\mathscr{A}_{2}$ as
modules over the Steenrod algebra $\mathscr{A}$ in dimensions $<20$, thus

$\mathrm{Ext}^{s,t}_{\mathscr{A}}(H^{\ast}(\mathrm{MO}\langle
8\rangle),\mathbb{Z}_{2})\cong\mathrm{Ext}^{s,t}_{\mathscr{A}_{2}}(\mathbb{Z}_{2},\mathbb{Z}_{2})$

for $t-s\leq 17$, where $\mathscr{A}_{2}$ is a subalgebra of $\mathscr{A}$,
generated by $\mathrm{Sq}^{1}$, $\mathrm{Sq}^{2}$ and $\mathrm{Sq}^{4}$. By
Corollary 3.3 and the change of ring,

$\mathrm{Ext}^{s,t}_{\mathscr{A}}(H^{\ast}(\mathrm{MPL}\langle
8\rangle),\mathbb{Z}_{2})\cong\mathrm{Ext}^{s,t}_{\mathscr{A}_{2}}(C^{\ast}(\mathrm{PL}),\mathbb{Z}_{2})\quad
t-s\leq 17.$

$0$$s$$t-s$123456789101112131415161712345678$h_{0}$$h^{2}_{0}$$h_{1}$$h^{2}_{1}$${h^{3}_{1}=h^{2}_{0}h_{2}}$${h_{2}}$${h^{2}_{2}}$${c_{0}}$${\omega}$${h_{1}^{3}\omega=h^{2}_{0}h_{2}\omega}$${h_{2}\omega}$${h^{2}_{2}\omega=h^{2}_{0}d_{0}}$${\tau}$${d_{0}}$${h^{2}_{0}\kappa=h_{1}d_{0}}$${\kappa}$
Fig 1.  $\mathrm{Ext}^{s,t}_{\mathscr{A}}(H^{\ast}(\mathrm{MO}\langle
8\rangle),\mathbb{Z}_{2})$

For
$\mathrm{Ext}^{s,t}_{\mathscr{A}_{2}}(C^{\ast}(\mathrm{PL}),\mathbb{Z}_{2})$,
there is an AAHSS

$E_{1}^{s,m,n}(\mathrm{PL}8)=\mathrm{Ext}^{s,s+m}_{\mathscr{A}_{2}}(\mathbb{Z}_{2},\mathbb{Z}_{2})\otimes
C^{n}(\mathrm{PL})$
$E_{1}^{s,m,n}(\mathrm{PL}8)\Longrightarrow\mathrm{Ext}_{\mathscr{A}_{2}}^{s,s+m+n}(C^{\ast}(\mathrm{PL}),\mathbb{Z}_{2})$
(3.1) $d_{r}:E_{r}^{s,m,n}\to E_{r}^{s+1,m+r-1,n-r}$

######  Lemma 3.5.

The natural homomorphism

$\mathscr{D}\langle
8\rangle_{\ast}:\mathrm{Ext}^{s,t}_{\mathscr{A}}(H^{\ast}(\mathrm{MO}\langle
8\rangle),\mathbb{Z}_{2})\to\mathrm{Ext}^{s,t}_{\mathscr{A}}(H^{\ast}(\mathrm{MPL}\langle
8\rangle),\mathbb{Z}_{2})$

is injective for $t-s\leq 11$.

###### Proof.

Since $\mathscr{D}\langle 8\rangle:\mathrm{MO}\langle
8\rangle\to\mathrm{MPL}\langle 8\rangle$ is $7$-equivalent, it is easy to get
the lemma for $t-s\leq 7$.

Note that $C^{n}(\mathrm{PL})=0$ for $n<8$. By the degrees of differentials of
the AAHSS (3.1), the lemma follows for $8\leq t-s\leq 11$. ∎

By the $\mathrm{Sq}^{1}$, $\mathrm{Sq}^{2}$ and $\mathrm{Sq}^{4}$ actions on
$C^{\ast}(\mathrm{PL})$ (shown in Section 2) and the AAHSS (3.1), we have

######  Lemma 3.6.

Partial cokernel of $\mathscr{D}\langle 8\rangle_{\ast}$ is shown as follows

$t-s$ | 8 | 9 | 9 | 10 | 10 | 11
---|---|---|---|---|---|---
s |  $\geq$ 0 |  $\geq$0 | 0 | 0 | 1 | 2
cok$(\mathscr{D}\langle 8\rangle_{\ast})$ | $h_{0}^{s}p_{8}$ | $h_{0}^{s}p_{9,1}$ | $p_{9,2}$ | $p_{10,3}$ |  $h_{1}p_{9,1}$, $h_{1}p_{9,2}$ |  $h^{2}_{1}p_{9,1}$, $h^{2}_{1}p_{9,2}$

######  Lemma 3.7.

In the $\mathrm{ASS}$

$\mathrm{Ext}^{s,t}_{\mathscr{A}}(\mathrm{PL}\langle
8\rangle)=\mathrm{Ext}^{s,t}_{\mathscr{A}}(H^{\ast}(\mathrm{MPL}\langle
8\rangle),\mathbb{Z}_{2})\Longrightarrow{{}_{2}\pi}_{t-s}(\mathrm{MPL}\langle
8\rangle)$ (3.2)

$d_{2}(p_{9,1})=h^{2}_{0}p_{8}$, $d_{3}(p_{9,2})=c_{0}$,
$d_{4}(h_{1}p_{9,1})=h_{1}\omega$.

###### Proof.

It is easy to see that $d_{2}(p_{9,1})=h^{2}_{0}p_{8}$ follows from the proof
of Lemma 2.5 and the map $\mathrm{MPL}\langle 8\rangle\to\mathrm{MSPL}$.

Since that $c_{0}\in\pi_{8}(\mathrm{MO}\langle 8\rangle)$ denotes an exotic
sphere (see Lemma 3.4), and
$c_{0}\in\mathrm{Ext}^{3,11}_{\mathscr{A}}(\mathrm{PL}\langle 8\rangle)$ (see
Lemma 3.5), we have that $c_{0}$ must be killed by a differential in the ASS
(3.2). So $d_{3}(p_{9,2})=c_{0}$.

Similarly, $h_{1}^{s}\omega$ for $s=1,2$ must also be killed by certain
differentials. By
$h_{1}p_{10,3}=0\in\mathrm{Ext}^{1,12}_{\mathscr{A}}(\mathrm{PL}\langle
8\rangle)$, $d_{5}(p_{10,3})\neq h_{1}\omega$. Since
$h^{2}_{0}h_{1}p_{8}=0\in\mathrm{Ext}^{3,12}_{\mathscr{A}}(\mathrm{PL}\langle
8\rangle)$, $d_{2}(h_{1}p_{9,1})=0$. So $d_{4}(h_{1}p_{9,1})=h_{1}\omega$. ∎

######  Proposition 3.8.

${\pi}_{8}(\mathrm{MPL}\langle 8\rangle)=\mathbb{Z}_{4}\oplus\mathbb{Z},$

${\pi}_{9}(\mathrm{MPL}\langle 8\rangle)=0,$ ${\pi}_{10}(\mathrm{MPL}\langle
8\rangle)=\mathbb{Z}_{2}.$

###### Proof.

By Lemma 3.5, 3.6 and 3.7, we get the ${{}_{2}\pi}_{n}(\mathrm{MPL}\langle
8\rangle)$. By Lemma 3.1 and the spectral sequence

$E_{2}^{p,q}=H_{p}(\mathrm{MPL}\langle
8\rangle;\pi_{q}(S^{0}))\Longrightarrow\pi_{p+q}(\mathrm{MPL}\langle
8\rangle)$

we see that ${\pi}_{n}(\mathrm{MPL}\langle 8\rangle)$ does not have odd
torsion for $n=8,9$.

$\textstyle{{E}_{2}^{p,q}(\mathrm{O}8)=H_{p}(\mathrm{MO}\langle
8\rangle;\pi_{q}(S^{0}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{p+q}(\mathrm{MO}\langle
8\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{E}_{2}^{p,q}=H_{p}(\mathrm{MPL}\langle
8\rangle;\pi_{q}(S^{0}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{p+q}(\mathrm{MPL}\langle
8\rangle)}$

Note that $\pi_{10}(\mathrm{MO}\langle
8\rangle)={E}_{\infty}^{0,10}(\mathrm{O}8)={E}_{2}^{0,10}(\mathrm{O}8)=\mathbb{Z}_{6}$.
By Lemma 3.4, $\mathscr{D}\langle
8\rangle_{\ast}({E}_{\infty}^{0,10}(\mathrm{O}8))=0$. Thus,
${E}_{\infty}^{0,10}=0$. By Lemma 3.1 and the spectral sequence,
${\pi}_{10}(\mathrm{MPL}\langle 8\rangle)$ does not have odd torsion. ∎

It is very difficult to compute ${\pi}_{11}(\mathrm{MPL}\langle
8\rangle),{\pi}_{13}(\mathrm{MPL}\langle 8\rangle)$ by the algebraic method.
Next, we consider them by surgery theory.

Recall the first stability theorem

######  Theorem 3.9.

[19] Let $\mathrm{CAT}=\mathrm{O}$ or $\mathrm{PL}$. For $i\leq m+1$, $m\geq
5$,

$\pi_{i}(\mathrm{TOP/CAT},\mathrm{TOP_{m}/CAT_{m}})=0.$

As $\pi_{i}(\mathrm{O}_{m})\cong\pi_{i}(\mathrm{O})$ for $i\leq m-2$, we have

######  Corollary 3.10.

$\pi_{i}(\mathrm{TOP_{m}})\cong\pi_{i}(\mathrm{TOP})$,
$\pi_{i}(\mathrm{PL_{m}})\cong\pi_{i}(\mathrm{PL})$ for $4\leq i\leq m-2$.

###### Proof.

Consider the following exact sequences

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{i}(\mathrm{CAT}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{i}(\mathrm{TOP_{m}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{i}(\mathrm{TOP_{m}/CAT_{m}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{i}(\mathrm{CAT})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{i}(\mathrm{TOP})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{i}(\mathrm{TOP/CAT})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

If $\mathrm{CAT}=\mathrm{O}$,
$\pi_{i}(\mathrm{TOP_{m}})=\pi_{i}(\mathrm{TOP})$ for $4\leq i\leq m-2$ by the
Five-Lemma. For $m\geq 5,$
$\mathrm{TOP_{m}/PL_{m}}=\mathrm{K}(\mathbb{Z}_{2},3)$ [19]. Hence
$\pi_{i}(\mathrm{PL_{m}})=\pi_{i}(\mathrm{PL})$ for $4\leq i\leq m-2$. ∎

######  Proposition 3.11.

${\pi}_{11}(\mathrm{MPL}\langle 8\rangle)={\pi}_{13}(\mathrm{MPL}\langle
8\rangle)=0.$

###### Proof.

Let $n=5$ or $6$. For any bordism class

$\\{M\\}\in{\Omega}_{2n+1}^{\mathrm{PL}\langle
8\rangle}\cong{\pi}_{2n+1}(\mathrm{MPL}\langle 8\rangle),$

following the surgery theory, we can assume that $H_{i}(M;\mathbb{Z})$ is
nontrivial only for $i=0,n,n+1,2n+1$. Let $f:M\to\mathrm{BPL}\langle 8\rangle$
be a $\mathrm{BPL}\langle 8\rangle$-structure of $M$. Since $\pi_{n}(M)\cong
H_{n}(M;\mathbb{Z})$, any generator $x\in H_{n}(M;\mathbb{Z})$ can be
represented by a PL embedding $g:S^{n}\to M$ [33]. Note that $g\circ f\simeq
0$. By Corollary 3.10, $\pi_{n}(\mathrm{BPL}_{n+1})=0$. The embedding $g$ can
be represented by the PL embedding $\bar{g}:S^{n}\times D^{n+1}\to M$. Let $F$
be the homotopy fibre of the map $\mathrm{BPL}\langle
8\rangle\to\mathrm{BPL}$. Since
$\pi_{n}(F)\cong\pi_{n+1}(\mathrm{BPL})\cong\pi_{n+1}(\mathrm{BO})=0$, the
$\mathrm{BPL}\langle 8\rangle$-structure of $S^{n}$ is unique. Hence we can
kill the elements in $H_{n}(M;\mathbb{Z})$ by surgery. In other words, $M$ is
$\mathrm{BPL}\langle 8\rangle$-bordant to $S^{2n+1}$. So
${\Omega}_{2n+1}^{\mathrm{PL}\langle 8\rangle}=0$. ∎

##  4\. Kervaire invariant

By the basic results of [19], we have a PL version of the Theorem 1 of [11] or
the Corollary 4 in Section 7 of [20].

######  Theorem 4.1.

Let $\xi$ be a $\mathrm{PL}$ linear bundle over a simply connected
$\mathrm{CW}$ complex $X$. Let $M^{2n}_{i}\stackrel{{\scriptstyle
f_{i}}}{{\to}}X$, $i=0,1$, $n\geq 5$, be normal maps from closed $\mathrm{PL}$
manifolds, i.e. $f^{\ast}_{i}\xi=\nu(M_{i})$, the stable normal bundle over
$M_{i}$. Suppose that $n$ is odd, $f_{0}$ and $f_{1}$ are normally bordant,
$f_{i}$ is $n$-connected, and that $B_{n}(M_{0})=B_{n}(M_{1})$ where $B_{n}$
denotes the $n$-th Betti number. Then $M_{0}$ and $M_{1}$ are
$\mathrm{PL}$-homeomorphic.

Here, we take $X=\mathrm{BPL}\langle 8\rangle$ associated with the universal
PL bundle $\xi=\gamma_{8}^{pl}$. Let $M^{10}$ be a closed 4-connected
manifold. So $H^{5}(M;\mathbb{Z})$ is free abelian of even rank $2s$. By the
fibration

$\mathrm{BPL}\to\mathrm{BTOP}\to\mathrm{K}(\mathbb{Z}_{2},4)$

$M$ admits a unique PL-manifold structure. Moreover, by obstruction theory,
there exists a normal map $f:M\to\mathrm{BPL}\langle 8\rangle$, unique up to
homotopy. Then, by Theorem 4.1, we have

######  Corollary 4.2.

Let $M^{10}_{i}$, $i=0,1$ be closed $4$-connected manifolds with the same
$B_{5}(M_{i})=2s$. They are $\mathrm{PL}$-homeomorphic if $M_{0}$ is normally
$\mathrm{BPL}\langle 8\rangle$-bordant to $M_{1}$.

In [17], Kervaire presented an invariant for such closed 4-connected
manifolds. Denote it by $\Phi(M)\in\mathbb{Z}_{2}$.

######  Lemma 4.3.

If a $4$-connected closed manifold $M$ of dimension $10$ is
$\mathrm{BPL}\langle 8\rangle$-bordant to $0$, then $\Phi(M)=0$.

###### Proof.

The assumption implies the existence of a $\mathrm{BPL}\langle
8\rangle$-manifold $V^{11}$ with boundary $M$. By classic surgery theory, we
can find a new $V$ so that it is 4-connected. Then the lemma follows from the
proof of Lemma 2.2 in [17]. ∎

By observing that $\Phi$ is additive with respect to the connected sum of
manifolds, we have

######  Corollary 4.4.

For two $4$-connected closed $10$-dimensional $\mathrm{PL}$ manifolds $M$ and
$M^{\prime}$, if $M$ is $\mathrm{BPL}\langle 8\rangle$-bordant to
$M^{\prime}$, $\Phi(M)=\Phi(M^{\prime})$.

######  Lemma 4.5.

A $4$-connected closed manifold $M$ of dimension $10$ represents a nontrivial
bordism class in $\Omega_{10}^{\mathrm{PL}\langle 8\rangle}$ iff $\Phi(M)=1$.

###### Proof.

Note that the PL sphere $S^{10}$ represents a trivial bordism class,
$\Phi(S^{10})=0$. From [17], there exists a closed 4-connected manifold
$M_{0}$ of dimension $10$ for which $\Phi(M_{0})=1$. Thus the theorem follows
by Corollary 4.4 and $\Omega_{10}^{\mathrm{PL}\langle
8\rangle}=\mathbb{Z}_{2}$. ∎

By Corollary 4.2 and Lemma 4.5, we have

######  Theorem 4.6.

The $B_{5}(M)$ and $\Phi(M)$ are a complete set of invariants of the
$\mathrm{PL}$-homeomorphism type of the $4$-connected closed manifold $M$ of
dimension $10$.

##  5\. Classification for certain 13-manifolds

In this section, let $\mathcal{M}$ be a PL manifold of dimension $13$ so that
$H^{\ast}(\mathcal{M};\mathbb{Z})\cong H^{\ast}(\mathrm{CP}^{3}\times S^{7})$.
As in [35], we define the normal $B$-structure for the manifolds with
restriction lift  (see Definition 1.4).

By Remark 1.6, denote the first Pontryagin class of $\mathcal{M}$ by
$p_{1}(\mathcal{M})=sx^{2}$ $s\in\mathbb{Z}$, the second Stiefel-Whitney class
by $w_{2}(\mathcal{M})=sx\mod{2}$. Let $\mathcal{H}$ denote the Hopf (vector)
bundle over $\mathrm{CP}^{4}$. If $s\geq 0$, let $\xi$ be the complementary
bundle of the Whitney sum $s\mathcal{H}$; if $s<0$, let $\xi$ be the Whitney
sum $-s\mathcal{H}$. Formally $\xi=-s\mathcal{H}$.

Let $\pi:{B}=\mathrm{BPL}\langle 8\rangle\times\mathrm{CP}^{4}\to\mathrm{BPL}$
be the classifying map of the bundle $\gamma_{8}^{pl}\times\xi$ where
$\gamma^{pl}_{8}$ is the universal bundle over $\mathrm{BPL}\langle 8\rangle$,
$\xi$ as a PL bundle forgets its vector bundle structure. Obviously, a
manifold $\mathcal{M}$ with restriction lift  has a normal $B$-structure i.e.
its Gauss map has a lift to $B$.

All $n$-dimensional manifolds with a normal ${B}$-structure form a bordism
group $\Omega_{n}^{\mathrm{PL}\langle 8\rangle}(\xi)$ which is isomorphic to
the homotopy group $\pi_{n}(\mathrm{M{B}})$ by the Pontryagin-Thom
isomorphism. Here, $\mathrm{M{B}}=\mathrm{MPL}\langle
8\rangle\wedge\mathrm{M}\xi$; $\mathrm{M}\xi$ and $\mathrm{MPL}\langle
8\rangle$ are the Thom spectra of the bundles $\xi$ and $\gamma_{8}$. Note
that if $p_{1}(\mathcal{M})=p_{1}(\mathcal{M}^{\prime})$, $\mathcal{M}$ and
$\mathcal{M}^{\prime}$ will be located in the same bordism group
$\Omega_{13}^{\mathrm{PL}\langle 8\rangle}(\xi)$.

By the same arguments as [35] and basic results of [19], we have

######  Lemma 5.1.

Let $p_{1}(\mathcal{M})=p_{1}(\mathcal{M}^{\prime})$. If $\mathcal{M}$ is
$B-$coborant to $\mathcal{M}^{\prime}$, then there is a $\mathrm{PL}$
$h$-cobordism $(W,\mathcal{M},\mathcal{M}^{\prime}).$

Now, consider the computation of $\Omega_{13}^{\mathrm{PL}\langle
8\rangle}(\xi)$. There is an Atiyah-Hirzebruch spectral sequence (AHSS) for
$\pi_{\ast}(\mathrm{MB})$

$E_{2}^{p,q}\cong H_{p}(\mathrm{M}\xi;\pi_{q}(\mathrm{MPL}\langle
8\rangle))\Longrightarrow\mathrm{MPL}\langle
8\rangle_{\ast}(\mathrm{M}\xi)\cong\pi_{\ast}(\mathrm{MPL}\langle
8\rangle\wedge\mathrm{M}\xi).$

By Proposition 1.11, $E_{2}^{p,13-p}=0$ for $0\leq p\leq 13$. Hence
$\Omega_{13}^{\mathrm{PL}\langle 8\rangle}(\xi)=0$.

The (1) of Theorem 1.5 follows from Lemma 5.1. Thus, any $\mathcal{M}$ with
restriction lift can be regarded as the total space of the sphere bundle of a
$8$-dimensional vector bundle over $\mathrm{CP}^{3}$ (see Remark 1.7). The
homotopy classification of the manifolds transforms to the fibre homotopy
classification of $S^{7}$-bundles over $\mathrm{CP}^{3}$. Subsequently, the
(2) of Theorem 1.5 follows from the lemma 5.2 of [35].

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Quantitative real-time measurements of dose and dose rate in UHDR proton
pencil beams via scintillation imaging system
Megan Clark1, Joseph Harms2, Roman Vasyltsiv1, Austin Sloop1, Jakub Kozelka3,
Bill Simon3, Rongxiao Zhang1, David Gladstone1, Petr Bruza1

1Thayer School of Engineering, Dartmouth College, Hanover, NH
2University of Alabama at Birmingham, Department of Radiation Oncology,
Birmingham, AL
3Sun Nuclear Inc., Melbourne FL 32940, US
Version typeset

Author to whom correspondence should be addressed is Megan Clark.

email<EMAIL_ADDRESS>
###### Abstract

Background: Ultra-high dose rate radiotherapy (UHDR-RT) has demonstrated
normal tissue sparing capabilities, termed the FLASH effect; however,
available dosimetry tools make it challenging to characterize the UHDR beams
with sufficiently high concurrent spatial and temporal resolution. Novel
dosimeters are needed for safe clinical implementation and improved
understanding of the effect of UHDR-RT.
Purpose: Ultra-fast scintillation imaging has been shown to provide a unique
tool for spatio-temporal dosimetry of conventional cyclotron and
synchrocyclotron pencil beam scanning (PBS) deliveries, indicating the
potential use for characterization of ultra-UHDR PBS proton beams. The goal of
this work is to introduce this novel concept and demonstrate its capabilities
in recording complex dose rate maps at FLASH-capable proton beam currents, as
compared to log-based dose rate calculation, internally developed UHDR beam
simulation, and a fast point detector (EDGE diode).
Methods: The light response of a scintillator sheet located at isocenter and
irradiated by pencil beam scanning proton fields (40-210 nA, 250 MeV) was
imaged by an ultra-fast iCMOS camera at 4.5-12 kHz sampling frequency. Camera
sensor and image intensifier gain were optimized to maximize the dynamic
range; the camera acquisition rate was also varied to evaluate the optimal
sampling frequency. Large field delivery enabled flat field acquisition for
evaluation of system response homogeneity. Image intensity was calibrated to
dose with film and the recorded spatio-temporal data was compared to a PPC05
ion chamber, log-based reconstruction, and EDGE diode. Dose and dose rate
linearity studies were performed to evaluate agreement under various beam
conditions. Calculation of full-field mean and PBS dose rate maps were
calculated to highlight the importance of high resolution, full-field
information in UHDR studies.
Results:Camera response was linear with dose (R2 = 0.997) and current (R22 =
0.98) in the range from 2-22 Gy and 40-210 nA, respectively, when compared to
ion chamber readings. The deviation of total irradiation time calculated with
the imaging system from the log file recordings decreased from 0.07% to 0.03%
when imaging at 12 kfps versus 4.5 kfps. Planned and delivered spot positions
agreed within 0.2 ± 0.1 mm and total irradiation time agreed within 0.2 ± 0.2
ms when compared with the log files, indicating the high concurrent spatial
and temporal resolution. For all deliveries, the PBS dose rate measured at the
diode location agreed between the imaging and the diode within 3 ± 2% and with
the simulation within 5 ± 3%
Conclusions: Full-field mapping of dose and dose rate is imperative for
complete understanding of UHDR PBS proton dose delivery. The high linearity
and various spatiotemporal metric reporting capabilities confirm the continued
use of this camera system for UHDR beam characterization, especially for
spatially resolved dose rate information.

##  I. Introduction

Proton therapy delivered at ultra-high dose rates has indicated a potential
reduction in normal tissue toxicity[1, 2, 3, 4, 5, 6], termed the FLASH
effect, leading to the first in-human clinical trial: The FAST-01
Nonrandomized Trial [7]. Some studies suggest a dose-rate threshold of 40-100
Gy/s is needed to elicit this effect, thus requiring that treatment planning
considers dose rate in addition to traditional dose volume histogram-based
planning [8, 9]. Despite the rapid push towards UHDR clinical implementation,
currently available dosimetry tools make it challenging to fully characterize
and monitor the dose and dose rates being delivered [10].
Ionization chambers used in proton FLASH studies, namely the PPC05 parallel-
plate ionization chamber, exhibit dose rate dependencies, thus requiring
extensive calibration of the recombination factor [3, 11, 12, 13].
Additionally, ionization chambers only provide information on a small
sensitive volume (for PPC05 0.05 cm2), making dose-rate validation of the
entire scanned field impossible. A 2D strip ionization chamber has reported
strong spatial-temporal mapping of UHDR proton PBS deliveries at a sampling
rate of 20 kHz, but requires implementation of 2D dose reconstructions and a
fixed geometry setup [14]. EDGE detector diodes and scintillator fibers have
demonstrated dose and dose rate (mean and instantaneous) independence at sub
millisecond resolution but lack concurrent spatial information [15, 16, 17].
Other novel detection methods include PET imaging and silicon carbon
detectors, which suffer from similar challenges [18, 19]. In summary, there
are no existing devices that allow for full-field spatial resolution at the
temporal resolution required for real-time monitoring of UHDR beams.
Luminescent detectors have been shown to provide adequate spatial and temporal
resolution for the demands of UHDR-RT [20, 21, 22, 23, 24]. Specifically,
scintillation-imaging has been used to acquire accurate dose and dose rate
measurements of clinical cyclotron and synchrocyclotrons, indicating potential
use for high-speed mapping of the scanning beam [25, 26]. It also enables
remote monitoring of the beam during delivery and accommodation of different
imaging surfaces and geometric setups. Advancements to the camera system
previously described by Clark et al, allow for over 12 kHz frame rate
acquisitions, thus extending the application of this imaging system to ultra-
high dose rate capable PBS systems [25].
The intensified camera used in this study is the first-of-its-kind to image an
UHDR cyclotron beam at up to 12 kHz frame rate and with concurrent
submillimeter (0.22×0.22 mm2) spatial resolution. In this study, the camera
sensor and image intensifier gain were optimized to maximize the dynamic
range, and the camera acquisition rate was varied to evaluate the optimal
sampling frequency. Image intensity was calibrated to dose with film and
camera angular corrections, flat and dark field, and background subtraction
image corrections were performed in MATLAB. The recorded spatio-temporal data
was compared to a PPC05 ion chamber, internally developed simulation, and EDGE
diode for validation of the reconstruction model and evaluation of the imaging
system performance. Dose and dose rate linearity studies were performed to
evaluate agreement under various beam conditions. Calculation of full-field
mean, instantaneous, and PBS dose rate maps were calculated to highlight the
importance of high resolution, full-field information in UHDR studies and
demonstrated the importance of various critical reporting parameters for UHDR
PBS deliveries.

##  II. Methods

###  II.A. Experimental Setup

An intensified CMOS camera (BeamSite UTLRA, DoseOptics) was set up on a tripod
to image the light response of scintillator sheet during irradiation of pencil
beam scanning proton fields (30-99 nA, 250 MeV) in the treatment vault of a
Varian ProBeam cyclotron system at the University of Alabama Birmingham Proton
Center in Birmingham, Alabama, Figure 1. The acquisition frame rated ranged
from 4.5-12 kHz, and all frames were streamed and saved to a fast computer
solid state drive (SSD). A Gen2 blue-sensitive intensifier with P46 fast-decay
phosphor was specifically chosen to avoid temporal blur due to luminescence
decay and afterglow in more standard phosphors. Camera analog gain of 1V and
image intensifier gain of approximately 1.8V were selected to maximize the
dynamic range. The camera acquisition rate was varied to evaluate the impact
on dose rate calculations and to optimize the sampling frequency used in later
studies. Large field delivery enabled flat field acquisition for evaluation of
spatial homogeneity.
A beam target consisting of an 8x11 in, 0.2 mm thick scintillator sheet
(Rapidex by Scintacor, Cambridge UK) placed at isocenter and laid over stacks
of solid water. Image intensity was calibrated to dose with film (Gafchromic
EBTXD, Ashland), which was placed and aligned directly under the scintillator
sheet. An EDGE diode detector and PPC05 ion chamber were placed 5 cm and 3 cm
below the film for temporal and cumulative dose comparisons, respectively.
When operating in UHDR mode, the VarianProbeam log file system only reports
cumulative dose, total irradiation time, dose rate, and percent of the field
receiving a certain dose rate. There is limited publicly available information
regarding the parameters required for accurate calculation of dose rate, as is
done by the log file system. Therefore, an internally developed simulation
(Section II.E.) was developed in MATLAB for a more comprehensive comparison to
the imaging data. One of the secondary aims of this work was to validate this
beam simulation for improved treatment planning of UHDR deliveries at UAB.

###  II.B. Treatment Plans

The baseline treatment plan for this study was designed to deliver 15 Gy to
the treatment isocenter with a spot spacing of 5 mm and single spot dwell
times around 3-4 ms, based on recommendations from the vendor. A diamond
pattern was used to maximize PBS dose rate, which will be defined in Section
II.F. below. The fields were 5 cm across the central axis in both x and y to
ensure lateral charged particle equilibrium at the center of the field for
accurate verification with an independent ion chamber. To characterize
linearity of the camera and scintillation system response with varying dose,
the MU/spot were either decreased or increased from the nominal plan to
deliver fields from 2 Gy up to 22 Gy. Dose rate linearity was studied by
delivering the 15 Gy baseline plan with various scanning nozzle currents
ranging from 30 nA to 99 nA. The system responded linearly with dose and
current (R2 = 0.997, R2 = 0.98), and so these fields were used for validation
of the simulation, as shown in Figure 1. UHDR PBS fields were delivered with a
Varian Probeam system converted to ultra-high dose rate mode after clinical
hours.

###  II.C. Image Processing

All image processing was performed in MATLAB (Version 2022b). Image stacks
were offset-corrected with background subtraction, divided by the flat-field
image for spatial and camera nonlinearity correction, spatially transformed
and pixel-size calibrated to account for the camera angle, and the image
intensity was calibrated to absolute dose with film [25]. Each individual
image of the scanning spot was used for evaluation of beam position, dose, and
dose rate. To identify the center of the scanning spot, each frame was fit to
a gaussian and the center of the gaussian was localized for comparison to the
treatment plan.
Following the image correction methods previously defined by Clark et al in
2022 for imaging at 1 kHz of a conventional synchrocyclotron, the camera was
focused to a spatial calibration grid located at isocenter under traditional
room lighting and low intensifier gain [25]. Imaging of this calibration grid
provided a geometric transform (to account for the oblique angle of the camera
relative to the scintillating sheet/incident beam), pixel size calibration,
and spatial resolution measurements to be used in image analysis. A large,
10×20 cm2 (flat) field was delivered at 250 MeV and imaged at 4.5 kHz
concurrently with EBTXD film measurement located under the scintillator sheet.
A flat-field correction map was calculated by dividing the geometry-corrected
cumulative camera image by the 2D dose distribution measured by film. All
subsequent image stacks were divided by this correction map to minimize
spatial response inhomogeneity of the scintillator and camera image forming
components, including lens and intensifier.

###  II.D. Frame Rate Optimization

A single spot plan was delivered and imaged at 10, 50, 70, and 99 nA with
film, acting as the reference, to offer insight into camera spatial accuracy.
The imaging system provided 0.22×0.22 mm2 spatial resolution, such that the 99
nA single spot full width half maximum agreed with the film within 0.5 mm in
both the x and y directions. The intensifier and sensor electronic gains were
set and kept constant for this entire study to accommodate high current
deliveries, as ultra-high dose rates were the beams of interest; this leads to
the higher dose rate linearity error and spot disagreement at low currents. To
find the optimal combination of electronic gains, we repeatedly imaged a 99 nA
beam delivery while varying the intensifier gain (IG) from 3V down to 0.8 V
and sensor gain (“analog” gain, AG) from 1x to 4x. At low IG/high AG the
amplified sensor noise limited the low photon response, while at extremely
high IG we observed temporary (millisecond time scale) charge depletion
leading to locally decreased sensitivity of the camera. An optimal combination
of IG = 1.8V and AG = 1V was set and kept constant for all measurements here.
This updated camera system can image at over 12,000 frames per second. A frame
rate study was performed by imaging the same beam at 4.5k, 7k, 10k, and 12k to
investigate improvements in temporal accuracy with higher frame rates. For all
frame rates, deviation from the log files was under 0.32 ms, with deviation
from log file total irradiation time decreasing with increasing frame rate
(Figure 1). This highlighted the potential capabilities of the camera system
for collecting accurate dose rate information, but a 4.5kHz frame rate was
chosen for this study to capture sufficiently accurate per-spot information
while minimizing data size for processing and storage purposes.

Figure 1:  a.) Camera facing scintillator sheet, diode, and PPC05 at
isocenter. b.) Results of dose and dose rate linearity studies from 2-22Gy and
30-220nA, respectively. c.) Impact of frame rate on imaged dose rate deviation
from log file. d.) Agreement between imaged and film data of single spot
profile at 99nA.

###  II.E. Simulation and Log-file Comparison

To acquire a better understanding of the dynamics of UHDR PBS deliveries, an
independent simulation was developed in MATLAB to mimic machine delivery and
provide further validation of the scintillator imaging system. This simulation
takes as input either a measured single spot map or functional spot model
(i.e. a 2D Gaussian distribution), planned spot map including MU/spot, and
scanning nozzle current. The user can also input parameters dose, dose rate,
and thresholds for calculation of PBS dose rate. The output of the simulation
are dose and dose rate maps for the field in question. From these maps,
parameters of interest such as mean dose rate or PBS dose rate for a given
pixel can be extracted. For initial validation, the total delivery time for
each field was used to determine scanning speed in the X and Y directions. For
comparison to the simulation, gamma analysis was performed on the full field
images at 3%/2mm, per proton characterization standards.

###  II.F. Dose Rate Definitions

Unlike point detectors, this imaging system enables full-field implementation
of different dose rate definitions. Pencil beam scanning dose rate, $D_{PBS}$,
defined by Folkerts et al. in 2021, was used for comparison of the diode, log-
based reconstruction, and log file recordings, as per convention in UHDR
proton beam studies and the FAST-01 trial [7, 27]. This dose rate definition
only considers the amount of time that dose, within a certain dose threshold
$d_{\dagger}$ of the min/max dose $D(\vec{x})$, is delivered to a certain
point in the field, $\vec{x}$:

$\dot{D}_{PBS}(\vec{x})=\frac{D(\vec{x})-2d_{\dagger}}{T(\vec{x})}.$ (1)

In addition to PBS dose rate maps, full-field maps of mean and maximum
instantaneous dose rate were calculated to highlight the variability of dose
rate with location in the field and definition of dose rate used. The mean
dose rate to a point $\vec{x}$ is defined as the total dose to that point,
$D(\vec{x})$, divided by the total delivery time, ${T_{tot}}$:

$\dot{D}_{mean}(\vec{x})=\frac{D(\vec{x})}{T_{tot}}.$ (2)

The maximum instantaneous dose rate is defined as the maximum dose to a point
in the field over the time between sampling frames:
$\dot{D}_{inst}=\frac{D_{max}(\vec{x})}{dt}$.
As is further discussed in the results section, reporting of dose thresholds
used for PBS dose rate definitions is imperative for meaningful comparison of
dose rate. For this work, a threshold dose, $d_{\dagger}$ of 0.5 Gy was chosen
and held constant for image analysis of all deliveries. While Varian reports a
$d_{\dagger}$ equal to 0.01 Gy in the calculation of PBS dose rate, the
limited dynamic range of the camera, which leads to uncertainty in the low-
dose region of each spot, required the higher chosen threshold. An additional
10% maximum dose threshold was applied when calculating the percent of the
field greater than 40Gy/s, such that only regions of the field receiving
greater than 10% of the maximum dose were considered. Investigation into the
impact of $d_{\dagger}$ on the calculated dose rate is reported and compared
to that simulated in the original publication by Folkerts. Additional study
into the impact on percent of the field greater than 40% is provided in the
results here. While Folkerts et al. simulate the dependency on dose rate on
$d_{\dagger}$, we believe this is the first measured dose rate area histogram
demonstrating this relationship.

##  III. Results

###  III.A. Spot Tracking and Beam Dynamics Characterization

An acquisition rate of 4,500 frames/second enabled accurate estimates of
dynamic scanning spot parameters including spot positions, spot dwell time,
spot scanning speed, and total irradiation time. Imaged spot positions agreed
within 0.2 ± 0.1 mm to the planned spot positions (Figure 2). Spot dwell time
increased from 3.61 ms for the 2 Gy delivery to 56.2 ms for the 15 Gy
delivery, with an average standard deviation in dwell time of only 0.2 ms. A
5x5 cm2 field was delivered with increased, 8 mm, spot spacing to enable
improved estimating of spot scanning speeds. This provided measurements of
spot scanning speeds of 12 ± 1 m/s in the x direction and 24 ± 9 m/s in the y
direction. Total irradiation times for the cumulative dose delivery for the 15
Gy, 99nA case was 792 ms and agreed within 0.1% with the simulation and 0.01%
with the log file. For the data accumulated during the dose rate linearity
study the total irradiation times agreed within an average 0.2 ± 0.2 ms when
compared with the log file recordings and within 1 ± 1 ms compared to the
simulated runs.

Figure 2: Illustration of scanning spot localization agreement to plan and
ability to capture multiple frames per spot.

###  III.B. Dose and Dose Rate Agreement

Calculating PBS dose-rate distribution is non-trivial due to complicated
spatio-temporal dynamics of PBS delivery. Here we compare the dose rate
quantities (see Table included in Figure 3) as reported by 1.) the
scintillator camera system, 2.) the log-based dose rate value, 3.) the UHDR
beam simulation, and 4.) the EDGE diode point detector in Table 1. For each
field used in the dose and dose rate linearity studies, full-field dose and
dose rate maps were compared between the imaging data and simulation, as shown
with the example 15 Gy, 99nA case in Figure 4. To compare to the diode, an
image was taken of the diode location without the scintillation sheet placed
on top and under standard lighting conditions, allowing localization of the
sensitive area of the diode within the images.

Figure 3: Example of potential comparison metrics between the image data,
simulation, log file recordings, and diode, highlighting the strong agreement
across the data modalities.

The dose and dose rate values at the diode location, the center of the field,
and the edge of the field (where higher dose rates are expected due to shorter
delivery times), were compared to highlight the sensitivity of dose rate to
the location of the field due to the scanning beam. Between the diode location
and center of the field, though less than 1 cm apart, there are significant
differences in the spatial temporal dose distribution, resulting in different
calculations of PBS dose rate (55 vs 58 Gy/s or 5% deviation). This highlights
the importance of full-field information for better understanding of the dose
rate distribution, as is elaborated on in the discussion. Cumulative 2D dose
maps agreed with film measurements and simulation cumulative maps with a mean
gamma passing rate at 3%/2mm of 100% and 99.3%, respectively, thus confirming
the spatial accuracy of the imaging system. The PBS dose rate maps agreed with
the simulation with a mean gamma passing rate at 3%/2mm of 98.7%. The PBS dose
rate measured at the diode location agreed between the imaging and the diode
within 3±2% and with the simulation within 5±3%. The simulation agreed with
the diode within 5±4%. Mean dose rates were calculated with imaging system to
be 18.7±2 Gy/s from the measured cumulative dose and total irradiation time
and agreed with the log file within 2±1%, with the simulation within 6±3%, and
the simulation agreed with the log files within 5±4%. The percentage of the
field receiving greater than 40Gy/s measured with the imaged maps agreed with
simulation 8±8%.

Figure 4: Comparison of full-field dose and dose rate maps between imaging and
simulated 15Gy delivery at 99nA, with temporal and representative dose rate
measurements at three points of interest within the field.

###  III.C. PBS Dose Rate Thresholds

As mentioned in Section II.F., dose rate, and thus the percent of the field
receiving a certain dose rate, is highly dependent on input threshold
parameters by the user. When defining PBS dose rate, Folkerts et al. simulated
the dose rate dependency on $d_{\dagger}$, or the dose threshold at which the
accumulation of dose is recognized, as shown in Figure 5. Here, we present the
first validation of this simulation, where PBS dose rate is demonstrated to
range from 35 Gy/s for a $d_{\dagger}$ of 0.01 Gy compared to 100 Gy/s for
$d_{\dagger}$ equal to 1 Gy, as shown in Figure 5. Because of the clear
dependency of dose rate on this threshold dose, it can be expected that the
reported the percentage of the field achieving the FLASH dose rate threshold,
which was set at 40 $Gy/s$ for this study to mimic the dose rate threshold set
in FAST-01, would also be highly dependent on this dose threshold. Figure 5
demonstrates that dependency, with deviations of 16% between the percentage of
fields, when only changing dose threshold.

Figure 5: a.) Visualization from imaging data of Folkert et al PBS dose rate
definition with 0.5Gy $d_{\dagger}$. b.) Demonstration of dependency of PBS
dose rate at the center of the field on the dose threshold, $d_{\dagger}$. c.)
Dependency of the percentage of the field achieving the FLASH dose rate
threshold on the dose threshold, $d_{\dagger}$.

##  IV. Discussion

Concurrent sub-millimeter ($<$0.5 mm) and sub-millisecond (±1 ms)
spatiotemporal resolution is a unique capability of this camera-scintillator
system, validated here. Additionally, this system enables remote, passive,
real-time beam monitoring without interfering with the beam path. Existing,
novel tools that have demonstrated the ability for temporal characterization
of UHDR PBS systems include scintillating point detectors, diodes, and
calibrated ionization chambers ; however, these systems lack spatial
information important to the dynamics of PBS deliveries[15, 13, 17]. The 2-D
strip ionization chamber proposed by Yang et has reported similar spatial-
temporal mapping of UHDR PBS deliveries at a sampling rate of 20 kHz, but
requires implementation of 2D dose reconstructions and a fixed geometry setup
[14]. Therefore, this system is the only available to provide external
validation of UHDR PBS deliveries.
The strongly inhomogeneous dose rate maps typical of UHDR PBS fields require
high spatial resolution for quantitative assessment. With delivery times on
the timescale of milliseconds to seconds, and spot dwell times on the order of
milliseconds, maximum instantaneous dose rates were measured to be over 450
Gy/s in the case of 15Gy delivered at 99nA. This means, that when calculating
PBS dose rate, a small deviation in temporal accuracy may lead to a high
deviation in calculated PBS dose rate. The highest current, low dose case (5
Gy, 99nA) delivered with the shortest measured spot dwell times, at 3.8 ± 0.5
ms. Therefore, to accurately image this case with $<$3% error in dose rate,
assuming accurate capture of dose distributions, a dosimetry system would need
to operate at over 10 kHz. This resolution requirement is highlighted by
looking at the results of the 15Gy/99nA delivery again, where even a deviation
by 0.001 seconds for the measured $T(\vec{x})$, as described in Figure 4, can
lead to a deviation in the calculated PBS dose rate at the center of the field
of 3.5 Gy/s or 7.5%. Not only do UHDR-capable beams require dosimeters with
high spatial resolution for full characterization, but they also require
concurrent high temporal resolution, such as that provided by this imaging
system.
Spatio-temporal agreement between the imaging system and various independent
measurement devices was found to be within 5%, confirming the accuracy and
utility of ultra-fast imaging for future dose and dose rate evaluation of UHDR
PBS beams. The ability to resolve per-spot and inter-spot information of the
scanning beam at kHz repetition rate is extremely useful for estimating
parameters, such as spot scanning speed and spot dwell time, which may inform
treatment planning algorithms for further UHDR studies. The spot scanning
speeds here reported agree within reasonable measurement resolution with that
previously published by Kanouta et al who reported 10 ± 0.8 m/s and 25. 5 ±
5.1m/s estimates and that from Poulsen et al who measured 7 and 32 m/s [15,
28], in the same respective directions.
This work also demonstrates the importance of dose thresholds used in dose
rate estimations, something currently not of convention in FLASH-RT research.
While using Folkert et al.’s PBS dose rate definition, we noted a strong
dependence of resulting dose rate quantities on the chosen threshold. To our
knowledge, this is the first time the relationship between dose rate and the
parameters of Equation 1 has been demonstrated with measured PBS field data.
Unfortunately, the exact value of $d_{\dagger}$ is rarely used in FLASH dose
rate reports in existing literature. The chosen $d_{\dagger}$ depends not only
on thresholds applied during post-processing, but also will be limited by
detector sensitivity and additional thresholds applied at lower level in
dosimeter readout electronics. Going forward, such dose thresholds should be
thoroughly assessed and reported, especially in the pre-clinical and clinical
studies to establish consistency and comparability of the biological outcome-
based results across different centers. Especially in the event some dose rate
is needed to elicit the FLASH effect, treatment planning results will be
highly dependent on this choice. In lieu of radiobiological data indicating
the impact of the $d_{\dagger}$ threshold on the presence of the FLASH effect,
we recommend investigators set $d_{\dagger}$ depending on the resolution of
their measurement system. For this work, we found that a threshold of 50 cGy
lead to repeatable measurements based on the dynamic range of our measurement
system. While the main limitation of this system is the restricted dynamic
range, optimized here for UHDR deliveries, future work can be done to automate
and extend this. Additionally, despite the small-buildup seen for shoot-
through proton beams at such a shallow depth, some of the discrepancies
between the PPC05/log/simulation and imaging and diode can be attributed to
this buildup factor and the geometry of the required setup. Next steps to this
work will involve additional comparison methods, improved system independence,
and imaging of non-rigid surfaces.

##  V. Conclusion

Spatio-temporal agreement with film and diode measurements confirms our
ability to use this scintillation imaging system to accurately map dose rates
of a UHDR PBS system. Additional comparison to the internally developed log-
file-based reconstruction validated the potential to use this simulation for
future study planning. Deviation of measured dose rate at the center, diode
location, and edges of the field highlight the importance of full-field maps
for UHDR PBS beam characterization, and dependency on dose thresholds in
defining dose rate indicate the importance of clarifying and establishing
standards for accurate comparisons across beams. Going forward, this camera
system can be used to validate UHDR PBS beams via its unique, passive beam
monitoring capabilities. This system is the only device available to provide
concurrent high spatial and temporally resolved information of UHDR PBS
deliveries.

## Acknowledgments

The authors acknowledge SBIR R44CA268466, NIH R44 CA268466, and the Norris
Cotton Cancer Center Shared Resources with NCI cancer center support grant
5P30 CA023108-41.

## Conflict of Interest

This work has been funded by NIH grant R44 CA268466 and SBIR R44CA268466. Petr
Bruza is affiliated with DoseOptics LLC, which provided hardware support for
this study.

##

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