1\rangle)}$${H_{5}(\mathcal{B}\pi;\mathbb{L}^{\varepsilon}_{\bullet})}$$\scriptstyle{\theta^{\varepsilon}}$$\scriptstyle{\cong}$$\scriptstyle{a}$$\scriptstyle{\cong}$$\scriptstyle{b}$$\scriptstyle{\cong}$$\scriptstyle{g}$$\scriptstyle{c}$$\scriptstyle{\sigma^{\varepsilon}}$$\scriptstyle{d}$$\scriptstyle{\cong}$$\scriptstyle{e}$

The isomorphism $a$ arises via the definition of cohomology with coefficients
in a spectrum. Since we have that $(\mathbb{L}_{\bullet}\langle
1\rangle)_{0}\simeq G/\operatorname{TOP}$ we have that

$H^{0}(X\times I,\partial;\mathbb{L}_{\bullet})\cong[X\times
I,\partial;G/\operatorname{TOP}]\cong\mathcal{N}_{\operatorname{TOP}}(X\times
I,\partial).$

The isomorphism $b$ is given by Sullivan-Ranicki duality. To define the map
$c$, we factor it as the composition of maps given in the diagram below where
$\widetilde{X\times I}$ denotes the universal cover of $X\times I$.

${H_{5}(X\times I;\mathbb{L}^{\varepsilon}_{\bullet}\langle
1\rangle)}$${H^{\pi}_{5}(\widetilde{X\times
I};\mathbb{L}^{\varepsilon}_{\bullet}\langle
1\rangle)}$${H_{5}(\mathcal{E}\pi\times_{\pi}(\widetilde{X\times
I});\mathbb{L}^{\varepsilon}_{\bullet}\langle
1\rangle)}$${H_{5}(\mathcal{B}\pi;\mathbb{L}^{\varepsilon}_{\bullet}\langle
1\rangle)}$${H_{5}(\mathcal{E}\pi\times_{\pi}\\{\operatorname{pt}\\};\mathbb{L}^{\varepsilon}_{\bullet}\langle
1\rangle)}$$\scriptstyle{c}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{c^{\prime}}$$\scriptstyle{\cong}$

To define the map $d$, recall that we have a map
$\mathbb{L}^{\varepsilon}_{\bullet}\langle
1\rangle\to\mathbb{L}^{\varepsilon}_{\bullet}$ by the definition of a
$1$-connective cover. Then $d$ is the induced map on homology via this map.
The definition of the isomorphism $e$ follows from the definition of the
$\pi$-equivariant homology of the $\operatorname{Orb}(\pi)$-space
$\ast_{\mathcal{TRIV}}$ with coefficients in a spectrum. For this details on
this, see [DL98, Ex 5.5]. The isomorphism $g$ is again given by the definition
of the $\pi$-equivariant homology. The map $\sigma^{\varepsilon}$ is what we
will call the $\varepsilon$_-assembly map_. Since we need to use the
$s$-cobordism theorem to construct our mapping cylinders, we will need to
consider the case $\varepsilon=s$, for which this assembly map is certainly
not an isomorphism in general. Composing all of these maps together we get a
factorisation of $\theta^{s}$.

Now we want to use this formalism to try to understand $\theta^{s}$ for some
groups for which we cannot use 4.13. Let $\Gamma$ be a finite cyclic group
such that we have a non-trivial homomorphism $h\colon\Gamma\to\pi$. In what
follows we will condense the ‘assembly map’ by setting
$\overline{\sigma}:=g\circ\sigma^{s}\circ e$.

Consider the following diagram.

${\mathcal{N}_{\operatorname{TOP}}(X\times
I,\partial)}$${L_{5}^{s}(\mathbb{Z}[\pi])}$${L_{5}^{s}(\mathbb{Z}[\Gamma])=0}$${H_{5}(X\times
I;\mathbb{L}^{s}_{\bullet}\langle
1\rangle)}$${H_{5}(\mathcal{B}\pi;\mathbb{L}^{s}_{\bullet})}$${H_{5}(\mathcal{B}\Gamma;\mathbb{L}^{s}_{\bullet})}$$\scriptstyle{\theta^{s}}$$\scriptstyle{\cong}$$\scriptstyle{a}$$\scriptstyle{\overline{\sigma}}$$\scriptstyle{h_{*}}$$\scriptstyle{\overline{\sigma}}$

By naturality of the assembly maps, if a normal invariant is hit by the map
$h_{*}$ after being mapped down to
$H_{5}(\mathcal{B}\pi;\mathbb{L}^{s}_{\bullet})$, then its surgery obstruction
must be zero. It follows that if the map $h_{*}$ is surjective then
$\theta^{s}$ is the zero map. To try to understand this map $h_{*}$ better we
will use the Atiyah-Hirzebruch spectral sequence (AHSS). Recall that the AHSS
is a homology spectral sequence that computes the generalised homology of a
space in terms of the regular homology of that space with coefficients in the
generalised homologies of a point. In our particular case it computes, for a
space $K$,

$E^{2}_{p,q}=H_{p}(K;\pi_{q}(\mathbb{L}_{\bullet}\langle 1\rangle))\implies
E^{\infty}_{p,q}=H_{p+q}(K;\mathbb{L}_{\bullet}\langle 1\rangle).$

Recall that $\pi_{q}(\mathbb{L}_{\bullet}\langle 1\rangle)=L_{q}(\mathbb{Z})$
for $q>0$ and is zero otherwise. We first wish to compute
$H_{5}(\mathcal{B}G;\mathbb{L}_{\bullet}\langle 1\rangle)$ for the cases
$G=\pi$ and $G=\Gamma$. We show the relevant terms below of the $E^{3}$-page,
along with the relevant third-page differentials, in Figure 2.

Figure 2. The $E^{3}$-page of the spectral sequence for computing
$H_{*}(\mathcal{B}G;\mathbb{L}_{\bullet}\langle 1\rangle)$ with the relevant
terms and differentials shown for computing
$H_{5}(\mathcal{B}G;\mathbb{L}_{\bullet}\langle
1\rangle)$.$0$$1$$2$$3$$4$$5$$p$$0$$1$$2$$3$$4$$5$$q$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$0$$H_{3}(\mathcal{B}G;\mathbb{Z}/2)$$H_{4}(\mathcal{B}G;\mathbb{Z}/2)$$0$$0$$0$$0$$0$$0$$H_{0}(\mathcal{B}G;\mathbb{Z})$$H_{1}(\mathcal{B}G;\mathbb{Z})$$0$$0$$0$$0$$0$$0$$\boldsymbol{d_{3}^{3,2}}$$\boldsymbol{d_{3}^{4,3}}$

The conclusions we can draw from this spectral sequence are contained in the
following lemma.

######  Lemma 4.16.

Let $\Gamma$ be a finite cyclic group of order $2n$, let
$\pi\cong\mathbb{Z}\times\Gamma$ and let $d_{G}$ denote the differential
$d^{4,3}_{3}$ denoted above. Then we have the following commutative diagram
where the rows are exact.

${0}$${\operatorname{coker}d_{\Gamma}}$${H_{5}(\mathcal{B}\Gamma;\mathbb{L}_{\bullet}\langle
1\rangle)}$${H_{3}(\mathcal{B}\Gamma;\mathbb{Z}/2)}$${0}$${0}$${\operatorname{coker}d_{\pi}}$${H_{5}(\mathcal{B}\pi;\mathbb{L}_{\bullet}\langle
1\rangle)}$${H_{3}(\mathcal{B}\pi;\mathbb{Z}/2)}$${0}$

and $H_{3}(\mathcal{B}\Gamma;\mathbb{Z}/2)\cong\mathbb{Z}/2$,
$\operatorname{coker}d_{\Gamma}\cong\mathbb{Z}/2n$ or $\mathbb{Z}/n$,
$H_{3}(\mathcal{B}\pi;\mathbb{Z}/2)\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$, and
$\operatorname{coker}d_{\pi}\cong\mathbb{Z}\oplus\mathbb{Z}/2n$ or
$\mathbb{Z}\oplus\mathbb{Z}/n$.

###### Proof.

First note that for any group $G$ the differential $d^{3,2}_{3}$ vanishes
since it is map from a torsion group to a torsion-free group. Then recall that
a model for $\mathcal{B}\Gamma$ is given by the infinite lens space $L(2n)$,
and that a model for $\mathcal{B}\pi$ is given by the space $S^{1}\times
L(2n)$. We have that $d_{\Gamma}\colon\mathbb{Z}/2\to\mathbb{Z}/2n$ is either
the zero map or is injective. Similarly, we get that
$d_{\pi}\colon\mathbb{Z}/2\oplus\mathbb{Z}/2\to\mathbb{Z}\oplus\mathbb{Z}/2n$
is either the zero map or the map $(a,b)\mapsto(0,nb)$ (the other
possibilities may be ruled out by further comparing with the spectral sequence
for $G=\mathbb{Z}$ and using naturality). From this we can conclude that the
isomorphism types of the cokernels are as described in the lemma.

Finally, the diagonal $p+q=5$ computes the associated graded for
$H_{5}(\mathcal{B}G;\mathbb{L}_{\bullet}\langle 1\rangle)$ and hence we have
the short exact sequences which form the rows of the stated commutative
diagram. The vertical maps are then induced by the obvious inclusion map
$\Gamma\to\pi$ and the diagram commutes by naturality of spectral sequences.
This completes the proof. ∎

We have that $H^{4}(X\times I,\partial;\mathbb{Z}/2)\cong
H^{3}(X;\mathbb{Z}/2)\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$ and the aim is now
to realise the element $(0,1)$ as the Casson-Sullivan invariant of a
homeomorphism $f\colon X\to X$. Following the realisation procedure laid out
in Section 4.2 and Section 4.3, by 4.10 and 4.12 it suffices to show that
there exists an element $y\in\mathcal{N}_{\operatorname{TOP}}(X\times
I,\partial)\cong[(X\times I,\partial),G/\operatorname{TOP}]$ such that the map
$m$ from 4.9 sends $y$ to $(0,1)\in H^{4}(X\times I,\partial;\mathbb{Z}/2)$
and such that $\theta(y)=0$.

######  Lemma 4.17.

Let $\Gamma$ be a finite cyclic group of order $2n$,
$\pi=\mathbb{Z}\times\Gamma$ and let $\iota\colon\Gamma\to\pi$ be the obvious
inclusion map. The leftmost vertical map in the commutative diagram from 4.16
is then $\iota_{*}$ and is given by

$\displaystyle\iota_{*}\colon\begin{cases}\mathbb{Z}/2n\\\
\mathbb{Z}/n\end{cases}\to$
$\displaystyle\begin{cases}\mathbb{Z}\oplus\mathbb{Z}/2n\\\
\mathbb{Z}\oplus\mathbb{Z}/n\end{cases}$ $\displaystyle a\mapsto$
$\displaystyle(0,a).$

###### Proof.

We see this from the following commutative diagram, where we have identified
$H_{1}(\mathcal{B}G;\mathbb{Z})\cong\operatorname{ab}(G)$ for any group $G$,
where $\operatorname{ab}(G)$ denotes the abelianisation of $G$.

${\Gamma}$${\operatorname{ab}(\Gamma)}$${\operatorname{coker}d_{\Gamma}}$${\pi}$${\operatorname{ab}(\pi)}$${\operatorname{coker}d_{\pi}}$$\scriptstyle{\iota}$$\scriptstyle{\cong}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\cong}$

Since $\iota(a)=(0,a)$, the commutativity of the above diagram gives the
result regardless of the isomorphism types of the cokernels given in 4.16. ∎

######  Proposition 4.18.

Let $X$ be a closed, connected, smooth, orientable $4$-manifold with
$\pi_{1}(X)\cong\pi=\mathbb{Z}\times\Gamma$. Then
$H^{3}(X;\mathbb{Z}/2)\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$ and there exists a
self-homeomorphism $f\colon X\to X$ such that $\operatorname{cs}(f)=(0,1)\in
H^{3}(X;\mathbb{Z}/2)$.

###### Proof.

First, note that we can similarly use the AHSS to compute $H_{5}(X\times
I;\mathbb{L}_{\bullet}\langle 1\rangle)$. Of course, we already know the
answer since $H_{5}(X\times I;\mathbb{L}_{\bullet}\langle 1\rangle)\cong
H_{1}(X;\mathbb{Z})\oplus H_{3}(X;\mathbb{Z}/2)$ by 4.8, but this means that
we can fit this data into a larger commutative diagram with the one from 4.16,
which we do now below.

${\operatorname{coker}d_{\Gamma}}$${H_{5}(\mathcal{B}\Gamma;\mathbb{L}_{\bullet}\langle
1\rangle)}$${H_{3}(\mathcal{B}\Gamma;\mathbb{Z}/2)}$${L_{5}^{s}(\mathbb{Z}[\Gamma])}$${\operatorname{coker}d_{\pi}}$${H_{5}(\mathcal{B}\pi;\mathbb{L}_{\bullet}\langle
1\rangle)}$${H_{3}(\mathcal{B}\pi;\mathbb{Z}/2)}$${L_{5}^{s}(\mathbb{Z}[\pi])}$${H_{1}(X;\mathbb{Z})}$${H_{5}(X\times
I;\mathbb{L}_{\bullet}\langle
1\rangle)}$${H_{3}(X;\mathbb{Z}/2)}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\theta^{s}}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\iota_{*}}$$\scriptstyle{\theta^{s}}$$\scriptstyle{a}$

Note that we have omitted the zeroes from the ends of the rows, but the rows
are still split exact sequences. Let $x\in H_{5}(X\times
I;\mathbb{L}_{\bullet}\langle 1\rangle)$ be the element corresponding to

$((0,1),0)\in H^{4}(X\times I,\partial;\mathbb{Z})\oplus H^{2}(X\times
I,\partial;\mathbb{Z}/2)$

via the isomorphism from 4.8. If we can show that the surgery obstruction
$\theta^{s}(x)=0$, then by 4.10 we can construct a homeomorphism $f\colon X\to
X$ which has $\operatorname{cs}(f)=m((0,1),0)=(0,1)$.

Using the fact that $x$ is hit by an element from $H_{1}(X;\mathbb{Z})$ and
the description of the leftmost map denoted by $\iota_{*}$ from 4.17, a simple
diagram chase tells us that $a(x)$ is in the image of $\iota_{*}$. Hence by
naturality of assembly maps, and the fact that
$L_{5}^{s}(\mathbb{Z}[\Gamma])=0$, we have that $\theta(x)=0$, completing the
proof. ∎

4.18 tells us that we can also find examples of homeomorphisms that are not
stably (pseudo-)smoothable but are homotopic to the identity for $4$-manifolds
with fundamental group $\mathbb{Z}\times\mathbb{Z}/2n$. Note also that the
above arguments would have worked just as well with $\mathbb{Z}$ replaced with
$\mathbb{Z}^{n}$, so we can also find non-smoothable homeomorphisms in those
cases as well.

##  5\. An application to stable isotopy of surfaces

This section is devoted to proving 1.2. We begin with a definition.

######  Definition 5.1.

Let $X$ be a connected, compact, orientable smooth $4$-manifold and let
$\Sigma_{1},\Sigma_{2}\subset X$ be a pair of smoothly embedded surfaces, such
that $\partial\Sigma_{1}=\partial\Sigma_{2}=L\subset\partial X$ a fixed link
in $\partial X$ (which may be disconnected). We say that $\Sigma_{1}$ and
$\Sigma_{2}$ are _topologically isotopic_ {smoothly isotopic} if there exists
a homeomorphism {diffeomorphism} of pairs
$F\colon(X,\Sigma_{1})\to(X,\Sigma_{2})$ such that $F$ is isotopic {smoothly
isotopic} to the identity. We say that $\Sigma_{1}$ and $\Sigma_{2}$ are
_externally stably smoothly isotopic_ if there exists $n\geq 0$ such that
$\Sigma_{1}$ and $\Sigma_{2}$ become smoothly isotopic in
$X\\#(\\#^{n}S^{2}\times S^{2})$, where we perform the connect-sums in the
complement of $\Sigma_{1}\cup\Sigma_{2}$.

The idea of proving 1.2 is as follows: first, we find a stable diffeomorphism
between the exteriors of the surfaces. Then, we show that we can modify this
diffeomorphism to find one such that when we glue back in the tubular
neighbourhoods of the surfaces, we have a diffeomorphism which is the identity
on $H_{2}(X;\mathbb{Z})$. Then we use the following result of Saeki and Orson-
Powell which says when a diffeomorphism of a simply-connected $4$-manifold is
stably isotopic to the identity.

######  Theorem 5.2 ([Sae06, OP23]).

Let $X$ be a connected, compact, simply-connected smooth $4$-manifold with
$($potentially disconnected$)$ boundary $\partial X$ and let $f\colon X\to X$
be a diffeomorphism (restricting to the identity on $\partial X$) such that
the variation $($see [OP23, Definition 2.11]$)$ induced by $f$ is trivial, and
such that the induced map on relative spin structures $($see [OP23, Definition
2.5]$)$ is trivial. Then $f$ is stably smoothly isotopic to the identity.

The above theorem crucially relies on the theorem of Quinn [Qui86] that smooth
pseudo-isotopy implies stable smooth isotopy (see also [Gab22, Section 2], and
[GGH+23]).

Since in 1.2 we assume that our surfaces are topologically isotopic, this
means that we have a homeomorphism of the surface exteriors, and we will take
this as our starting point. The following lemma will handle most of the
technical aspects of the proof of 1.2.

######  Lemma 5.3.

Let $X$, $\Sigma_{1}$ and $\Sigma_{2}$ be as in the statement of 1.2 and let
$\widehat{G}\colon X\to X$ be a homeomorphism which sends $\Sigma_{1}$ to
$\Sigma_{2}$. Then $\widehat{G}$ is isotopic relative $\partial
X\cup\Sigma_{1}$ to a homeomorphism which sends $\nu\Sigma_{1}$ to
$\nu\Sigma_{2}$ by a diffeomorphism.

###### Proof.

Let $U:=\widehat{G}(\nu\Sigma_{1})$. We begin by isotoping
$\widehat{G}|_{\Sigma_{1}}$ to a diffeomorphism, which is always possible
since homeomorphisms of surfaces are isotopic to diffeomorphisms [Eps66] (see
also [Hat14]). This isotopy extends to $\widehat{G}$ by extending it first to
a tubular neighbourhood (perform the isotopy less and less as you extend
radially from $\Sigma_{1}$) and then as the constant isotopy on the complement
of the tubular neighbourhood of $\Sigma_{1}$ (of course, it was clear, by the
isotopy extension theorem [EK71], that this extended, but we can see the
extension explicitly and easily in this case). Denote the result of this
isotopy still by $\widehat{G}$. By the uniqueness of topological tubular
neighbourhoods [FQ90, Chapter 9.3], $U$ and $\nu\Sigma_{2}$ are isotopic by an
isotopy that fixes $\Sigma_{2}$, and hence we can isotope $\widehat{G}$
relative to $\partial X\cup\Sigma_{1}$ such that
$\widehat{G}(\nu\Sigma_{1})=\nu\Sigma_{2}$ as bundles. We now smooth the map
on the fibres of the normal bundles relative to the $0$-section $\Sigma_{1}$,
which we can do since any map $\Sigma_{1}\to\operatorname{O}(2)$ can be
isotoped to a smooth map. The result of this isotopy now sends $\nu\Sigma_{1}$
to $\nu\Sigma_{2}$ via a diffeomorphism. ∎

######  Remark 5.4.

It should be noted that one cannot always smooth homeomorphisms of
$4$-manifolds near arbitrary codimension $2$ submanifolds. In general, one can
only smooth a homeomorphism near a codimension $2$ submanifold after a small
topological isotopy (see [FQ90, Theorem 8.1A]). 5.3 does not contradict this
since we have a much stronger hypothesis (it is a self-homemorphism of a
smooth manifold and our homeomorphism already maps the submanifold in question
to another smooth submanifold).

######  Lemma 5.5.

Let $X$, $\Sigma_{1}$ and $\Sigma_{2}$ be as in the statement of 1.2 and let
$G\colon X\setminus\nu\Sigma_{1}\to X\setminus\nu\Sigma_{2}$ be the
homeomorphism of the surface exteriors and $\widehat{G}\colon X\to X$ the
extension of $G$. Then for some $k\geq 0$ there exists a diffeomorphism

$F\colon(X\setminus\nu\Sigma_{1})\\#(\\#^{k}S^{2}\times
S^{2})\to(X\setminus\nu\Sigma_{2})\\#(\\#^{k}S^{2}\times S^{2})$

which restricts to the identity on $\partial X$. Furthermore, $F$ extends to a
diffeomorphism

$\widehat{F}\colon X\\#(\\#^{k}S^{2}\times S^{2})\to X\\#(\\#^{k}S^{2}\times
S^{2}),$

and the induced maps on second homology fit into the following commutative
diagram.

${H_{2}(X\setminus\nu\Sigma_{1})\oplus\mathbb{Z}^{2}\oplus\mathbb{Z}^{2k-2}}$${H_{2}(X\setminus\nu\Sigma_{2})\oplus\mathbb{Z}^{2}\oplus\mathbb{Z}^{2k-2}}$${H_{2}((X\setminus\nu\Sigma_{1})\\#(\\#^{k}S^{2}\times
S^{2}))}$${H_{2}((X\setminus\nu\Sigma_{2})\\#(\\#^{k}S^{2}\times
S^{2}))}$${H_{2}(X\\#(\\#^{k}S^{2}\times
S^{2}))}$${H_{2}(X\\#(\\#^{k}S^{2}\times
S^{2}))}$${H_{2}(X)\oplus\mathbb{Z}^{2}\oplus\mathbb{Z}^{2k-2}}$${H_{2}(X)\oplus\mathbb{Z}^{2}\oplus\mathbb{Z}^{2k-2}}$$\scriptstyle{\cong}$$\scriptstyle{(G_{*},A,\operatorname{Id})}$$\scriptstyle{\cong}$$\scriptstyle{(i_{1})_{*}}$$\scriptstyle{F_{*}}$$\scriptstyle{(i_{2})_{*}}$$\scriptstyle{\cong}$$\scriptstyle{(\widehat{F})_{*}}$$\scriptstyle{\cong}$$\scriptstyle{(\operatorname{Id},A,\operatorname{Id})}$
(5.1)

where $i_{1}$ and $i_{2}$ denote the inclusions of the exteriors, and
$A\colon\mathbb{Z}^{2}\to\mathbb{Z}^{2}$ is either the identity map if
$\operatorname{cs}(G)=0$, or is given by the map sending $(x,y)\mapsto(-y,-x)$
if $\operatorname{cs}(G)\neq 0$. Furthermore, $\widehat{F}$ is topologically
pseudo-isotopic to a map which differs from the stabilisation of $\widehat{G}$
only on a neighbourhood of a curve in $X\setminus\nu\Sigma_{1}$ and a single
$S^{2}\times S^{2}$ summand.

###### Proof.

We assume, by 5.3, that $\widehat{G}\colon X\to X$ already restricts to a
diffeomorphism of the tubular neighbourhoods
$\widehat{G}|_{\nu\Sigma_{1}}\colon\nu\Sigma_{1}\to\nu\Sigma_{2}$.

We begin by showing the existence of a stable diffeomorphism of the exteriors,
and that the top square in Equation 5.1 commutes. Assume that
$\operatorname{cs}(G)\neq 0$, and let $\gamma\subset X\setminus\nu\Sigma_{1}$
be a framed embedded curve such that $[\gamma]\in
H_{1}(X\setminus\nu\Sigma_{1};\mathbb{Z}/2)$ is dual to
$\operatorname{cs}(G)$. By 3.2, the connect-sum homeomorphism (using the
notation therein)

$G^{\prime}:=G\\#_{\gamma=\theta}\widehat{\sigma}\colon(X\setminus\nu\Sigma_{1})\\#(S^{2}\times
S^{2})\to(X\setminus\nu\Sigma_{2})\\#(S^{2}\times S^{2})$

has
$\operatorname{cs}(G^{\prime})=\operatorname{cs}(G)+\operatorname{cs}(G)=0$,
where $\widehat{\sigma}=\sigma$ or $\sigma\circ t$, depending on $G$.

By 2.23, $G^{\prime}$ is stably pseudo-isotopic to a diffeomorphism

$F\colon(X\setminus\nu\Sigma_{1})\\#(\\#^{k}S^{2}\times
S^{2})\to(X\setminus\nu\Sigma_{2})\\#(\\#^{k}S^{2}\times S^{2})$

for some $k\geq 1$. By the computation in [Lee70] (see [Gal24, Lemma 2.3]),
$\widehat{\sigma}$ induces the map sending $(x,y)\mapsto(-y,-x)$ on
$H_{2}((S^{1}\times S^{3})\\#(S^{2}\times S^{2}))$ and hence the top square in
Equation 5.1 commutes.

If $\operatorname{cs}(G)=0$ then we may immediately apply 2.23 and similarly
obtain a diffeomorphism $F$, though potentially needing no stabilisations, and
the top half of Equation 5.1 commutes with $A$ given by the identity map.

Now $F$ extends to a diffeomorphism on $X\\#(\\#^{k}S^{2}\times S^{2})$ since
we already assumed that $\widehat{G}$ restricted to a diffeomorphism
$\nu\Sigma_{1}\to\nu\Sigma_{2}$. More specifically, we can fill back in the
tubular neighbourhoods of the surfaces $\Sigma_{1}$ and $\Sigma_{2}$ to obtain
a diffeomorphism $\widehat{F}$ which fits into the following diagram.

${(X\setminus\nu\Sigma_{1})\\#(\\#^{k}S^{2}\times
S^{2})}$${(X\setminus\nu\Sigma_{2})\\#(\\#^{k}S^{2}\times
S^{2})}$${X\\#(\\#^{k}S^{2}\times S^{2})}$${X\\#(\\#^{k}S^{2}\times
S^{2})}$$\scriptstyle{F}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{\widehat{F}}$

This means that the middle square in Equation 5.1 must commute, and the
commutativity of the bottom square then follows immediately. The final
statement is clear by the construction. ∎

###### Proof of 1.2.

By 5.5, we have that for some $k\geq 0$ there exists a diffeomorphism

$\widehat{F}\colon X\\#(\\#^{k}S^{2}\times S^{2})\to X\\#(\\#^{k}S^{2}\times
S^{2})$

such that on second homology it induces the map

$(\operatorname{Id},A,\operatorname{Id})\colon
H_{2}(X)\oplus\mathbb{Z}^{2}\oplus\mathbb{Z}^{2k-2}\to
H_{2}(X)\oplus\mathbb{Z}^{2}\oplus\mathbb{Z}^{2k-2}$

where $A$ is either the identity map or the map sending $(x,y)\mapsto(-y,-x)$.
Since $\widehat{F}$ is an extension of a diffeomorphism of the exteriors, it
must send $\Sigma_{1}$ to $\Sigma_{2}$.

We want to apply 5.2, and so we need to show that the variation induced by
$\widehat{F}$, as well as the induced map on relative spin structures, is
trivial. By the last statement of 5.5, the variation induced by $\widehat{F}$
can only differ by the variation induced by $\widehat{G}$ by its action on
relative homology classes which cannot be represented by a relative surface
disjoint from the neighbourhood of a curve union a single ($S^{2}\times
S^{2}$)-connect-summand. Such a class $x\in H_{2}(X\\#(\\#^{k}S^{2}\times
S^{2}),\partial)$ can be represented as $x=x^{\prime}+x^{\prime\prime}$, where
$x^{\prime}$ is represented by a relative surface disjoint from a curve union
a single ($S^{2}\times S^{2}$)-connect-summand, and $x^{\prime\prime}$ is a
(potentially trivial) homology class on that ($S^{2}\times S^{2}$)-connect-
summand. Since the variation induced by $\widehat{G}$ is topologically
isotopic to the identity, its induced variation is trivial, and hence the
action of the variation induced by $\widehat{F}$ on all relative classes
disjoint from a curve union a single ($S^{2}\times S^{2}$)-connect-summand is
trivial. Putting these facts together, this means that the variation induced
by $\widehat{F}$ can only differ from the trivial variation if its induced map
on homology is non-trivial.

If $A$ is the identity map, then $\widehat{F}$ acts trivially on homology, and
hence (by the above) its induced variation is trivial. If $A$ is not the
identity map, then post-compose $\widehat{F}$ with the map

$a:=\operatorname{Id}\\#a^{\prime}\\#\operatorname{Id}\colon X\\#(S^{2}\times
S^{2})\\#(\\#^{k}S^{2}\times S^{2})\to X\\#(S^{2}\times
S^{2})\\#(\\#^{k}S^{2}\times S^{2}),$

where $a^{\prime}\colon S^{2}\times S^{2}\to S^{2}\times S^{2}$ is the map
defined as the antipodal map on both $S^{2}$-factors, composed with the
diffeomorphism that swaps the two $S^{2}$-factors, and obtain a diffeomorphism
$a\circ\widehat{F}$. This new diffeomorphism still sends $\Sigma_{1}$ to
$\Sigma_{2}$, since $a$ is supported away from $\Sigma_{1}$, but now induces
the trivial map on homology. In either case, this means the map we have
created now induces the trivial variation.

That the induced map on relative spin structures by $\widehat{F}$ is trivial
follows again from the last statement in 5.5 and that every arc between two
distinct boundary components of $X$ can be made disjoint from a curve union a
single $(S^{2}\times S^{2})$-summand.

Now apply 5.2 to obtain that $\widehat{F}$ is stably smoothly isotopic to the
identity, and hence $\Sigma_{1}$ and $\Sigma_{2}$ are stably smoothly
isotopic. ∎

As was mentioned in Remark 1.5, 1.2 has the following consequence when paired
with certain results concerning when embedded surfaces are (stably)
topologically isotopic.

######  Corollary 5.6.

Let $X$ be as in 1.2. Let $\Sigma_{1}$, $\Sigma_{2}$ be a pair of homologous,
smoothly embedded surfaces in $X$ with the same genus and same boundaries,
such that
$\pi_{1}(X\setminus\nu\Sigma_{1})\cong\pi_{1}(X\setminus\nu\Sigma_{2})$. Then
if any of the following conditions are satisfied, the surfaces are stably
smoothly isotopic relative to their boundaries $($below $b_{2}(X)$ denotes the
second Betti number of $X$ and $\operatorname{sig}(X)$ denotes the signature
of $X$$)$.

  1. (i)

$\Sigma_{1}$ and $\Sigma_{2}$ are both spheres and
$\pi_{1}(X\setminus\nu\Sigma_{1})\cong\mathbb{Z}/d$ for some $d\geq 0$.

  2. (ii)

$\pi_{1}(X\setminus\nu\Sigma_{1})$ is trivial.

  3. (iii)

$\pi_{1}(X\setminus\nu\Sigma_{1})\cong\mathbb{Z}$ and
$b_{2}(X)\geq\lvert\operatorname{sig}(X)\rvert+6$.

  4. (iv)

$\pi_{1}(X\setminus\nu\Sigma_{1})\cong\mathbb{Z}/d$ for some $d\geq 2$ and
$b_{2}(X)>\lvert\operatorname{sig}(X)\rvert+2$ and the genus of $\Sigma_{1}$
is not the minimal genus needed to represent that homology class among
surfaces with the same fundamental group of the exterior.

###### Proof.

If case (i) is satisfied then [HK93, Theorem 4.8] (or [LW90, Corollary 1.3] if
$d$ is odd and $X$ is closed) shows that the surfaces are stably topologically
isotopic. Then 1.2 applied to the stabilisation shows that the surfaces are
stably smoothly isotopic. If any of the other cases are satisfied, then
[Sun15, Theorem 7.1, 7.2, 7.4] shows that the surfaces are topologically
isotopic, and then applying 1.2 shows that they are stably smoothly isotopic.
∎

##  6\. Application to isotopy classification of smooth structures

This section is devoted to writing up the correspondence between non-isotopic
but diffeomorphic smooth structures and non-smoothable homeomorphisms. We will
then interpret the results from the rest of the paper to show that there exist
many examples of non-isotopic but diffeomorphic smooth structures on
$4$-manifolds that remain non-isotopic after arbitrarily many stabilisations
(in contrast to instances of this phenomenon detected via gauge theory).

###  6.1. Non-isotopic but diffeomorphic smooth structures

We start with the basic definitions.

######  Definition 6.1.

Let $\mathscr{S}_{1},\mathscr{S}_{2}$ be a pair of smooth structures on a
topological manifold $X$, restricting to a fixed smooth structure
$\mathscr{S}_{\partial}$ on $\partial X$. We denote the resulting smooth
manifolds from these structures by $X_{\mathscr{S}_{i}}$. A _diffeomorphism_
$f\colon X_{\mathscr{S}_{1}}\to X_{\mathscr{S}_{2}}$ is a self-homeomorphism
$f$ of $X$, restricting to the identity on $\partial X$, such that
$f^{*}(\mathscr{S}_{2})=~{}\mathscr{S}_{1}$. We say that these two smooth
structures are _isotopic_ {_pseudo-isotopic_} if there exists an isotopy
$f_{t}\colon X\to X$ {pseudo-isotopy $F\colon X\times I\to X\times I$} such
that $f_{0}=\operatorname{Id}$, $f_{1}^{*}(\mathscr{S}_{2})=\mathscr{S}_{1}$
and $(f_{t}|_{\partial X})^{*}(\mathscr{S}_{\partial})=\mathscr{S}_{\partial}$
{$F|_{X\times\\{0\\}}=\operatorname{Id}$, $(F|_{\partial X\times
I})^{*}(\mathscr{S}_{\partial}\times I)=\mathscr{S}_{\partial}\times I$ and
$(F|_{X\times\\{1\\}})^{*}(\mathscr{S}_{2})=\mathscr{S}_{1}$}. Note that, in
the case of isotopy, $f_{t}^{*}(\mathscr{S}_{2})$ is then a $1$-parameter
family of smooth structures which continuously deforms from $\mathscr{S}_{2}$
into $\mathscr{S}_{1}$.

Throughout we will assume that $X$ already has a smooth structure on its
boundary. For all of our applications, $X$ will be a $4$-manifold, and hence
its boundary admits a unique smooth structure up to isotopy.

######  Definition 6.2.

Let $\operatorname{Homeo}(X,\partial X)$ be the group of self-homeomorphisms
of $X$ restricting to the identity map on the boundary and let
$\operatorname{Diff}(X_{\mathscr{S}})$ be the group of self-diffeomorphisms of
$X_{\mathscr{S}}$ restricting to the identity near $\partial X$. We topologise
these using the compact-open topology and the $C^{\infty}$ topology,
respectively.

Let $\pi_{0}\operatorname{Homeo}(X,\partial X)$
{$\widetilde{\pi}_{0}\operatorname{Homeo}(X,\partial X)$} denote the
_topological mapping class group of_ $X$ {_topological pseudo-mapping class
group of_ $X$} which is defined as the quotient of
$\operatorname{Homeo}(X,\partial X)$ via the equivalence relation: $f\sim g$
if $f$ is isotopic {pseudo-isotopic} to $g$. We can analogously define the
_smooth mapping class group_ {_smooth pseudo-mapping class group_}, which we
will denote by $\pi_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial X)$
{$\widetilde{\pi}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial X)$}.

######  Remark 6.3.

There is a separate definition for the pseudo-mapping-class group, where we
instead first define groups
$\widetilde{\operatorname{Diff}}(X_{\mathscr{S}},\partial X)$ and
$\widetilde{\operatorname{Homeo}}(X,\partial X)$, called the _block-
diffeomorphism group_ and _block-homeomorphism group_ , respectively. These
are defined as semi-simplicial complexes, such that the pseudo-mapping-class
groups $\widetilde{\pi}_{0}\operatorname{Homeo}(X,\partial X)$ and
$\widetilde{\pi}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial X)$ naturally
occur as $\pi_{0}\widetilde{\operatorname{Homeo}}(X,\partial X)$ and
$\pi_{0}\widetilde{\operatorname{Diff}}(X_{\mathscr{S}},\partial X)$,
respectively. We will not develop this viewpoint further in this paper.

There is then an inclusion $\operatorname{Diff}(X_{\mathscr{S}},\partial
X)\hookrightarrow\operatorname{Homeo}(X,\partial X)$ given by forgetting the
smooth structure. This induces the following map on the level of mapping class
groups {pseudo-mapping class groups}.

$\displaystyle\begin{split}\Phi\colon\pi_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial
X)&\to\pi_{0}\operatorname{Homeo}(X,\partial X),\\\
\widetilde{\Phi}\colon\widetilde{\pi}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial
X)&\to\widetilde{\pi}_{0}\operatorname{Homeo}(X,\partial X).\end{split}$ (6.1)

We are particularly interested in the cokernel of this map which we will write
as the quotient

$\operatorname{coker}\Phi=\frac{\pi_{0}\operatorname{Homeo}(X,\partial
X)}{\pi_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial X)},\
\operatorname{coker}\widetilde{\Phi}=\frac{\widetilde{\pi}_{0}\operatorname{Homeo}(X,\partial
X)}{\widetilde{\pi}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial X)}$

which corresponds to self-homeomorphisms of $X$ which are not topologically
isotopic {pseudo-isotopic} to any self-diffeomorphism of $X_{\mathscr{S}}$. We
have that this is a well-defined group, by the following lemma.

######  Lemma 6.4.

Let $\Phi$ {$\widetilde{\Phi}$} be the map from 6.2. Then the subgroup
$\operatorname{im}\Phi$ {$\operatorname{im}\widetilde{\Phi}$} is a normal
subgroup in $\pi_{0}\operatorname{Homeo}(X,\partial X)$
{$\widetilde{\pi}_{0}\operatorname{Homeo}(X,\partial X)$}.

###### Proof.

We show that the image of $\operatorname{Diff}(X_{\mathscr{S}})$ in
$\operatorname{Homeo}(X,\partial X)$ is normal, and the lemma then follows
immediately. Let $f\colon X_{\mathscr{S}}\to X_{\mathscr{S}}$ be a
diffeomorphism and let $g\colon X\to X$ be a homeomorphism. Then we want to
show that $(g^{-1}\circ f\circ g)^{*}(\mathscr{S})=\mathscr{S}$. It suffices
to show that $f^{*}(g^{*}(\mathscr{S}))=g^{*}(\mathscr{S})$, i.e. that $f$ is
also a diffeomorphism with respect to the smooth structure induced by $g$.
This is true, since $f$ being smooth with respect to $g^{*}(\mathscr{S})$ is
equivalent to the function $\psi\circ g\circ f\circ g^{-1}\circ\varphi^{-1}$
being a (classically) smooth map for any charts $\psi$,$\varphi$. By
maximality of smooth structures, $\psi\circ g$ and $\varphi\circ g$ are both
charts for $\mathscr{S}$, and $f$ being a diffeomorphism with respect to
$\mathscr{S}$ finishes the proof. ∎

######  Proposition 6.5.

Let $X_{\mathscr{S}}$ be a smooth manifold, $f\colon X\to X$ be a self-
homeomorphism and let
$\Phi\colon\pi_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial
X)\to\pi_{0}\operatorname{Homeo}(X,\partial X)$
{$\widetilde{\Phi}\colon\widetilde{\pi}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial
X)\to\widetilde{\pi}_{0}\operatorname{Homeo}(X,\partial X)$} be the map from
6.2. Then the smooth structures $f^{*}(\mathscr{S})$ and $\mathscr{S}$ are
isotopic {pseudo-isotopic} if and only if
$[f]\in\operatorname{im}\Phi${$[f]\in\operatorname{im}\widetilde{\Phi}$}.

###### Proof.

We prove this only for $\Phi$, with the proof being exactly the same for
$\widetilde{\Phi}$.

The reverse implication is straightforward. Let $f$ be a smoothable
homeomorphism relative to $\mathscr{S}$, i.e. $f$ is topologically isotopic to
a diffeomorphism $f^{\prime}\colon X_{\mathscr{S}}\to X_{\mathscr{S}}$. Then
the smooth structure $f^{*}(\mathscr{S})$ is isotopic to
$(f^{\prime})^{*}(\mathscr{S})$, and we have
$(f^{\prime})^{*}(\mathscr{S})=\mathscr{S}$ since $f^{\prime}$ is a
diffeomorphism.

Now for the forwards implication. Let $f$ be such that $\mathscr{S}$ and
$f^{*}(\mathscr{S})$ are isotopic. Then by the definition we have that there
exists a diffeomorphism $g\colon X_{\mathscr{S}}\to X_{f^{*}(\mathscr{S})}$
such that $g$ is topologically isotopic to the identity. More specifically,
there exists a continuous path of homeomorphisms $g_{t}\colon X\to X$ such
that $g_{0}=\operatorname{Id}_{X}$ and
$(g_{1}^{-1})^{*}(f^{*}(\mathscr{S}))=\mathscr{S}$. Consider the composition
homeomorphism $f_{t}:=f\circ g_{t}\colon X\to X$ and note that $f_{0}=f$ and

$f_{1}=f\circ g\colon X_{\mathscr{S}}\to X_{\mathscr{S}}$

is a diffeomorphism by construction. ∎

######  Corollary 6.6.

Let $X_{\mathscr{S}}$ be a smooth manifold and let
$\mathcal{S}(X_{\mathscr{S}},\partial X)$
{$\widetilde{\mathcal{S}}(X_{\mathscr{S}},\partial X)$} denote the set of
isotopy {pseudo-isotopy} classes of smooth structures on $X$ diffeomorphic to
$\mathscr{S}$ restricting to the given smooth structure on $\partial X$. Then
there is a bijection as defined below

$\displaystyle\frac{\pi_{0}\operatorname{Homeo}(X,\partial
X)}{\pi_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial X)}$
$\displaystyle\longrightarrow\mathcal{S}(X_{\mathscr{S}},\partial X)$
$\displaystyle[f]$ $\displaystyle\longmapsto f^{*}(\mathscr{S}),$
$\displaystyle\frac{\widetilde{\pi}_{0}\operatorname{Homeo}(X,\partial
X)}{\widetilde{\pi}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial X)}$
$\displaystyle\longrightarrow\widetilde{\mathcal{S}}(X_{\mathscr{S}},\partial
X)$ $\displaystyle[f]$ $\displaystyle\longmapsto f^{*}(\mathscr{S}).$

###### Proof.

The fact that the map written in the statement is well-defined and only maps
the trivial element to $\mathscr{S}$ follows directly from 6.5. That the
second condition is enough to ensure that the map is injective follows from
6.4, That the map is surjective follows from the definition of
$\mathcal{S}(X_{\mathscr{S}},\partial X)$
{$\widetilde{\mathcal{S}}(X_{\mathscr{S}},\partial X)$}. ∎

Note that since the quotient is a group (by 6.4), this means that we get a
group structure on $\mathcal{S}(X_{\mathscr{S}})$
{$\widetilde{\mathcal{S}}(X_{\mathscr{S}})$} by 6.6.

######  Remark 6.7.

There is an alternative interpretation of this section given recently by Lin-
Xie [LX23], where we instead construct a space of a smooth structures, and now
interpret our $\mathcal{S}(X_{\mathscr{S}},\partial X)$ as $\pi_{0}$ of this
space. We make this more precise. We have a fibration

$\mathcal{B}\\!\operatorname{Diff}(X_{\mathscr{S}},\partial
X)\to\mathcal{B}\\!\operatorname{Homeo}(X,\partial X)$

induced by the inclusion and we denote the homotopy fibre of this map by
$F(X_{\mathscr{S}})$. From the long exact sequence of the fibration and the
fact that $\pi_{i}(\mathcal{B}G)=\pi_{i-1}(G)$ for any group $G$, we get the
exact sequence

$\pi_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial
X)\to\pi_{0}\operatorname{Homeo}(X,\partial
X)\to\pi_{0}(F(X_{\mathscr{S}}))\to 0.$

Since it is clear that the first map in this sequence is $\Phi$ we have that
$\pi_{0}(F(X_{\mathscr{S}}))=\operatorname{coker}\Phi$. We call
$F(X_{\mathscr{S}})$ the _space of smooth structures on_ $X$ _diffeomorphic
to_ $\mathscr{S}$, and $\operatorname{coker}\Phi$ corresponds to its path
components. Using block diffeomorphisms and block homeomorphisms (see Remark
6.3), we can similarly define a space $\widetilde{F}(X_{\mathscr{S}})$ where
$\pi_{0}(\widetilde{F}(X_{\mathscr{S}}))$ corresponds to
$\operatorname{coker}\widetilde{\Phi}$. We will not explore this
interpretation further in this paper.

###  6.2. Stably non-isotopic but diffeomorphic smooth structures

Similarly, all of Section 6.1 can be considered stably. Let $(S^{2}\times
S^{2})_{\mathscr{T}}$ be $S^{2}\times S^{2}$ with smooth structure
$\mathscr{T}$, the standard smooth structure on $S^{2}\times S^{2}$ given as
the product of the standard smooth structures on $S^{2}$. Further let
$X_{\mathscr{S}}$ be a smooth $4$-manifold, and note that, up to isotopy, we
may assume that $\mathscr{S}$ is standard on some topologically embedded disc
$D\subset X$. Hence we may define a smooth structure (independent of choices)
denoted by $\mathscr{S}\\#\mathscr{T}$ on the topological connect-sum
$X\\#S^{2}\times S^{2}$. This motivates the following definitions.

######  Definition 6.8.

Let $X_{\mathscr{S}_{i}}$ for $i=1,2$ be two smooth manifolds with smooth
structures $\mathscr{S}_{i}$, respectively. Then a _stable diffeomorphism_
from $X_{\mathscr{S}_{1}}$ to $X_{\mathscr{S}_{2}}$ is a diffeomorphism (in
the sense of Definition 6.1)

$f\colon X_{\mathscr{S}_{1}}\\#(\\#^{k}(S^{2}\times S^{2})_{\mathscr{T}})\to
X_{\mathscr{S}_{2}}\\#(\\#^{k}(S^{2}\times S^{2})_{\mathscr{T}})$

for some non-negative integer $k$. In this case, we say that the smooth
structures $\mathscr{S}_{i}$ are _stably isotopic_ {_stably pseudo-isotopic_}
if $\mathscr{S_{1}}\\#(\\#^{k}\mathscr{T})$ is isotopic {pseudo-isotopic} to
$\mathscr{S_{2}}\\#(\\#^{k}\mathscr{T})$.

Analogously to the unstable case, we then have the stable mapping class
groups.

######  Definition 6.9.

Let $\pi^{\operatorname{Stab}}_{0}\operatorname{Homeo}(X,\partial X)$
{$\widetilde{\pi}^{\operatorname{Stab}}_{0}\operatorname{Homeo}(X,\partial
X)$} denote the _stable topological mapping class group of_ $X$ {_stable
topological pseudo-mapping class group of_ $X$} which is defined as the
quotient of $\operatorname{Homeo}(X,\partial X)$ via the equivalence relation:
$f\sim g$ if $f$ is stably isotopic {pseudo-isotopic} to $g$.

Continuing the analogy with the unstable case, there is then the obvious
inclusion map which induces a map of the stable mapping class groups {pseudo-
mapping class groups}

$\displaystyle\begin{split}\Phi^{\operatorname{Stab}}\colon\pi^{\operatorname{Stab}}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial
X)&\to\pi^{\operatorname{Stab}}_{0}\operatorname{Homeo}(X,\partial X),\\\
\widetilde{\Phi}^{\operatorname{Stab}}\colon\widetilde{\pi}^{\operatorname{Stab}}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial
X)&\to\widetilde{\pi}^{\operatorname{Stab}}_{0}\operatorname{Homeo}(X,\partial
X).\end{split}$ (6.2)

Again we are interested in the cokernel of this map which we write as the
quotient of the stable mapping class groups {stable pseudo-mapping class
groups} in the obvious manner:

$\operatorname{coker}\Phi^{\operatorname{Stab}}=\frac{\pi^{\operatorname{Stab}}_{0}\operatorname{Homeo}(X,\partial
X)}{\pi^{\operatorname{Stab}}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial
X)},\
\operatorname{coker}\widetilde{\Phi}^{\operatorname{Stab}}=\frac{\widetilde{\pi}^{\operatorname{Stab}}_{0}\operatorname{Homeo}(X,\partial
X)}{\widetilde{\pi}^{\operatorname{Stab}}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial
X)}.$

This cokernel corresponds to stable isotopy {pseudo-isotopy} classes of self-
homeomorphisms of $X$ which are not stably topologically isotopic {pseudo-
isotopic} to any stable self-diffeomorphism of $X_{\mathscr{S}}$. The proofs
of 6.4, 6.5 and 6.6 follow through unchanged for $\Phi^{\operatorname{Stab}}$,
and hence we have the following corollary.

######  Corollary 6.10.

Let $X_{\mathscr{S}}$ be a smooth manifold and let
$\mathcal{S}^{\operatorname{Stab}}(X_{\mathscr{S}},\partial X)$
{$\widetilde{\mathcal{S}}^{\operatorname{Stab}}(X_{\mathscr{S}},\partial X)$}
be the set of stable isotopy classes of smooth structures on $X$ {stable
pseudo-isotopy classes of smooth structures on $X$} stably diffeomorphic to
$\mathscr{S}$. Then there are bijections as defined below

$\displaystyle\frac{\pi^{\operatorname{Stab}}_{0}\operatorname{Homeo}(X,\partial
X)}{\pi^{\operatorname{Stab}}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial
X)}$
$\displaystyle\longrightarrow\mathcal{S}^{\operatorname{Stab}}(X_{\mathscr{S}},\partial
X)$ $\displaystyle[f]$ $\displaystyle\longmapsto f^{*}(\mathscr{S}).$
$\displaystyle\frac{\widetilde{\pi}^{\operatorname{Stab}}_{0}\operatorname{Homeo}(X,\partial
X)}{\widetilde{\pi}^{\operatorname{Stab}}_{0}\operatorname{Diff}(X_{\mathscr{S}},\partial
X)}$
$\displaystyle\longrightarrow\widetilde{\mathcal{S}}^{\operatorname{Stab}}(X_{\mathscr{S}},\partial
X)$ $\displaystyle[f]$ $\displaystyle\longmapsto f^{*}(\mathscr{S}).$

###  6.3. Proofs of 1.11 and 1.12

We now reinterpret the results from the rest of the paper in terms of smooth
structures. By 2.22, we know that the Casson-Sullivan invariant is an
obstruction to a homeomorphism $f\colon X_{\mathscr{S}}\to X_{\mathscr{S}}$
being stably pseudo-isotopic to a diffeomorphism. Hence, if $f\colon X\to X$
has $\operatorname{cs}(f)\neq 0$, then $f$ represents a non-trivial element in
$\operatorname{coker}\widetilde{\Phi}^{\operatorname{Stab}}$ and by 6.10 $f$
gives rise to a non-trivial element in
$\mathcal{S}^{\operatorname{Stab}}(X_{\mathscr{S}},\partial X)$. We now give
the formal proofs of the theorems.

###### Proof of 1.11.

We first recap the hypothesis of the theorem. Let $X$ be a $4$-manifold such
that $X=X^{\prime}\\#(S^{2}\times S^{2})$ for some compact, connected, smooth,
orientable $4$-manifold $X^{\prime}$ and let $\mathscr{S}$ denote the smooth
structure on $X$ induced from $X^{\prime}$.

Let $\eta\in H^{3}(X,\partial X;\mathbb{Z}/2)$ be non-trivial. Then by 1.1
there exists a homeomorphism $f\colon X\to X$ such that
$\operatorname{cs}(f)=\eta$. By 2.22 we have that $f$ represents a non-trivial
element in $\operatorname{coker}\widetilde{\Phi}^{\operatorname{Stab}}$ and
hence by 6.10 we see that $\mathscr{S}_{\eta}:=f^{*}(\mathscr{S})$ is a non-
trivial element in
$\widetilde{\mathcal{S}}^{\operatorname{Stab}}(X_{\mathscr{S}},\partial X)$.
Repeating this for all non-trivial $\eta\in H^{3}(X,\partial X;\mathbb{Z}/2)$
generates the family of smooth structures $\\{\mathscr{S}_{\eta}\\}$ and these
all must be distinct in
$\widetilde{\mathcal{S}}^{\operatorname{Stab}}(X_{\mathscr{S}},\partial X)$
since they correspond (via 6.10) to homeomorphisms which all have different
Casson-Sullivan invariants. ∎

The proof for 1.12 is analogous.

###### Proof of 1.12.

Again we recap the hypothesis of the theorem. Let $X$ be a closed, connected,
smooth, orientable $4$-manifold with $\pi_{1}(X)\cong\pi$ where $\pi$ which
satisfies the Casson-Sullivan realisation condition (see Definition 4.11). Let
$X=X_{\mathscr{S}}$ for some smooth structure $\mathscr{S}$.

Let $\eta\in H^{3}(X,\partial X;\mathbb{Z}/2)$ be non-trivial. Then by 1.7
there exists a homeomorphism $f\colon X\to X$ such that
$\operatorname{cs}(f)=\eta$. The proof is then exactly the same as the proof
of 1.11. ∎

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# The gravitational baryogenesis and a new higher-order extended uncertainty
principle with parameter adaptability for the minimum length

Song-Shan Luo1,2 Zhong-Wen Feng1,2 Email<EMAIL_ADDRESS>1 School of
Physics and Astronomy, China West Normal University, Nanchong, 637009, China
2 Institute of Theoretical Physics, China West Normal University, Nanchong,
637009, China

###### Abstract

In this manuscript, we explore the baryon asymmetry of the universe by
employing a novel higher-order extended uncertainty principle (EUP) that
maintains a minimum length ${\rm{\Delta}}{x_{\rm
min}}=4\sqrt{\left|{\rm{\beta_{0}}}\right|}\ell_{p}$ for both positive and
negative deformation parameters. Our results demonstrate that the influence of
the EUP noticeably modifies the Friedmann equations, leading to a
transformation in the characteristics of the pressure and density of the
Universe, and subsequently disrupting its thermal equilibrium. Additionally,
by amalgamating the adapted Friedmann equations with the conventional theory
of gravitational baryogenesis, one can derive a non-zero factor of baryon
asymmetry $\eta$, indicating that the quantity of matter in the universe
surpasses that of antimatter. Finally, we also utilized astronomical
observations to constrain the bounds for both the positive and negative
deformation parameters.

Higher-order extended uncertainty principle Gravitational baryogenesis
Friedmann equations

##  I Introduction

Advancements in the field of physics have revealed that nature encompasses not
just regular matter, but also its counterpart, antimatter. In the standard
model of particle physics, it was previously hypothesized that an equal
quantity of matter and antimatter existed. However, numerous astronomical
observations, such as the Big Bang Nucleosynthesis (BBN) 1 , high-precision
measurements of the Cosmic Microwave Background (CMB) 2 , and the Wilkinson
Microwave Anisotropy Probe (WMAP) experiment, have revealed the opposite, our
Universe is predominantly composed of matter with only a minuscule amount of
antimatter. This has resulted in an enigmatic puzzle in particle cosmology,
known as the baryon asymmetry of the universe (BAU) 3 .

Although the exact mechanism behind BAU remains elusive, researchers have
endeavored to formulate numerous theories in an attempt to elucidate this
phenomenon, such as electroweak baryogenesis 4 , gravitational waves
baryogenesis 5 , and so on. Of all these theories, the gravitational
baryogenesis introduced by Davoudiasl 6  is noteworthy, as it proposes that a
gravitational interaction can dynamically break CPT in an expanding Universe
and lead to baryon asymmetry. The theory of gravitational baryogenesis reveals
gravitational coupling as a means of generating baryon asymmetry and has
attracted widespread attention and research. For instance, in the
$f(R)$-theory of gravity 7 , the baryon asymmetry in the universe is explained
by the gravitational baryogenesis. In Ref. 8 , the gravitational baryogenesis
is studied in the framework of $f(P)$ gravity, and the applicability of such
modified gravity in solving the cosmic baryon asymmetry is obtained.

Despite the significant role played by gravitational baryogenesis in
comprehending BAU, it still has some shortcomings, the most notable of which
is its inability to fulfill all of the Sakharov conditions. It is widely
acknowledged that the universe displays baryon asymmetry only if it fulfills
the Sakharov conditions 9 , namely, (1) interactions that violate baryon-
number conservation; (2) CP and C violation; and (3) a deviation from thermal
equilibrium. In the original theory of gravitational baryogenesis, the first
two Sakharov conditions are satisfied by introducing a coupling term that
couples space-time to the baryon current. However, the theory fails to fulfill
the third condition as it assumes thermal equilibrium, which violates the
third Sakharov condition. Additionally, the original theory is incapable of
generating a baryon asymmetry in the radiation-dominated universe. In order to
fix this issue, some approaches have been proposed, see e.g., Refs. 10 ; 11 ;
12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 . In particular, considering that the effect
of generalized uncertainty principle (GUP hereafter, which is a kind of
quantum gravity (QG) model 19 ; 20 ; 21 ; 21++ ; a1 ; a2 ; a3 ; a4 ; b1 ; b2 ;
g2 ; g3 ; g4 ) can modify the properties of the universe and change its
thermodynamic state, Das, Vagenas _et al_. 22 ; 22x  used the KMM model model
and ADV model of GUP to investigate BAU, their results demonstrated that the
universe in the GUP framework at non-thermal equilibrium, leading to a non-
zero factor $\eta$ for measuring the number of baryonic matter exceeding the
antibaryonic matter. Subsequently, Ref. 23  have extended this approach to the
case of higher-order GUP and showed that the higher order QG correction terms
have a significant effect on the baryon asymmetry.

On the other hand, while initial investigations mainly focused on the impact
of positive GUP parameters, there has been a growing body of research
examining the effects of negative GUP parameters. For instance, the authors in
Ref. 24  derived the negative GUP parameter (for a GUP on lattice) for the
first time. Then, Scardigli and Casadio derived a negative GUP parameter by
assuming that the GUP-corrected Hawking temperature can be obtained through a
Wick rotation of an effective Schwarzschild-like metric 24+ . In Ref. 25 , the
properties of sub-Planckian black holes with negative GUP parameters were
analysed. Ong pointed out that the Chandrasekhar limit no longer exists in the
framework of positive GUP parameter 26 , whereas the negative GUP parameter
solves this contradiction 27 . In Ref. 28 , the thermodynamic evolution and
phase transition of a static black hole were investigated considering both
positive and negative GUP parameters. The findings indicated distinct
behaviors in the thermodynamic properties of black holes depending on whether
the GUP parameters were positive or negative. Based on the aforementioned
research, it is believed that the GUP model with negative parameters is as
important for the correction of the physical theory as the case with positive
parameters. However, a contradiction arises when the parameter is negative as
the GUP no longer includes a minimum length 29 ; 30 , which contradicts the
model-independent existence of the minimum length 31 . For addressing this
issue, Du and Long 32  recently introduced a novel uncertainty principle
model, which can be expressed as

$\displaystyle\Delta x\Delta
p\geq\frac{\hbar}{2}\frac{1}{{1\pm\left({{{16{\beta_{0}}\ell_{p}^{2}}\mathord{\left/{\vphantom{{16{\beta_{0}}\ell_{p}^{2}}{\Delta{x^{2}}}}}\right.\kern-1.2pt}{\Delta{x^{2}}}}}\right)}},$
(1)

with the deformation parameter ${\beta_{0}}$ and the Planck length
${\ell_{p}}$. $\Delta x$ and $\Delta p$ are the uncertainties for position and
momentum, respectively. It is important to highlight that in Ref. 32 , the
equation mentioned above is referred to as the GUP. However, it is crucial to
note that the RHS of the equation is no longer related to $\Delta x$ but to
$\Delta x_{0}$. Therefore, in accordance with the definition provided in Refs.
b5 ; b5+ ; b5++ , Eq. (1) should be more appropriately labeled as the extended
uncertainty principle (EUP). Now, Eq. (1) exhibits two primary
characteristics: one is the deformation parameter ${\beta_{0}}$ (EUP
parameter) in the model can be assigned either positive or negative values,
with “$+$” representing the positive EUP parameter and “$-$” representing the
negative EUP parameter. The other one is it maintains a fixed and uniform
minimum length ${{\Delta}}{x_{\rm
min}}=4\sqrt{\left|{\rm{\beta_{0}}}\right|}\ell_{p}$, irrespective of whether
the parameter is positive or negative. These advantages guarantee the efficacy
of the QG effect within the model, enabling us to analyze the impact of both
positive and negative EUP parameters on the identical physical system. In
addition, if ignoring the higher order correction terms, Eq. (1) reduce to
$\Delta x\Delta
p\geq\hbar{{\left[{1+{{16\ell_{p}^{2}{\beta_{0}}}\mathord{\left/{\vphantom{{16\ell_{p}^{2}{\beta_{0}}}{\Delta{x^{2}}}}}\right.\kern-1.2pt}{\Delta{x^{2}}}}}\right]}\mathord{\left/{\vphantom{{\left[{1+{{16\ell_{p}^{2}{\beta_{0}}}\mathord{\left/{\vphantom{{16\ell_{p}^{2}{\beta_{0}}}{\Delta{x^{2}}}}}\right.\kern-1.2pt}{\Delta{x^{2}}}}}\right]}2}}\right.\kern-1.2pt}2}$,
when comparing with the quadratic GUP (i.e., $\Delta x\Delta
p\geq{{\hbar\left[{1+\beta_{0}^{{\text{KMM}}}\ell_{p}^{2}\Delta{p^{2}}/{\hbar^{2}}}\right]}\mathord{\left/{\vphantom{{\hbar\left[{1+\beta_{0}^{{\text{KMM}}}\ell_{p}^{2}\Delta{p^{2}}/{\hbar^{2}}}\right]}2}}\right.\kern-1.2pt}2}$),
it is found that $\left|{{\beta_{0}}}\right|\sim\beta_{0}^{{\text{KMM}}}$.

In this paper, our aim is to integrate the new higher-order EUP with
gravitational baryogenesis in order to explore the BAU. By incorporating Eq.
(1) into the first law of thermodynamics, we derive modified Friedmann
equations. These equations reveal that effects of QG play a crucial role in
enhancing the energy density and pressure during the radiation-dominated era,
thereby disrupting the thermal equilibrium of the universe and ultimately
leading to the emergence of a non-zero baryon asymmetry factor $\eta$.
Finally, we estimate the bounds of the EUP parameters for both positive and
negative cases.

This paper is organized as follows. In section II, we briefly review the
original scheme of gravitational baryogenesis. In SectionIII, by using the new
higher-order EUP (1) under different parameter signs, we derive modified
Bekenstein-Hawking entropy and Friedmann equations. Then, combining the
modifications with gravitational baryogenesis, a non-zero time derivative of
the Ricci scalar curvature $\dot{R}$ and a non-zero baryon asymmetry factor
$\eta$ are obtained. In section IV, based on the data of experiments and
observations, we constrained the boundaries of both positive and negative
deformation parameters. Conclusions and discussion are presented in section V.
For later convenience, we use the units $\hbar=c={k_{B}}=1$.

##  II Gravitational baryogenesis in the standard cosmological model

In order to break the CPT dynamically in an expanding Universe, Davoudiasl _et
al_. constructed the following interaction 6 :

$\displaystyle\frac{1}{{M_{*}^{2}}}\int{{d^{4}}x\sqrt{-g}\left({{\partial_{\mu}}R}\right){J^{\mu}}},$
(2)

where ${J^{\mu}}$ denotes the baryon current, ${M_{\rm{*}}}$ is the cut-off
scale of the effective theory, $g$ and $R$ stand for the metric and the Ricci
curvature scalar, respectively. The baryon asymmetry is usually characterised
by the baryon asymmetry factor (BAF), that is
$\eta={{{n_{B}}}\mathord{\left/{\vphantom{{{n_{B}}}s}}\right.\kern-1.2pt}s}$
with the baryon number density $n_{B}$ and the entropy density for the
universe
$s={{2{\pi^{2}}{g_{\text{*}}}{T^{3}}}\mathord{\left/{\vphantom{{2{\pi^{2}}{g_{\text{*}}}{T^{3}}}{45}}}\right.\kern-1.2pt}{45}}$.
Notably, during the expansion of the universe, if the temperature of universe
$T$ drops below the critical temperature ${T_{D}}$ for baryon asymmetric
interactions to occur, the BAF can be expressed as 7 ; 11 ; 12 ; 13 ; 14 ; 15
; 16

$\displaystyle\eta=\frac{{{n_{B}}}}{s}\simeq-{\left.{\frac{{15{g_{b}}}}{{4{\pi^{2}}{g_{*}}}}\frac{{\dot{R}}}{{M_{*}^{2}T}}}\right|_{{T_{D}}}},$
(3)

where $\dot{R}={{\partial R}\mathord{\left/{\vphantom{{\partial R}{\partial
t}}}\right.\kern-1.2pt}{\partial t}}$ is the time derivative of the Ricci
scalar curvature of the universe. ${g_{b}}$ and ${g_{\rm{*}}}$ are the number
of intrinsic degrees of freedom of baryons and the degrees of freedom of
particles that contribute to the entropy of universe, respectively. In the
standard cosmological model with the energy density $\rho$ and pressure $p$,
the Ricci curvature scalar is given by

$\displaystyle R=-8\pi G\left({\rho-3p}\right).$ (4)

Since the matter source in the universe can be seen as a perfect fluid, the
equation of state parameter of standard cosmological model becomes
$w={p\mathord{\left/{\vphantom{p\rho}}\right.\kern-1.2pt}\rho}$, and Eq. (4)
evolves as

$\displaystyle R=-8\pi G\rho\left({1-3w}\right).$ (5)

Notably, when considering the radiation-dominated era, one has
$w={1\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}3}$, which leads to the
Ricci scalar curvature $R$ and its derivative $\dot{R}$ vanish, and then
resulting in $\eta=0$. However, according to many astronomical observations,
it is shown that the BAF is not equal to zero, implying that there is more
matter than antimatter in the universe. For instance, Particle Data Group
gives $\eta\leq 8.6\times{10^{-11}}$ 33 , and BBN shows
$3.4\times{10^{-10}}\leq\eta\leq 6.9\times{10^{-10}}$ 34 . To solve this
problem, we will investigate the gravitational baryogenesis with higher-order
EUP corrections.

##  III Gravitational baryogenesis in the framework of higher-order EUP

###  III.1 The EUP corrected entropy

According to the holographic principle, when a gravitational system absorbs a
particle, the area of its apparent horizon and the total energy within it
increase. The minimal change of the area ${\rm{\Delta}}A$ can be expressed as
35 ; 36

$\displaystyle\Delta A\sim Xm,$ (6)

where $X$ and $m$ denote the size and mass of the particle, respectively.
However, in quantum mechanics, the standard deviation of the $X$ distribution
is used to describe the width of a particle’s wave packet (the position
uncertainty $\Delta x$) and the momentum uncertainty $\Delta p$ is not allowed
to be larger than the mass. Thus, Eq. (6) can be rewritten as

$\displaystyle\Delta A\geq\Delta x\Delta p.$ (7)

Implying that the minimum increment of the area of the gravitational system is
limited by the momentum uncertainty $\Delta p$ and the position uncertainty
$\Delta x$ of quantum mechanics. According to Eq. (1), the momentum
uncertainty is obtained as

$\displaystyle\Delta p\geq\frac{\hbar}{2}\frac{1}{{\Delta
x\pm{{16\beta_{0}\ell_{p}^{2}}\mathord{\left/{\vphantom{{16\beta\ell_{p}^{2}}{\Delta
x}}}\right.\kern-1.2pt}{\Delta x}}}}.$ (8)

For a static spherical gravitational system, the position uncertainty is
approximately equal to the radius of the apparent horizon 37 , which is
$\Delta x\approx 2r$. Combining with the above equations, the minimal change
of the area is

$\displaystyle\Delta A\geq\chi\tilde{\hbar}\left(\beta_{0}\right),$ (9)

where
$\tilde{\hbar}\left(\beta_{0}\right)={1\mathord{\left/{\vphantom{1{\left[{2+\left({{{32\pi\beta\ell_{p}^{2}}\mathord{\left/{\vphantom{{32\pi\beta\ell_{p}^{2}}A}}\right.\kern-1.2pt}A}}\right)}\right]}}}\right.\kern-1.2pt}{\left[{2+\left({{{32\pi\beta_{0}\ell_{p}^{2}}\mathord{\left/{\vphantom{{32\pi\beta_{0}\ell_{p}^{2}}A}}\right.\kern-1.2pt}A}}\right)}\right]}}$
is the effective Planck constant, and $\chi=4\ln 2$ is the calibration factor
38 . When $\beta_{0}=0$, one has
$\tilde{\hbar}\left(\beta_{0}\right)={1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}$.
Based on the information theory, the minimal increase of entropy is related to
the value of the area

$\displaystyle\frac{{\rm{d}}S}{{\rm{d}}A}\simeq\frac{{\Delta{S_{\min}}}}{{\Delta{A_{\min}}}}=\frac{1}{{8\tilde{\hbar}\left(\beta_{0}\right)}}.$
(10)

In the classical limit, the original entropy of a gravitational system can be
expressed as
${S_{0}}={A\mathord{\left/{\vphantom{A{4G}}}\right.\kern-1.2pt}{4G}}$.
Typically, if one wants to modify the entropy, the area $A$ becomes a function
of $A$, i.e., $f\left(A\right)$. So when the effect of EUP is considered, the
general expression of entropy becomes
${S_{0}}={{f\left(A\right)}\mathord{\left/{\vphantom{{f\left(A\right)}{4G}}}\right.\kern-1.2pt}{4G}}$.
The relationship between entropy and area is obtained by calculating the
derivative of this entropy with respect to area $A$ 39

$\displaystyle\frac{{{\rm d}S}}{{{\rm
d}A}}=\frac{{f^{\prime}\left(A\right)}}{{4G}},$ (11)

where
$f^{\prime}\left(A\right)={{{\text{d}}f\left(A\right)}\mathord{\left/{\vphantom{{{\text{d}}f\left(A\right)}{{\text{d}}A}}}\right.\kern-1.2pt}{{\text{d}}A}}$.
Now, by comparing Eq. (10) with Eq. (11), for $\beta_{0}>0$ case, one has

$\displaystyle
f^{\prime}\left(A\right)=\frac{1}{{2\tilde{\hbar}\left(\beta_{0}\right)}}=1+\frac{{16\pi\beta_{0}\ell_{p}^{2}}}{A}.$
(12)

For $\beta_{0}<0$ case, this relationship becomes

$\displaystyle
f^{\prime}\left(A\right)=\frac{1}{{2\tilde{\hbar}\left(\beta_{0}\right)}}=1-\frac{{16\pi\beta_{0}\ell_{p}^{2}}}{A}.$
(13)

However, when $\beta_{0}=0$, one has $f^{\prime}\left(A\right)=1$, which
consistent with the standard result in classical limit. Then, by integrating
Eq. (12), the EUP corrected the entropy with positive parameter is given by

$\displaystyle{S_{{\text{EUP}}}}=\int{\frac{{f^{\prime}\left(A\right)}}{{4G}}{\text{d}}A}=\frac{A}{{4G}}+\frac{{4\pi}}{G}\beta_{0}\ell_{p}^{2}\ln
A,$ (14)

whereas the negative EUP parameter case is

$\displaystyle{S_{{\text{EUP}}}}=\frac{A}{{4G}}-\frac{{4\pi}}{G}\beta_{0}\ell_{p}^{2}\ln
A.$ (15)

Clearly, the EUP yields entropy with the parameter $\beta_{0}$ and a product
of logarithmic terms. The incorporation of logarithmic corrections to entropy
holds validity in 4 (or 3+1) dimensions. Nevertheless, it is worth noting that
in varying dimensions, the modified entropy is recognized to exhibit power law
behavior, namely, polynomial form, rather than logarithms b3 ; b4 . In the
next subsection, we will use Eq. (14) and Eq. (15) to derive the modified
Friedmann equations.

###  III.2 The EUP corrected Friedmann equations

In Refs. 38 ; 39 ; 40 ; 41 ; 42 , it is found that the Friedman equations can
be derived from Bekenstein-Hawking entropy and the first law of
thermodynamics. Therefore, based on the modified entropy formula, we will
derive the EUP-corrected Friedman equations that deviate from the thermal
equilibrium. In the homogeneous and isotropic spacetime, the FRW universe is
described by the line element as follows:

$\displaystyle{\rm{d}}{s^{2}}={h_{\mu\nu}}{\rm{d}}{x^{\mu}}{\rm{d}}{x^{\nu}}+{\tilde{r}^{2}}\left({{\rm{d}}{\theta^{2}}+{{\sin}^{2}}\theta{\rm{d}}{\varphi^{2}}}\right),$
(16)

where ${x^{\mu}}=\left({t,r}\right),\tilde{r}=ra\left(t\right)$ with the scale
factor $a\left(t\right)$,
${h_{\mu\nu}}={\rm{diag}}\left[{-1,\frac{{{a^{2}}}}{{\left({1-k{r^{2}}}\right)}}}\right]$
is the two-dimensional metric with the spatial curvature constant $k$ and
$\mu,\nu=0,1$ , respectively. According to
${h^{\mu\nu}}{\partial_{u}}\tilde{r}{\partial_{\nu}}\tilde{r}=0$, one can re-
express the dynamical apparent of FRW universe as
$\tilde{r}=ar={\left({{H^{2}}+{k\mathord{\left/{\vphantom{k{{a^{2}}}}}\right.\kern-1.2pt}{{a^{2}}}}}\right)^{-\frac{1}{2}}}$
with the Hubble parameter
$H={{\dot{a}}\mathord{\left/{\vphantom{{\dot{a}}a}}\right.\kern-1.2pt}a}$.
Since the matter of the FRW universe can be considered as a perfect fluid, the
energy-momentum tensor of the FRW universe is given by

$\displaystyle{T_{\mu\nu}}=\left({\rho+p}\right){u_{\mu}}{u_{\nu}}+p{g_{\mu\nu}},$
(17)

where ${\rho}$, $p$, ${u_{\mu}}$ and ${g_{\mu\nu}}$ represent the energy
density, pressure, the four velocity of the fluid, and the spacetime metric of
FRW universe, respectively. According to the law of conservation of energy-
momentum $T_{;\nu}^{\mu\nu}=0$, the continuity equation can be obtained as

$\displaystyle\dot{\rho}+3H\left({\rho+p}\right)=0.$ (18)

Next, we derive the Friedmann equation from a thermodynamic point of view in
terms of the modified entropy (13). According to Ref. 43 , the first law of
thermodynamics for the matter content within the apparent horizon takes the
following form

$\displaystyle{\rm{d}}E=T{\rm{d}}S+W{\rm{d}}V,$ (19)

where $E=\rho V$ denotes the total of all the matter energy in the apparent
horizon,
$V={{4\pi{{\tilde{r}}^{3}}}\mathord{\left/{\vphantom{{4\pi{{\tilde{r}}^{3}}}3}}\right.\kern-1.2pt}3}$
is the 3-dimensional sphere’s volume, and
$W={{\left({\rho-p}\right)}\mathord{\left/{\vphantom{{\left({\rho-p}\right)}2}}\right.\kern-1.2pt}2}$
is the density of work. Based on the Eq. (18), the differential form of energy
can be expressed as

$\displaystyle{\rm{d}}E=\rho{\rm{d}}V+V{\rm{d}}\rho=4\pi{\tilde{r}^{2}}\rho{\rm{d}}\tilde{r}+\frac{{4\pi{{\tilde{r}}^{3}}}}{3}{\rm{d}}\rho,$
(20)

Then, the temperature of the apparent horizon is
$T={\kappa\mathord{\left/{\vphantom{\kappa{2\pi}}}\right.\kern-1.2pt}{2\pi}}$,
and the surface gravity for the metric FRW universe is
$\kappa={{-\left({1-{{\mathop{\tilde{r}}\limits^{.}}\mathord{\left/{\vphantom{{\mathop{\tilde{r}}\limits^{.}}{2H\tilde{r}}}}\right.\kern-1.2pt}{2H\tilde{r}}}}\right)}\mathord{\left/{\vphantom{{-\left({1-{{\mathop{\tilde{r}}\limits^{.}}\mathord{\left/{\vphantom{{\dot{\tilde{r}}}{2H\tilde{r}}}}\right.\kern-1.2pt}{2H\tilde{r}}}}\right)}{\tilde{r}}}}\right.\kern-1.2pt}{\tilde{r}}}$
with
$\mathop{\tilde{r}}\limits^{.}={{\partial\tilde{r}}\mathord{\left/{\vphantom{{\partial\tilde{r}}{\partial
t}}}\right.\kern-1.2pt}{\partial t}}$. Thus, the equation for the first law of
thermodynamics on the right can be rewritten as

$\displaystyle
T{\rm{d}}S=-\frac{1}{{2\pi\tilde{r}}}\left({1-\frac{{\mathop{\tilde{r}}\limits^{.}}}{{2H\tilde{r}}}}\right)\frac{{f^{\prime}\left(A\right)}}{{4G}}{\rm{dA}},$
(21) $\displaystyle
W{\rm{d}}V=2\pi{\tilde{r}^{2}}\left({\rho-p}\right){\rm{d}}\tilde{r}.$ (22)

Combining the FRW cosmic dynamics horizon $\tilde{r}$, and because the radius
of the horizon horizon is fixed within an infinitesimal time interval, that is
$\mathop{\tilde{r}}\limits^{.}=0$. Then the first Friedman equation is

$\displaystyle-4\pi
G\left({\rho+p}\right)=\left({\dot{H}-\frac{k}{{{a^{2}}}}}\right)f^{\prime}\left(A\right).$
(23)

Substituting the continuity equation (18) into the Eq. (23), and integrating,
the second Friedmann equation is obtained as follows:

$\displaystyle\frac{8}{3}\pi
G\rho=-4\pi\int{f^{\prime}\left(A\right)\frac{{{\text{d}}A}}{{{A^{2}}}}}.$
(24)

By putting Eq. (12) into Eq. (23) and Eq. (24), the EUP corrected Friedmann
equations are given by

$\displaystyle-4\pi
G\left({\rho+p}\right)=\left({\dot{H}-\frac{k}{{{a^{2}}}}}\right)\left({1+\frac{{16\pi\beta_{0}\ell_{p}^{2}}}{A}}\right),$
(25) $\displaystyle\frac{8}{3}\pi
G\rho=\frac{{4\pi\left({A+8\pi\beta_{0}\ell_{p}^{2}}\right)}}{{{A^{2}}}}+\mathcal{C}.$
(26)

In the vacuum energy era, the boundary conditions determine the integration
constant $\mathcal{C}$ 22 . When the apparent horizon A tends to infinity, the
energy density becomes a cosmological constant $\rho={\rm{\Lambda}}$, which
leads to $\mathcal{C}={{8\pi G\Lambda}\mathord{\left/{\vphantom{{8\pi
G\Lambda}3}}\right.\kern-1.2pt}3}$. Considering
$A=4\pi{\tilde{r}^{2}}={{4\pi}\mathord{\left/{\vphantom{{4\pi}{\left({{H^{2}}+\frac{k}{{{a^{2}}}}}\right)}}}\right.\kern-1.2pt}{\left({{H^{2}}+\frac{k}{{{a^{2}}}}}\right)}}$,
the modified Friedmann equations becomes

$\displaystyle-4\pi
G\left({\rho+p}\right)=\left({\dot{H}-\frac{k}{{{a^{2}}}}}\right)\left[{1+4\beta_{0}\ell_{p}^{2}\left({{H^{2}}+\frac{k}{{{a^{2}}}}}\right)}\right],$
(27) $\displaystyle\frac{8}{3}\pi
G\left({\rho-\Lambda}\right)=\frac{{\frac{{4\pi}}{{\left({{H^{2}}+{k\mathord{\left/{\vphantom{k{{a^{2}}}}}\right.\kern-1.2pt}{{a^{2}}}}}\right)}}+8\pi\beta_{0}\ell_{p}^{2}}}{{{{\left[{{{4\pi}\mathord{\left/{\vphantom{{4\pi}{\left({{H^{2}}+\frac{k}{{{a^{2}}}}}\right)}}}\right.\kern-1.2pt}{\left({{H^{2}}+\frac{k}{{{a^{2}}}}}\right)}}}\right]}^{2}}}}.$
(28)

In line with our research objective, our focus will be on studying a flat
universe that is dominated by radiation. Therefore, the cosmological constant
$\Lambda$ and the spatial curvature constant $k$ should reduce to zero22 ,
resulting in the modified equations taking the following form

$\displaystyle-4\pi
G\left({\rho+p}\right)=\dot{H}\left({1+4\beta_{0}\ell_{p}^{2}{H^{2}}}\right),$
(29) $\displaystyle\frac{8}{3}\pi
G\rho={H^{2}}\left({1+2\beta_{0}\ell_{p}^{2}{H^{2}}}\right).$ (30)

It needs to be noted that Eq. (29) and Eq. (30) are valid on the basis of
$\beta_{0}>0$. With the same derivation, one can obtain the results for the
case $\beta_{0}<0$ as follow

$\displaystyle-4\pi
G\left({\rho+p}\right)=\dot{H}\left({1-4\beta_{0}\ell_{p}^{2}{H^{2}}}\right),$
(31) $\displaystyle\frac{8}{3}\pi
G\rho={H^{2}}\left({1-2\beta_{0}\ell_{p}^{2}{H^{2}}}\right).$ (32)

The aforementioned findings demonstrate that the EUP has a distinct impact on
the energy density and pressure in the early universe, irrespective of whether
the EUP parameters are positive or negative. However, when $\beta_{0}=0$, the
modifications return to the original cases. In the subsequent section, they
will be used to break thermodynamic equilibrium and explain baryon asymmetry
in the early universe.

###  III.3 The EUP corrected baryon asymmetry factor

From the above, it is clear that in the standard cosmological model, the Ricci
curvature scalar can be expressed by the energy density ${{\rho}}$ and the
pressure ${p}$. In order to investigate the universe’s deviation of thermal
equilibrium, it is indispensable to re-express the ${{\rho}}$ and ${{p}}$ as
$\rho_{{\rm{EUP}}}={\rho_{0}}+{{\Delta}}\rho$ and
$p_{{\text{EUP}}}={p_{0}}+{{\Delta}}p$, where $\rho_{0}$ and ${p_{0}}$ denote
the original energy density and pressure at thermal equilibrium, respectively.
${\rm{\Delta\rho}}$ and ${\rm{\Delta}}p$ are correction quantities.
Substituting $\rho_{{\rm{EUP}}}$ and $p_{{\text{EUP}}}$ into the Eq. (29) and
Eq. (30), considering $\dot{H}=-4\pi G\left({{\rho_{0}}+{p_{0}}}\right)$ and
${H^{2}}={{8\pi G{\rho_{0}}}\mathord{\left/{\vphantom{{8\pi
G{\rho_{0}}}3}}\right.\kern-1.2pt}3}$, one yields

$\displaystyle{\rho_{{\rm{EUP}}}}={\rho_{0}}+\frac{{16}}{3}G\pi\beta_{0}\ell_{p}^{2}{\rho_{0}}^{2},$
(33)
$\displaystyle{p_{{\text{EUP}}}}=w{\rho_{0}}+\frac{{16G\pi\beta_{0}\ell_{p}^{2}}}{3}\left({1+2w}\right){\rho_{0}}^{2}.$
(34)

The results demonstrate that the QG effect in the higher-order EUP can cause
changes in energy density and pressure in the radiation-dominated era,
breaking the universe’s thermal equilibrium and fulfilling the third Sakharov
condition. Next, we substitute the corrected energy density and pressure into
Eq. (5) to obtain the corrected Ricci curvature scalar:

$\displaystyle{R_{\rm EUP}}=$
$\displaystyle\frac{8}{3}G\pi{\rho_{0}}\left[{32G\pi\beta_{0}\ell_{p}^{2}{\rho_{0}}-3+w\left({9+96G\pi\beta_{0}\ell_{p}^{2}{\rho_{0}}}\right)}\right].$
(35)

Both energy density and pressure are functions with respect to time, and using
the continuity equation, the time derivative of the modified Ricci curvature
scalar is obtained as

$\displaystyle{\dot{R}_{{\text{EUP}}}}$
$\displaystyle=-16\sqrt{\frac{2}{3}}{\pi^{\frac{3}{2}}}\left({1+w}\right){\left({G{\rho_{0}}}\right)^{\frac{3}{2}}}$
$\displaystyle\times\left[{64G\pi\beta_{0}\ell_{p}^{2}{\rho_{0}}-3+3w\left({3+64G\pi\beta_{0}\ell_{p}^{2}{\rho_{0}}}\right)}\right].$
(36)

Here we set $w=1/3$ because the universe is in a radiation-dominated era, so
the above equation becomes:

$\displaystyle{\dot{R}_{\rm
EUP}}=-\frac{{8192}}{3}\sqrt{\frac{2}{3}}G{\pi^{\frac{5}{2}}}\beta_{0}\ell_{p}^{2}{\rho_{0}}{(G{\rho_{0}})^{\frac{3}{2}}}.$
(37)

This is the EUP corrected the time derivative of the Ricci curvature scalar in
the radiation-dominated era, substituting Eq. (37) into the expression of Eq.
(3), the modified BAF is given by

$\displaystyle{\eta_{\rm
EUP}}=\frac{{10240\sqrt{\frac{{2\pi}}{3}}\beta_{0}\ell_{p}^{2}{{(G{\rho_{0}})}^{\frac{5}{2}}}{g_{b}}}}{{{g_{\rm{*}}}M_{\rm{*}}^{2}{T_{D}}}}.$
(38)

Similarly, by using the Eq. (31) and Eq. (32), the expression of BAF for
$\beta_{0}<0$ becomes

$\displaystyle{\eta_{{\text{EUP}}}}=-\frac{{10240\sqrt{\frac{{2\pi}}{3}}\beta_{0}\ell_{p}^{2}{{\left({G{\rho_{0}}}\right)}^{\frac{5}{2}}}{g_{b}}}}{{{g_{*}}M_{*}^{2}{T_{D}}}}.$
(39)

Obviously, Eq. (38) and Eq. (39) are on longer zero, which are associated with
the deformation parameter ${\rm{\beta_{0}}}$. Our results demonstrate that the
EUP with positive and negative parameters can generate baryon asymmetry in the
radiation-dominated era.

##  IV The bounds of the deformation parameters $\beta_{0}$

Besides the numerous theoretical studies on the EUP, a popular study direction
in QG phenomenology focuses on attempting to quantify the size of EUP
corrections by constraining the deformation parameter 44 ; 45 ; 46 ; 47 ; 47+
; 47a+ ; 47b+ . The deformation parameters are theoretically constantly
assumed to be 1, allowing the QG effect to be valid when the energy is close
to the Planck scale. The bounds of the deformation parameters can be
determined with experimental data and observations when the assumption is
ignored. Therefore, in this section, we will compare the theoretical results
with the observed results to obtain the bounds of deformation parameter. Then,
we need to replace the gravitational mass with the Planck mass
$G={1\mathord{\left/{\vphantom{1{M_{p}^{2}}}}\right.\kern-1.2pt}{M_{p}^{2}}}$,
with $M_{p}\sim 1.22\times{10^{28}}{\rm eV}$. The density at thermal
equilibrium can be expressed as
${\rho_{0}}={{\pi{g_{\text{*}}}{T_{D}}^{4}}\mathord{\left/{\vphantom{{\pi{g_{\text{*}}}{T_{D}}^{4}}{30}}}\right.\kern-1.2pt}{30}}$,
when the deformation parameter takes positive, the BAF is rewritten as

$\displaystyle{\eta_{{\text{EUP}}}}=\frac{{512{\pi^{3}}\beta_{0}\ell_{p}^{2}{g_{b}}}}{{135\sqrt{5}{g_{\text{*}}}M_{\text{*}}^{2}{T_{D}}}}{\left({\frac{{{g_{\text{*}}}{T_{D}}^{4}}}{{{M_{p}}^{2}}}}\right)^{\frac{5}{2}}}.$
(40)

To further obtain the bounds of the EUP parameter, we set
${M_{*}}={{{M_{p}}}\mathord{\left/{\vphantom{{{M_{{p}}}}{\sqrt{8\pi}}}}\right.\kern-1.2pt}{\sqrt{8\pi}}}$,
$T_{D}\sim 2\times{10^{25}}{\rm{eV}}$, ${{g_{*}}}=106$, ${{g_{b}}}=1$. Then,
solving Eq. (40), the deformation parameter $\beta_{0}>0$ can be expressed as

$\displaystyle{{\rm{\beta}}_{0}}=8.1\times{10^{18}}{\eta_{\rm EUP}}.$ (41)

When the deformation parameter takes negative, the corresponding constraint is
obtained in the same way:

$\displaystyle-{{\rm{\beta}}_{0}}=8.1\times{10^{18}}{\eta_{\rm EUP}}.$ (42)

Obviously, the above equations show that the bounds of the deformation
parameter ${{\rm{\beta}}_{0}}$ are determined by the observation ${\eta_{\rm
EUP}}$. We list the range of ${\eta}$ given by astronomical observations over
the last four decades, which can be used to constrain the bounds of higher-
order generalized uncertainty principle parameters, as follows in Table 1.

Table 1:  The range of $\eta$ and the bounds of ${{\rm{\beta}}_{0}}$ from data
source.

Data source | $\eta$ | ${{\rm{\beta}}_{0}}>0$ | ${{\rm{\beta}}_{0}}<0$
---|---|---|---
Particle Data Group33  | $\eta\leq 8.6\times{10^{-11}}$ | ${{\rm{\beta}}_{0}}\leq 6.9\times{10^{8}}$ | ${{\rm{\beta}}_{0}}\geq-6.9\times{10^{8}}$
BBN34  | $3.4\times{10^{-10}}\leq\eta\leq 6.9\times{10^{-10}}$ | $2.7\times{10^{9}}\leq{{\rm{\beta}}_{0}}\leq 5.5\times{10^{9}}$ | $-5.8\times{10^{9}}\leq{{\rm{\beta}}_{0}}\leq-2.9\times{10^{9}}$
Planck observations48  | $\eta\leq 6.2\times{10^{-10}}$ | ${{\rm{\beta}}_{0}}\leq 5.0\times{10^{9}}$ | ${{\rm{\beta}}_{0}}\geq-5.0\times{10^{9}}$
Deuterium and 3He abundances49  | $5.7\times{10^{-11}}\leq\eta\leq 9.9\times{10^{-11}}$ | $4.6\times{10^{8}}\leq{{\rm{\beta}}_{0}}\leq 8.0\times{10^{8}}$ | $-8.0\times{10^{8}}\leq{{\rm{\beta}}_{0}}\leq-4.6\times{10^{8}}$
Acoustic peaks in CMB measured by WMAP50  | $\eta\leq 6.3\times{10^{-10}}$ | ${{\rm{\beta}}_{0}}\leq 5.1\times{10^{9}}$ | ${{\rm{\beta}}_{0}}\geq-5.1\times{10^{9}}$
Deuterium and Hydrogenium abundance 51 ; 52  | $5.7\times{10^{-10}}\leq\eta\leq 6.7\times{10^{-10}}$ | $4.6\times{10^{9}}\leq{{\rm{\beta}}_{0}}\leq 5.4\times{10^{9}}$ | $-5.4\times{10^{9}}\leq{{\rm{\beta}}_{0}}\leq-4.6\times{10^{9}}$

In Table 1, by utilizing the data from different experiments and observation
results, it is found that the bounds of positive deformation parameter is
$4.6\times{10^{8}}\sim 5.5\times{10^{9}}$, while the bounds of negative
deformation parameter is $-5.5\times{10^{9}}\sim-4.6\times{10^{8}}$.
Furthermore, we would like to compare our results with those in Ref. 22 . By
subjecting both models to the same conditions, specifically the Deuterium and
3He abundances 49 , one can directly compare their respective outcomes. The
result obtained from the KMM model is expressed as
$\beta_{0}^{{\text{KMM}}}=-4.63\times{10^{18}}\eta$, which subsequently yields
a constraint on the deformation parameter, i.e.,
$-4.58\times{10^{8}}\leq\beta_{0}^{{\text{KMM}}}\leq-2.64\times{10^{8}}$. It
is easy found that the absolute values of our results
$\left|{{\beta_{0}}}\right|$ have the same order of magnitude as
$\beta_{0}^{{\text{KMM}}}$, which is consistent with the relationship between
the two deformation parameter that we presented in section 1, that is
$\left|{{\beta_{0}}}\right|\sim\beta_{0}^{{\text{KMM}}}$. However, it should
be noted that the KMM model can only give bound for negative parameter, while
ours can give both positive and negative parameter cases.

##  V Conclusion

In the present work, we have investigated the gravitational baryogenesis
mechanism within the framework of a new higher-order EUP with a fixed and
uniform minimum length in the presence of positive and negative deformation
parameters. Our analysis involved deriving the corrected Bekenstein-Hawking
entropy and Friedmann equations by combining the higher-order EUP with the
first law of thermodynamics. These modifications allowed us to explore the
thermodynamic equilibrium of the universe during the radiation era,
considering the effects of QG. Finally, we calculated the EUP-corrected baryon
asymmetry factor, denoted as $\eta$. Our results revealed that the EUP plays
an important role in modifying the pressure ${p}$ and energy density $\rho$
terms in the Friedmann equations, thus disrupting the thermal equilibrium of
the universe. This modification addresses a limitation in the original theory
by satisfying Sakharov’s third condition. Moreover, we observed that the EUP
introduces non-zero values for the Ricci scalar $R$ and its time derivative
$\dot{R}$, leading to a non-zero $\eta$. This indicates that the EUP can
effectively generate baryon asymmetry during the radiation-dominated era.

Furthermore, by comparing our theoretical results with observational data, we
have successfully constrained the positive EUP parameter to the range of
$4.6\times{10^{8}}\sim 5.5\times{10^{9}}$, and the negative deformation
parameter to the range of $-5.5\times{10^{9}}\sim-4.6\times{10^{8}}$. It is
well known that current research has focused on EUP with positive parameter,
whereas EUP with negative deformation parameters has received relatively less
attention. It demonstrates that EUP with negative deformation parameters can
exert the same influence on baryon asymmetry as EUP with positive deformation
parameters. Moreover, we have successfully demonstrated the feasibility of
constraining both positive and negative deformation parameters using
observational data. In future research, our objective is to integrate the
newly proposed EUP (1) into various other physical theories. We plan to
explore its implications in fields such as the Chandrasekhar limit, black hole
thermodynamics, and the associated phase transitions. By incorporating the EUP
model into these diverse contexts, we aim to gain a deeper understanding of
its effects and potential applications. Additionally, we will endeavor to
refine the constraints on the parameters of the EUP model by conducting
rigorous analysis of experimental data. This data-driven approach will allow
us to further validate the model’s predictions and assess its accuracy and
reliability.

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