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Arf invariants of codimension one in a Wall group of the dihedral group P. M. Akhmet'evab, Yu. V. Muranovc a Tikhonov Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, Moscow, Russia b Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, Russian Academy of Sciences, Troitsk, Moscow, Russia c University of Warmia and Mazury in Olsztyn, Olsztyn, Poland
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     § 1. IntroductionLet $M^n$ be a closed smooth (piecewise linear ($\mathrm{PL}$) or topological) manifold of dimension $n$ with fundamental group $\pi=\pi_1(M)$ and orientation character $w$. (The latter is a homomorphism $w\colon \pi \to \{\pm\}$.) One of the main problems of geometric topology is to find manifolds within a given category that are simply homotopy equivalent to $M$. If $M$ is smooth, then the problem is extremely hard even in the simply connected case. For the definition of the Arf invariant and its connections with fundamental problems in homotopy theory, see [1] and [2]. A simple homotopy equivalence $f\colon X^n\to M^n$ is called a homotopy smoothing in the smooth case and a homotopy triangulation in the piecewise linear case. Two homotopy smoothings (triangulations) $f_i\colon X_i\to M$ ($i=1,2$) are equivalent if there exists a diffeomorphism (a piecewise linear isomorphism) $g\colon X_1\to X_2$ such that the maps $f_1$ and $f_2\circ g$ are homotopic. The set of equivalence classes of homotopy smoothings (triangulations) of a manifold $M$ is denoted by $hS(M)$ (by $hT(X)$). For ${n\geqslant 5}$ the set $hT(M^n)$ is included in the Browder-Novikov-Wall-Sullivan surgery exact sequence:
$$
\begin{equation}
\dots\to L_{n+1}(\pi_1(M))\xrightarrow{\nu} hT(X)\to [M, G/\mathrm{PL}]\xrightarrow{\theta} L_n(\pi_1(M)),
\end{equation}
\tag{1.1}
$$
where $G/\mathrm{PL}$ is the fibre of the natural map $\mathrm{BPL}\to \mathrm{BG}$ of classifying spaces, $[M, G/\mathrm{PL}]$ is the set of homotopy classes of maps from $M$ to $G/\mathrm{PL}$, and $L_{n}(\pi_1(M))$ is the Wall obstruction group, which depends on the group $\pi_1(M)$, the orientation $w$ and $n\bmod 4$ (see [3] and [4]). Note that the surgery obstruction groups $L_*(\pi_1(M), w)$ do not depend on the category of manifolds under consideration. In what follows we work in the PL category, and we always assume that an orientation homomorphism is chosen on $\pi$ when we consider the Wall groups $L_*(\pi)$. A map $f\colon X^n\to M^n$ of manifolds of the same dimension is called normal if the normal bundle of $X$ is the pullback of some bundle $\xi$ on the manifold $M$. The normal cobordism classes of normal maps to $M$ correspond bijectively to the elements of the set $[M, G/\mathrm{PL}]$. The map $\theta$ in (1.1) defines an obstruction to a surgery of a normal map to $M$ to a simple homotopy equivalence, and the image of $\theta$ contains only elements realised by normal maps of closed manifolds. Any element of the Wall group $L_n(\pi_1(M), w)$ can be realised by an obstruction to surgery of a normal map with boundary which is a simple homotopy equivalence on the boundary (see [3]). This construction is used to define the map $\nu$ in (1.1). The map $\nu$ is an action of the group $L_{n+1}(\pi_1(M^n))$ on the set $hT(X)$ (see [3]). Let $x\in L_{n+1}(\pi_1(M^n))$. For any homotopy triangulation $f\colon X\to M^n$ there exists a normal map $F\colon W\to M^n\times I$ of manifolds with boundary, where $\partial W= \partial_0W\cup \partial_1W$, ${\partial_0W=X}$, $F|_{\partial_0W}=f$, and $F|_{\partial_1W}\colon \partial_1W\to X\times \{1\}$ is a simple homotopy equivalence. Then the obstruction to a surgery of $F$ to a simple homotopy equivalence is equal to $x\in L_{n+1}(\pi)$. Therefore, $\nu(f)=F|_{\partial_1W}\colon \partial_1W\to X\times \{1\}$ is the homotopy triangulation obtained by the action of $x$ on the triangulation $f$. If $x\in L_{n+1}(\pi_1(M^n))$ acts trivially on some triangulation $f\colon X^n\to M^n$, then $x$ is realised by a normal map of closed manifolds (see, for instance, [5], the introduction, or [6], § 2). Therefore, elements that are not realised by normal maps of closed manifolds (do not lie in the image of $\theta$ in (1.1)) act nontrivially on any triangulation of $M^n$. In the case of a finite fundamental group the problem of finding the image of $\theta$ reduces to the same problem for finite $2$-groups (see [7]). The Browder-Livesay invariants provide an effective tool for proving the nonrealisability of elements of the Wall groups of finite $2$-groups. These invariants are obstructions to splitting a simple homotopy equivalence along a one-sided submanifold (see [5] and [8]–[12]). We recall briefly the basic concepts of the problem of splitting along a one-sided submanifold which are necessary for understanding the main results of this work. A pair of closed manifolds $(M, N)$ is called a Browder-Livesay pair if $N^{n-1}$ is a one-sided submanifold of codimension $1$, $n\geqslant 5$, and the embedding $N^{n-1}\to M^n$ induces an isomorphism of fundamental groups. Let $U$ denote a tubular neighborhood of $N$ in $M$ with boundary $\partial U$. Consider the following square diagram $F$ of fundamental groups and natural maps: All groups in the square $F$ are equipped with orientation, the upper horizontal map is an isomorphism of oriented groups, the vertical maps are inclusions of index $2$ subgroups, and the orientation homomorphisms on the groups $\pi_1(X)=\pi^+$ and $\pi_1(Y )=\pi^-$ coincide on the images of the vertical maps, and are different away from these images. In what follows we do not specify the ‘$+$’ orientation unless this leads to confusion.A simple homotopy equivalence $f\colon X\to M$ splits along a submanifold $N$ if it is homotopic to a map $g \colon X\to M$ that is transverse to $N$, and for $Y = g^{-1}(N)$ the restrictions
$$
\begin{equation*}
g|_Y\colon Y\to N\quad\text{and} \quad g|_{(X \setminus Y)}\colon X\setminus Y\to M\setminus N
\end{equation*}
\notag
$$
are simple homotopy equivalences. In this case the groups of obstructions to splitting are called the Browder-Livesay groups and denoted by $LN_{n-1}(\rho\to \pi)$. These groups depend functorially on the pair $(\rho\to \pi)$ and $(n-1)\bmod 4$. For a Browder-Livesay pair given by the square $F$, there is a braid of exact sequences connecting the Browder-Livesay groups and the Wall groups (see [3], [4], [11] and [13]): The map $i_*\colon L_*(\rho)\to L_*(\pi)$ in (1.2) is induced by the inclusion of groups and $i^!_-\colon L_*(\pi^-)\to L_*(\rho)$ is the transfer homomorphism. The map $\partial $ is the composition of the transfer and scaling homomorphism, and $c^-$ is the composition of the induction and scaling homomorphism. These maps will be defined algebraically in § 2. Rows of the diagram (1.2) are chain complexes, and the map $\Gamma $ defines an isomorphism of homology groups in the corresponding terms of the diagram. For $x\in L_{n}(\pi)$, the element $\partial (x)\in LN_{n-2}(\rho\to \pi)$ is called the (first) Browder-Livesay invariant. If $\partial (x)\ne 0$, then $x$ is not realised by a normal map of closed manifolds (see [10]). Let $\partial x=0$ in (1.2). Consider the diagram It consists of two lines of the diagram (1.2) and two lines of a similar diagram for the group $\pi^-$. If $\partial (x)=0$, then the homology class $[x]$ of $x$ is defined in $L_n(\pi)$ of the top row. If this class is distinct from zero, then we can consider the set $\partial^- \Gamma[x]\subset LN_{n-3}(\rho\to \pi^-)$. This set is called the second Browder-Livesay invariant. If $0\notin [x]$, then $x$ is not realised by a normal map of closed manifolds (see [11]). By iterating this construction we obtain an eight-line diagram, which can be extended in the vertical direction by simple repetition due to the 4-periodicity of Wall groups (see [5] and [12]). Definition (see [5] and [12]). An element $x\in L_n(\pi)$ is called an element of the first type with respect to the subgroup $\rho\subset \pi$ of index 2 if at some step of iterations the class $\partial^{\pm}\Gamma^k [x]$ does not contain zero. An element $x\in \operatorname{Ker} \partial$ that is not an element of the first type is called an element of the second type if $0\notin \Gamma^k[x]$ for all $k\geqslant 0$. Otherwise, the element $x\in L_n(\pi)$ is called an element of the third type. Theorem 1 (see [5] and [12]). Elements of the first and second type are not realised by normal maps of closed manifolds. In many cases classification by type allows one to describe the image of the map $\theta$ in the exact sequence (1.1) (see [5], [12] and [14]). For example, it was proved in [5], § 7, that, given an elementary 2-group $\pi$ with an arbitrary orientation, elements of the third type in the Wall groups $L_*(\pi )$ are realisable by normal maps, and these are the only realisable elements. According to the Morgan-Pardon conjecture, stated in the introduction to [5], in the Wall groups of a finite 2-group only ‘Arf invariants of various codimensions’ are realisable by normal maps of closed $\mathrm{PL}$-manifolds (the definition was given in [5], § 6; also see [12], § 4). These invariants are specified by elements of the third type. The main results in this work are Theorem 3 and the corollary in § 3, which assert that there exists an element $x$ of the third type in the Wall group $L_3(D_3)$ of the dihedral group of order $8$ with trivial orientation. This element is not realisable by a normal map of closed $\mathrm{PL}$-manifolds, and therefore it acts nontrivially on any homotopy triangulation of a closed $\mathrm{PL}$-manifold of dimension $4k + 2$ with trivial orientation. Furthermore, for any closed $\mathrm{PL}$-manifold $M^{4k+ 3}$ ($4k + 3 \geqslant 6$) with $\pi_1(M)=D_3$ there is a Browder-Livesay pair $(M^{4k+ 3}, N^{4k+2})$ such that in the two-line diagram for this pair the homology class $\Gamma [x]\in L_2(D_3^-)= L_2(\pi_1(N^{4k+2}))$ defined by $[x]\in L_3 (D_3)$ is the Arf invariant in codimension 1. According to the Morgan-Pardon conjecture, $x$ should be realisable by a normal map of closed manifolds. Therefore, the result on the unrealisability of $x$ disproves the well-known Morgan-Pardon conjecture. Theorem 4 proves the nontriviality of the Hasse-Witt torsion of the unique nontrivial element of the Browder-Livesay group $LN_3(\mathbb Z/2\oplus \mathbb Z/2\to D_3^-)$ in the group $Wh_2( \mathbb Z/2\oplus \mathbb Z/2)$. In § 4 we discuss some geometric questions relating to Theorem 4. § 2. Algebraic properties of Browder-Livesay groupsWe recall the algebraic description of the Browder-Livesay groups, which are the Wall groups of the ring $\mathbb Z\pi$ with anti-structure, and give the algebraic definitions of the maps in the diagram (1.3) (see [15] and [16]). Let $R$ be a ring with unity. An anti-structure is a triple $(R,\alpha, u)$, where $u$ is an invertible element of $R$ and $\alpha$ is an anti-automorphism of the ring $R$ satisfying the conditions $\alpha(u)=u^{-1}$ and $\alpha^2(x)= uxu^{-1}$ for any $x\in R$. The anti-automorphism $\alpha$ induces an involution on the group $K_1(R)$, and for any subgroup $X\subset K_1(R)$ that is invariant under this involution the Wall groups $L_n^X(R,\alpha, u)$ are defined. They depend only on the anti-structure $(R,\alpha, u)$, $X\in K_1(R)$ and $n\bmod 4$. Let $\pi$ be a group with an orientation homomorphism $w$. There is an involution $x\to \overline{x}$ on the group ring $\mathbb Z\pi$, which is given by the formula
$$
\begin{equation*}
\overline{\sum n_g\cdot g}=\sum n_g\cdot w(g)g^{-1} \quad\text{for } n_g\in \mathbb Z, \quad g\in \pi.
\end{equation*}
\notag
$$
The subgroup $\{\pm \pi\}\subset K_1(\mathbb Z\pi)$ is invariant under this involution. The triple $(\mathbb Z\pi, ^-, 1)$ is an anti-structure, and so the Wall groups $L_n^{\{\pm\pi\}}(\mathbb Z\pi, ^-, 1)$ are defined. For even $n$ the Wall groups $L_n^{\{\pm\pi\}}(\mathbb Z\pi, ^-, 1)$ coincide with the surgery obstruction groups $L_n(\mathbb Z\pi)$, and for $n=2k+1$ the group $L_{2k+1}(\mathbb Z\pi)$ is obtained by taking the quotient of the group $L_{2k+1}^{\{\pm\pi\}}(\mathbb Z\pi, ^-, 1)$ by the subgroup $\mathbb Z/2$ (see [17]). Note that this subgroup $\mathbb Z/2$ comes functorially from the group $L_{2k+1}^{\{\pm 1\}}(\mathbb Z, \operatorname{Id}, 1)$. Theorem 2 (see [11], Theorem 2, and [3]). Let $(\pi, w)$ be a group with orientation homomorphism, let $\rho\to \pi$ be an inclusion of an index 2 subgroup, and let $t\in \pi\setminus \rho$. Then there is an isomorphism where $\alpha(x)=t^{-1}\overline{x}t$. In odd dimensions the quotient by the subgroup $\mathbb Z/2$ must be taken on the right-hand side, similarly to the case of odd-dimensional Wall groups. Let $(R,\alpha, u)$ be an anti-structure and $v\in R$ be an invertible element. A scaling of the anti-structure $(R,\alpha, u)$ is the anti-structure $(R, \beta, w)$ such that $\beta(x)=v\alpha(x)v^{-1}$ and $w=v\alpha(v^{-1})u$. Scaling defines an isomorphism between the categories of quadratic forms over the anti-structures under consideration and, consequently, an isomorphism of $L$-groups $ (R,\alpha, u) \xrightarrow{v} (R, \beta, w)$, which is also referred to as scaling (see [18]). For an inclusion $\rho\to \pi$ of an index 2 subgroup, there is a natural inclusion of anti-structures $(\mathbb Z\rho, \alpha, u)\to (\mathbb Z\pi, \alpha, u)$, where $u=- w(t)t^{-2}$. Let $\gamma$ denote the automorphism of the ring $\mathbb Z\pi$ given by $\gamma(x+yt)=x-yt$, $x,y\in \mathbb Z\rho$. Then there is an anti-structure $(\mathbb Z\pi, \widetilde{\alpha}, \widetilde{u})$ such that $\widetilde{\alpha}(x+yt)= t[\gamma\alpha(x+yt) )]t^{-1}$ and $\widetilde{u}=-t\alpha(t^{-1})u$; it is obtained from the anti-structure $(\mathbb Z\pi, \gamma\alpha, u)$ by scaling to $t$ (see [18], p. 352). In this case there is an isomorphism of the Wall groups of anti-structures $(\mathbb Z\pi, \alpha, u)$ and $(\mathbb Z\pi, \gamma\widetilde{\alpha}, -\widetilde{u})$. Now we can describe the maps $\partial$ and $c$ in the diagram (1.3) using the language of rings with anti-structures. Let $t\in \pi\setminus \rho$. The map $\partial$ is the composite
$$
\begin{equation}
\begin{aligned} \, \notag &L_n^{\{\pm \pi\}}(\mathbb Z\pi, ^-, 1)\xrightarrow{t^{-1}} L_n^{\{\pm \pi\}}(\mathbb Z\pi, \alpha, w(t)t^{-2}) \\ &\qquad \xrightarrow{\cong} L_{n-2}^{\{\pm \pi\}}(\mathbb Z\pi, \alpha, -w(t)t^{-2})\xrightarrow{i^!} L_{n-2}^{\{\pm \rho\}}(\mathbb Z\rho, \alpha, -w(t)t^{-2}), \end{aligned}
\end{equation}
\tag{2.1}
$$
where $\alpha(x)=t^{-1}\overline{x}t$. The above isomorphism between Wall groups follows from the more general isomorphism $L_n(R,\alpha, u)\cong L_n(R, \alpha, -u)$, which holds for arbitrary anti-structures (see [18], p. 350), and $i^!$ is the transfer homomorphism. Similarly, the map $c$ is the composite
$$
\begin{equation}
\begin{aligned} \, \notag &LN_n(\rho\to \pi^-)=L_{n}^{\{\pm \rho\}}(\mathbb Z\rho,\alpha, -w(t)t^{-2}) \\ &\qquad \xrightarrow{i_*} L_n^{\{\pm \pi\}}(\mathbb Z\pi, \gamma\alpha, -w(t)t^{-2})\xrightarrow{t} L_{n}^{\{\pm \pi\}}(\mathbb Z\pi, ^-, 1), \end{aligned}
\end{equation}
\tag{2.2}
$$
where $i_*$ is induced by inclusion, and the orientation homomorphism on the two lower groups is twisted by $\gamma$ outside the subgroup $\rho$.
§ 3. The two-line Kharshiladze diagram for the dihedral group $D_3$ of order 8Here we consider the dihedral group of eight elements and recall the basic facts about the two-line Kharshiladze diagram (see [5] and [19]) given by the rows of the braided diagram (1.2). We denote by $\mathbf D=D_3$ the dihedral group of order $8$, which is defined by the presentation
$$
\begin{equation}
\{\mathbf a,\mathbf b\mid [\mathbf b,\mathbf a]=\mathbf b^2,\, \mathbf a^2=\mathbf b^4=1\}.
\end{equation}
\tag{3.1}
$$
The notation $D_3$ is due to the fact that this group is included in an infinite series of $2$-groups; see [20]. We consider the following subgroups of $\mathbf D$:
$$
\begin{equation*}
\{\mathbf a,\mathbf a\mathbf b^2=\dot{\mathbf a}\} = \mathbb I_{\mathbf a \times \dot{\mathbf a}}, \qquad \{\mathbf a\mathbf b=\mathbf d, \mathbf a\mathbf b^3=\dot{\mathbf d}\} = \mathbb I_{\mathbf d \times \dot{\mathbf d}}\quad\text{and} \quad \{\mathbf b\} = \mathbb I_{\mathbf b}.
\end{equation*}
\notag
$$
Clearly, $\mathbb I_{\mathbf b}\cong \mathbb Z/4$, a cyclic group of order $4$. Lemma. The subgroups $\mathbb I_{\mathbf d \times \dot{\mathbf d}}$ and $\mathbb I_{\mathbf a \times \dot{\mathbf a}}$ are isomorphic to $\mathbb Z/2 \times \mathbb Z/2$ and are therefore subgroups of index 2 of $\mathbf D$. There is an involutive automorphism $\chi\colon \mathbf D \to \mathbf D$ which is identical on the subgroup $\mathbb I_{\mathbf b}$ and satisfies $\chi(\mathbb I_{\mathbf a \times \dot {\mathbf a}})=\mathbb I_{\mathbf d \times \dot{\mathbf d}}$. Proof. Consider the coordinate square $K$ with vertices $A_1=(1,1)$, $A_2=(-1,1)$, $A_3=(-1,-1)$ and $A_4=(1,-1)$ on the Euclidean plane. Let $\mathbf D' \subset O(2)$ be the symmetry group of the square $K$. It consists of eight elements: the rotations of the square through angles of ${k\pi}/{2}$, $k=0,1,2,3$, the reflections in the coordinate axes, and the reflections in the bisectors of the coordinate angles passing through a pair of centrally symmetric vertices of the square. The rotations form a cyclic subgroup $\mathbb Z/4$. We denote by $\mathbf b'$ its generator given by the counterclockwise rotation through an angle of ${\pi}/{2}$: We denote the reflection in the horizonal (vertical) coordinate axis by $\mathbf d'$ (respectively, by $\dot{\mathbf d}'$) and the reflection in the bisector through $(A_1,A_3)$ (through $(A_2,A_4)$) by $\mathbf a'$ (respectively, by $\dot{\mathbf a}'$).
It is easy to see that the elements $\mathbf b'$ and $\mathbf a'$ of $\mathbf D'$ satisfy the same relations (3.1) as the elements $\mathbf b$ and $\mathbf a$ of $\mathbf D$. Therefore, an isomorphism $\mathbf D' \to \mathbf D$ is defined. It maps the element $\mathbf d'$ (the element $\dot{\mathbf d}'$) to $\mathbf d$ (respectively, to $\dot{\mathbf d}$). Thus, we can write $\mathbf D$ instead of $\mathbf D'$, $\mathbf b$ instead of $\mathbf b'$, etc. We claim that $\mathbb I_{\mathbf d \times \dot{\mathbf d}}$ is isomorphic to $\mathbb Z/2 \times \mathbb Z/2$. Indeed, this subgroup is generated by the reflections of the plane in coordinate axes, which are involutive and commute. In a similar way, $\mathbb I_{\mathbf a \times \dot{\mathbf a}}$ is isomorphic to $\mathbb Z/2 \times \mathbb Z/2$, as this subgroup is generated by the reflections in the bisectors of the coordinate angles. Let $\psi\colon \mathbf D \to \mathbf D$ be the automorphism defined by the conjugation of $\mathbf D \subset O(2)$ by the clockwise rotation of the $\tau$-plane through an angle of ${\pi}/{4}$: Clearly, $\psi(\mathbf d)=\mathbf a$, $\psi(\dot{\mathbf d})=\dot{\mathbf a}$ and $\psi(\mathbf b)=\mathbf b$. Hence $\psi(\mathbb I_{\mathbf a \times \dot{\mathbf a}})=\mathbb I_{\mathbf d \times \dot{\mathbf d}}$ and $\psi\vert_{\mathbb I_{\mathbf b}}=\operatorname{Id}$.Consider the group $\mathbf D$ with orientation character $w\colon \mathbf D \to \mathbb Z/2$. If $w$ is nontrivial, then it is uniquely characterised by its kernel, which is a subgroup of index $2$ in $\mathbf D$. For a subgroup $\pi$ of index 2 in $\mathbf D$, let $\mathbf D^-_{\pi}$ denote the group with the corresponding orientation character $w$, which is nontrivial outside $\pi$. There are only three nontrivial orientation characters on the group $\mathbf D$, which correspond to the subgroups of index 2 defined above. Therefore, we obtain the following groups with orientation characters: $\mathbf D^-_{\mathbf a \times \dot{\mathbf a}}$, $\mathbf D^-_{\mathbf d \times \dot{\mathbf d}}$ and $\mathbf D^-_{\mathbf b}$. Let $\mathbf D^+$ denote the group $\mathbf D$ with trivial orientation. When it is necessary to specify the trivial orientation character on a subgroup $\pi$, we write $\pi^+$. An orientation character on $\mathbf D$ induces an orientation character on any of its subgroups. In what follows we also consider orientation characters on the subgroups of $\mathbf D$ defined above. Let $\mathbb I_{\mathbf a \times \dot{\mathbf a}}^-$ denote the group with orientation character that is nontrivial on each of the generators $\mathbf a$ and $\dot{\mathbf a}$. This orientation character on $\mathbb I_{\mathbf a \times \dot{\mathbf a}}^-$ can be induced from the orientation character on $\mathbf D^-_{\mathbf d \times \dot {\mathbf d}}$ or from the orientation character on $\mathbf D^-_{\mathbf b}$. However, it cannot be induced from the orientation character on $\mathbf D^-_{\mathbf a \times \dot{\mathbf a}}$. The orientation character on $\mathbb I_{\mathbf d \times \dot{\mathbf d}}^-$ can be induced from the character on $\mathbf D^-_{\mathbf a \times \dot{\mathbf a}}$ or from the character on $\mathbf D^-_{\mathbf b}$, but cannot be induced from the character on $\mathbf D_{\mathbf d \times \dot{\mathbf d}}^-$. The latter induces a trivial character on $\mathbb I_{\mathbf d \times \dot{\mathbf d}}$. Similarly, the character on $\mathbb I_{\mathbf b}^-$ can be induced from the character on $\mathbf D^-_{\mathbf a \times \dot{\mathbf a}}$ or from the character on $\mathbf D^-_{\mathbf d \times \dot{\mathbf d}}$. The trivial character on $\mathbb I_{\mathbf b}$ is induced from the character on $\mathbf D_{\mathbf b}^-$ or from the character on $\mathbf D^+$. Let $i\colon \pi \to \mathbf D$ be an inclusion of index $2$ subgroup, let $w\colon \pi\to \mathbb Z/2$ be an orientation character on $\pi$, and let $t\in \mathbf D\setminus \pi$. Let $\mathbf D^-_{\pi}$ denote the group $\mathbf D$ with the orientation character $w$ that coincides with $w$ on $\pi$ and satisfies $w(t)=- 1$, and let $\mathbf D^+_{\pi}$ denote the group $\mathbf D$ with the orientation character that coincides with $w$ on $\pi$ and satisfies $w(t)= 1$. The Browder-Livesay groups $LN_{n}(\pi \to \mathbf D^{\pm}_{\pi})$ are defined for $n=0,1,2,3 \ \operatorname{mod}4$, and the two-line diagram in (1.3) takes the form Geometrically, the diagram (3.2) corresponds to a homotopy $\mathrm{PL}$-triangulation (or homotopy smoothing) of a high-dimensional oriented manifold $M^{n}$ with $\pi_1(M)=\mathbf D^+ _{\pi}$ (we will be interested in the case $\dim(M)+ 1=n+1=3\ \operatorname{mod} 4$) with a distinguished one-sided submanifold
$$
\begin{equation}
M^{n} \supset N_{\mathbf a \times \dot{\mathbf a}}^{n-1}
\end{equation}
\tag{3.3}
$$
of codimension $1$ whose fundamental class realises the Poincaré dual of the cohomology class of the cocycle defined by the homomorphism $\mathbf D \to \mathbb Z/2$ with kernel $\mathbb I_{\mathbf a \times \dot{\mathbf a}}$. Therefore, $(M, N)$ is a Browder-Livesay pair. We complement the pair $(M, N)$ to a sequence of one-sided submanifolds:
$$
\begin{equation}
M^{n} \supset N_{\mathbf a \times \dot{\mathbf a}}^{n-1} \supset K^{n-2}_{\mathbf a \times \dot{\mathbf a}},
\end{equation}
\tag{3.4}
$$
where the submanifold $K^{n-2}_{\mathbf a \times \dot{\mathbf a}}$ is distinguished in $N_{\mathbf a \times \dot{\mathbf a}}^{n- 1}$ by the same subgroup $\mathbb I_{\mathbf a \times \dot{\mathbf a}}$. Then $(N,K)$ also is a Browder-Livesay pair, and since orientation characters add up, the fundamental group $\pi_1(K)$ coincides with $\mathbf D$ and has the same orientation character as $\pi_1(M)$. Note that the choice of the second submanifold in the sequence (3.4) does not affect the result of the corollary (see below). This submanifold can be chosen otherwise, using the subgroup $\mathbb I_{\mathbf b}$ or $\mathbb I_{\mathbf d \times \dot{\mathbf d}}$. For $\mathbb I_{\mathbf b}$ we obtain a sequence of one-sided submanifolds
$$
\begin{equation}
M^{n} \supset N_{\mathbf a \times \dot{\mathbf a}}^{n-1} \supset K_{\mathbf d \times \dot{\mathbf d}}^{n-2}.
\end{equation}
\tag{3.5}
$$
In the case of $\mathbb I_{\mathbf d \times \dot{\mathbf d}}$ we obtain a sequence of one-sided submanifolds
$$
\begin{equation}
M^{n} \supset N_{\mathbf a \times \dot{\mathbf a}}^{n-1} \supset K_{\mathbf b}^{n-2}.
\end{equation}
\tag{3.6}
$$
We consider the case (3.4) below; the calculations in the other cases are quite similar. We get the following diagram: where $\mathbf D^+=\mathbf D^+_{\mathbf a \times \dot{\mathbf a}}$, the orientation character is trivial on $\mathbb I_{\mathbf a \times \dot{\mathbf a}}$, and $\mathbf b\in\mathbf D^+\setminus \mathbb I_{\mathbf a \times \dot{\mathbf a}}$. The groups and maps involved in (3.7) are known from [19], [20], [17], [21], p. 527, and [22]. Since the notation in these works differs from the notation introduced above, we describe the relationship between the Wall and Browder-Livesay groups using two different systems of notation. In the notation used by Wall and in [19], [20] and [22] we have In [17] the orientation $w\colon D_3\to \mathbb Z/2$ on the group $D_3$ is defined as follows:
$$
\begin{equation*}
\begin{aligned} \, D_3^{+,+} &\colon \quad w(x)=w(y)=1, \\ D_3^{-,+} &\colon \quad w(x)=-1, \quad w(y)=1, \end{aligned}
\end{equation*}
\notag
$$
that is, the first symbol ‘$\pm$’ corresponds to the value of $w$ at the generator $x$, and the second symbol corresponds to the value of $w$ at $y$. We identify $D_3$ with $\mathbf D$ by identifying the generator $b$ with $x$ and $a$ with $y$. This results in the identifications $\mathbf D^+=D_3^{+,+}$ and $\mathbf D_{\mathbf a\times\dot{\mathbf a}}^-=D_3^{-,+}$ of groups with orientation. Therefore, according to [17] we have the following:
$$
\begin{equation*}
\begin{array}{|c|c|c|c|c|} \hline {} & n=0 &n=1 &n=2 &n=3 \\ \hline L_n(\mathbf D^+)=L_n(D_3^{+,+}) &\mathbb Z^5 &0 &\mathbb Z/2 &(\mathbb Z/2)^4 \\ \hline L_n(\mathbf D_{\mathbf a\times\dot{\mathbf a}}^-)=L_n(D_3^{-,+}) &\mathbb Z\oplus \mathbb Z/2 &0 &\mathbb Z/2 &\mathbb Z/2 \\ \hline \end{array}
\end{equation*}
\notag
$$
Up to isomorphisms of groups with orientation, there are only two oriented groups, $[\mathbb Z/2\oplus \mathbb Z/2]^+$ and $[\mathbb Z/2\oplus \mathbb Z/2]^-$. According to [19], we have the following:
$$
\begin{equation*}
\begin{array}{|c|c|c|c|c|} \hline {} & n= 0 &n=1 &n=2 &n=3 \\ \hline L_n(\mathbb I_{\mathbf a\times\dot{\mathbf a}})=L_n([\mathbb Z/2\oplus \mathbb Z/2]^{+}) & \mathbb Z^4 & 0 & \mathbb Z/2 &(\mathbb Z/2)^3 \\ \hline L_n(\mathbb I_{\mathbf a\times\dot{\mathbf a}}^-)=L_n([\mathbb Z/2\oplus \mathbb Z/2]^{-}) &\mathbb Z/2 & 0 & \mathbb Z/2 &0 \\ \hline \end{array}
\end{equation*}
\notag
$$
Let $\sigma_1$ and $ \sigma_2$ be generators of the Wall group $\mathbb Z/2\oplus \mathbb Z/2$. The inclusion of groups $i\colon [\mathbb Z/2\oplus \mathbb Z/2]^{+}\to D_3^{\pm, +}$ given by $i(\sigma_1)=x^2$, $i_1(\sigma_2)=y$, $w(x)=\pm 1$ and $w(y)=1$ is isomorphic to the inclusion $\mathbb I_{\mathbf a\times \dot{\mathbf a}}\to \mathbf D_{\mathbf a\times \dot{\mathbf a}}^{\pm}$. According to [22], we have the following:
$$
\begin{equation*}
\begin{array}{|c|c|c|c|c|} \hline {} & n=0 &n=1 &n=2 &n=3 \\ \hline LN_n(\mathbb I_{\mathbf a\times \dot{\mathbf a}}\to \mathbf D^+) & \mathbb Z/2 & \mathbb Z/2 & \mathbb Z^2 &0 \\ \hline LN_n(\mathbb I_{\mathbf a\times \dot{\mathbf a}}\to \mathbf D^-_{\mathbf a\times \dot{\mathbf a}}) & \mathbb Z^2 & 0 & \mathbb Z/2 &\mathbb Z/2 \\ \hline \end{array}
\end{equation*}
\notag
$$
Therefore, the diagram (3.7) takes the following form: Theorem 3. (a) The homology group in the middle term of the second row of the diagram (3.8) is isomorphic to $\mathbb Z/2$. (b) An element $x \in L_3(\mathbf D^+)$ representing the nontrivial homology class of the second row homology group is not realisable by a normal map of closed manifolds. Proof. Statement (a) is proved using the isomorphism from [20], p. 247, the calculations of maps in [22], Case C, and the above results on the Wall and Browder-Livesay groups, since the second line of the diagram (3.8) can be written as
Statement (b) follows from Theorem 5 in [20]. Indeed, it implies that the elements realisable by closed submanifolds form a subgroup of index $4$ of $L_3(\mathbf D^+)$. The elements in the image of the left-hand homomorphism in the top row of (3.9) form a subgroup of index $4$ and are realisable by normal maps of closed manifolds. Therefore, $x$ is not realisable. Corollary. The element $x \in L_3(\mathbf D^+)$ is an element of the third type (Arf invariant in codimension $1$) with respect to any system of submanifolds defined by a system of subgroups beginning with the subgroup $\mathbb I_{\mathbf a\times \dot{\mathbf a}}$. Proof. Let $\pi \!\subset\! \mathbf D_{\mathbf a\times\dot{\mathbf a}}^-$ be an oriented subgroup of index $2$ in the oriented group $\mathbf D_{\mathbf a\times\dot{\mathbf a}}^-$. Then the induced inclusion
$$
\begin{equation*}
\mathbb Z/2=L_{2}(\pi) \to L_{2}(\mathbf D^-_{\mathbf a \times \dot{\mathbf a}}) =\mathbb Z/2,
\end{equation*}
\notag
$$
corresponding to the map $L_{2}(\mathbb I_{\mathbf a \times \dot{\mathbf a}}) \xrightarrow{i_{\ast}} L_{2}(\mathbf D^-_{\mathbf a \times \dot{\mathbf a}})$ in (3.8) is an isomorphism, as it preserves the Arf invariant. Theorem 4. The unique nonzero element $y \in LN_{3}(\mathbb I_{\mathbf a \times \dot{\mathbf a}} \to \mathbf D^{-}_{\mathbf a \times \dot{\mathbf a}})$ has a nontrivial torsion in $Wh_2(\mathbb Z/2 \times \mathbb Z/2)$. Proof. Consider the homomorphism
$$
\begin{equation}
LN_{3}(\mathbb I_{\mathbf a \times \dot{\mathbf a}} \to \mathbf D^{-}_{\mathbf a \times \dot{\mathbf a}}) \xrightarrow{\mathbf b \circ i_{\ast}} L_3(\mathbf D^+)
\end{equation}
\tag{3.10}
$$
in the diagram (3.8). To understand the structure of the obstruction group, we consider the simpler case $LN_3(\pi \to \pi \times \mathbb Z/2^-)$ first. Then the obstruction group corresponds to the obstruction to the splitting of the submanifold $\mathbb RP^2 \times M^5 \subset \mathbb RP^3 \times M^5$ for an orientable manifold $M^5$ with fundamental group $\pi$. This case was analysed in [19], p. 827, § 2. The splitting obstruction group is isomorphic to $L_1(\pi)$.
According to § 2, the group $LN_{3}(\mathbb I_{\mathbf a \times \dot{\mathbf a}} \to \mathbf D^{-}_{\mathbf a \times \dot{\mathbf a}})$ is isomorphic to the Wall group of the ring with anti-structure $(\mathbb Z[\mathbb I_{\mathbf a \times \dot{\mathbf a}}], \theta, u)$, where $\theta(x)=\mathbf b^{-1} x\mathbf b$ and ${u=-w(\mathbf b)\mathbf b^{-2}}$, since $\mathbf b \in \mathbf D^{-}_{\mathbf a \times \dot{\mathbf a}}\setminus \mathbb I_{\mathbf a \times \dot{\mathbf a}}$. Therefore, $u=\mathbf a\dot{\mathbf a}$ and the anti-automorphism $\theta$ acts on the generators according to the formulae $\theta(\mathbf a)=\dot{\mathbf a}$, $\theta(\dot{\mathbf a})=\mathbf a$ and extends to the whole group ring by linearity. In the case under consideration $\theta(\mathbf a \dot{\mathbf a})=\mathbf a \dot{\mathbf a} = (\mathbf a \dot{\mathbf a})^{-1}$, so the anti-automorphism $\theta$ is an involution. Note that $\theta$ coincides on $\mathbb I_{\mathbf a \times \dot{\mathbf a}}$ with the outer conjugation by the element $\mathbf b \in \mathbf D \setminus \mathbb I_{\mathbf a \times \dot{\mathbf a}}$. The nonzero element $y$ of the Browder-Livesay group $LN_{3}(\mathbb I_{\mathbf a \times \dot{\mathbf a}} \to \mathbf D^{-}_{\mathbf a \times \dot{\mathbf a}})$ is defined by the matrix of an automorphism of a quadratic form over the ring with anti-structure $(\mathbb Z[\mathbb I_{\mathbf a \times \dot{\mathbf a}}], \theta, u) =(\mathbb Z[\mathbb Z/2\oplus \mathbb Z/2], \theta, u)$, where $\mathbf a$ and $\dot{\mathbf a}$ are the generators of the $\mathbb Z/2$-summands. This means that the element $\mathbf b^2 = \mathbf a \dot{\mathbf a} \in \mathbb Z[\mathbb I_{\mathbf a \times \dot{\mathbf a}}]$ defines the generalised Hermitian sign for a quadratic form over the ring $\mathbb Z[\mathbb I_{\mathbf a \times \dot{\mathbf a}}]$ with anti-structure (see [3], p. 160). The automorphism matrix preserves the $\mathbf a \dot{\mathbf a}$-symmetric form $P$:
$$
\begin{equation}
\left[\begin{matrix} 0 & \mathbf a\dot{\mathbf a} \\ 1 & 0 \end{matrix}\right\rfloor.
\end{equation}
\tag{3.11}
$$
The matrix $A$ (with determinant $-\mathbf a \dot{\mathbf a}$) has the form
$$
\begin{equation}
\left[\begin{matrix} \mathbf a + \dot{\mathbf a} & \mathbf a + \dot{\mathbf a} + \mathbf a\dot{\mathbf a} \\ \dot{\mathbf a} + \mathbf a + 1 & 1 + \mathbf a + \dot{\mathbf a} + \mathbf a \dot{\mathbf a} \end{matrix}\right\rfloor.
\end{equation}
\tag{3.12}
$$
To calculate the Hasse-Witt invariant $LN_3(\mathbb I_{\mathbf a \times \dot{\mathbf a}} \to \mathbf D^-_{\mathbf a \times \dot{\mathbf a}}) \to Wh_2(\mathbb Z/2 \times \mathbb Z/2)/(x + x^{\ast})$ consider the relation which we complete to the relation of unimodular matrices as a result of stabilisation:
$$
\begin{equation}
\overline{A}\,\overline{P}\, \overline{A}^{\,\ast}=\overline{P}^{\,\ast}.
\end{equation}
\tag{3.14}
$$
In this formula $\overline{A}$ (and similarly for other matrices) is a high-dimensional unimodular matrix obtained as a result of stabilising $A$ and replacing $+1$ in one of the stabilising diagonal elements by the elementary element of the group ring that inverts the determinant of $A$. (Since all matrices under consideration have size $2\times2$, we use the convention that this diagonal element has index $3$.) The matrix $A^{\ast}$ (and similarly for other matrices) is the transposed matrix $A$, since in our case the anti-automorphism $\alpha$ is identical on the group ring $\mathbb Z[\mathbb Z/2 \times \mathbb Z /2]$. The elements $x,x^{\ast} \in Wh_2(\mathbb Z/2 \times \mathbb Z/2)$ are related by transposition, and the group operation corresponding to taking the disjoint union of classes of stable matrices is written additively.
Next we write the factorizations into elementary matrices for $\overline{A}$ and $\overline{P}$ (and the corresponding conjugate factorizations for $\overline{A}^{\,\ast}$ and $\overline{P}^{\,\ast}$). The calculations are very simple; it is sufficient to carry them out over the ring $\mathbb Z/2[\mathbb Z/2 \times \mathbb Z/2]$. We substitute the product into (3.14) and calculate the symbol in $K_2(\mathbb Z/2[\mathbb Z/2 \times \mathbb Z/2])$ to obtain $\langle \mathbf a + \dot{\mathbf a} + \mathbf a\dot{\mathbf a};\mathbf a \dot{\mathbf a}\rangle$. Indeed, over $\mathbb Z/2[\mathbb Z/2 \times \mathbb Z/2]$ the relation (3.13) is reduced to the equality $CDC^{-1}D^{- 1}=E$ by elementary transformations, where
$$
\begin{equation}
C=\left[ \begin{matrix} \mathbf a\dot{\mathbf a} & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & \mathbf a\dot{\mathbf a} \end{matrix} \right\rfloor
\end{equation}
\tag{3.15}
$$
and
$$
\begin{equation}
D=\left[ \begin{matrix} \mathbf a + \dot{\mathbf a} + \mathbf a\dot{\mathbf a} & 0 & 0\\ 0 & \dot{\mathbf a} + \mathbf a + 1 & 0 \\ 0 & 0 & \mathbf a\dot{\mathbf a} \end{matrix} \right\rfloor.
\end{equation}
\tag{3.16}
$$
It was proved in [23] that the element of $K_2(\mathbb Z/2 \times \mathbb Z/2)$ thus constructed is not equal to zero and cannot be expressed in terms of the symbols $\langle \pm p_1;\pm p_2 \rangle $, where $p_1,p_2 \in \mathbb Z/2 \times \mathbb Z/2$. The group $Wh_2(\mathbb Z/2 \times \mathbb Z/2)$ was calculated in [24]. In general, Hasse-Witt torsion depends on the factorization into elementary matrices. However, in the case under consideration there is no ambiguity since all elements in $Wh_{2}(\mathbb Z/2 \times \mathbb Z/2)$ have order $2$. § 4. DiscussionThe geometric meaning of the result obtained in Theorem 4 requires further examination. In fact, the image $z=\mathbf b \circ i_{\ast}(y) \in L_3(\mathbf D^+)$ of the nontrivial element ${y \in LN_{3}(\mathbb I_{\mathbf a \times \dot{\mathbf a}} \to \mathbf D^{-}_{\mathbf a \times \dot{\mathbf a}})=\mathbb Z/2}$ is nonzero, but lies in the image of the homomorphism $L_ {3}(\mathbb I_{\mathbf a \times \dot{\mathbf a}}) \xrightarrow{i_{\ast}} L_3(\mathbf D^+)$. The element $y$ also acts on the identical homotopy triangulation of $M^n$ for $n \equiv 3 \pmod{4}$ (as an element of the group of obstructions to splitting along $N_{\mathbf a \times \dot{\mathbf a}}^{n-1} \times I \subset M^n \times I$) and defines a homotopy triangulation $\nu(y) \in hT(M^{n} \times \{1\}) $. Consider the pair
$$
\begin{equation}
(M^n,N^{n-1}_{\mathbf a \times \dot{\mathbf a}}) = (M^{n-2}_1 \times D^2, N_{1}^{n-3} \times D^2),
\end{equation}
\tag{4.1}
$$
where $(M^{n-2}_1,N_{1}^{n-3})$ is a pair of manifolds that stabilises to a pair (3.3) of manifolds with boundary by multiplication by a $2$-disc. The submanifold $N^{n-3}_1 \times D^2 \subset M^{n-2}_1\times D^2$ represents a $2$-parameter family of submanifolds over the disc. Arguing as in [25], we can show that the homotopy triangulation of $M^{n-2}_1 \times D^2 \times \{1\}$ which restricts to a $\mathrm{PL}$-homeomorphism on the boundary $M^{n- 2}_1 \times S^1 \times \{1\}$ must have critical points in the $2$-parameter family of the inverse images of the submanifold $N^{n-3}_1 \times D^2 \times \{1\}$. Therefore, the element $\nu(y) \in hT(M^{n} \times \{1\})$ is nonzero. However, this cannot occur if the image $z=\mathbf b \circ i_{\ast}(y) \in L_3(\mathbf D^+)$ is zero, since otherwise the homotopy smoothing $\nu(z) \in hT(N^{n-1}_{\mathbf a \times \dot{\mathbf a}} \times \{1\})$ would be identical and the inverse image $N^{n-1}_{\mathbf a \times \dot{\mathbf a}} = N_1^{n-3} \times D^2 \times \{1\}$ would be fibred over $D^2$. On the other hand, according to Theorem 4, the inverse image of the manifold under consideration would necessarily contain critical points of the projection onto a disc. This result was presented at the seminar dedicated to the 85th anniversary of Professor A. V. Chernavsky. 
Citation in format AMSBIB
\Bibitem{AkhMur23} |
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