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An elementary approach to local combinatorial formulae for the Euler class of a PL spherical fibre bundle G. Yu. Panina St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 3, DOI: https://doi.org/10.4213/sm9737 
Bibliographic databases:
    § 1. IntroductionA combinatorial formula for a characteristic class is an algorithm whose output is a simplicial cochain representing this class. It is assumed that the base is a triangulated manifold or, more generally, a simplicial complex. There exists a (relatively simple) combinatorial formula for tangential Stiefel-Whitney classes, and an interesting long story of finding combinatorial formulae for the Pontryagin classes of triangulated manifolds, begun by Gabrielov, Gelfand, and Losik and completed by Gaifullin. A local combinatorial formula for a circle bundle appeared independently in Mnëv-Sharygin [1] and Igusa [2]. This formula is discussed in § 6, where we show that it fits in the general case of bundles with fibre $S^n$. Recently Mnëv presented a local combinatorial formula for the Euler class of a fibre-oriented $n$-dimensional PL spherical fibre bundle in terms of combinatorial Hodge-theoretic twisting cochains in Hirsch’s homology model [3]. Our paper introduces an elementary approach (that is, with minimal technique required) to local combinatorial formulae for the Euler class. Let $S^n\to E \xrightarrow{\pi} B$ be a locally trivial fibre bundle whose fibre is the oriented sphere $S^n$. We assume that $B$ and $E$ are finite simplicial complexes and $\pi$ is a simplicial map, that is, $\pi$ maps simplices to simplices, linearly on each simplex. We also assume that $\pi^{-1}(x)$ is a combinatorial sphere for all $x\in B$. This means, in particular, that the dual cell decomposition is well defined. A local combinatorial formula for the rational Euler class $e(E\xrightarrow{\pi}B)$ is an algorithm which assigns a rational number to every oriented $(n+1)$-dimensional simplex $\sigma^{n+1} $ in the base. The output of the algorithm is a cochain $\mathcal{E}$ representing the Euler class. The value of this cochain at a simplex $\sigma^{n+1}$ depends only on the restriction of the bundle $\pi^{-1}(\sigma^{n+1}) \to \sigma^{n+1}$. We present two formulae. The first (see § 3) coincides with that due to Mnëv. The second (see § 4) presents a shortcut on the last step of the algorithm. One can alter these algorithms even further (we explain how in § 6) and thus obtain various different local combinatorial formulae. In § 6 we explain that for $n=1$ the second formula (§ 4) coincides with the Igusa-Mnëv-Sharygin combinatorial formula for a circle bundle; see [2] and [1]. Here is the leading idea in short (it traces back to Kazarian’s multisections [4], also used in [5]). The following way to compute the Euler class exists. Proposition 1 (see [6] and [7]). Let $E \to B$ be a fibre-oriented spherical bundle over a triangulated base $B$. Assume that a continuous partial section $s$ of it is defined on the $n$-skeleton of $B$. Fix an orientation on each $(n+1)$-dimensional simplex $\sigma^{n+1}\in B$, and set $\mathcal{E}(s,\sigma^{n+1})$ to be the degree of the map $s\colon\partial \sigma^{n+1} \to S^n$. Then the integer cochain $\mathcal{E}(s,\sigma^{n+1})$ is a cocycle and represents the Euler class $e(E\xrightarrow{\pi}B)$. For bundles arising from vector bundles of rank $n+1$ over the base $B$ this proposition was proved in [6]. The general case was treated in Fomenko and Fuchs [7], § 23.5. Next, if we have several partial sections $s_1,\dots,s_r$, set $\mathcal{E}_i(\sigma^{n+1})=\mathcal{E}(s_i,\sigma^{n+1})$. The average $\frac{1}{r}\sum_i \mathcal{E}_i(\sigma^{n+1})$ is clearly a rational cochain representing the Euler class. Furthermore, if we have partial sections $s_1,\dots,s_M$ and $s_1',\dots,s_N'$, then the average of cochains
$$
\begin{equation*}
\frac{1}{M-N}\biggl(\sum_i \mathcal{E}_i-\sum_j \mathcal{E}_j'\biggr)
\end{equation*}
\notag
$$
also represents the rational Euler class, provided that $M \neq N$. The combinatorics of the triangulation of the bundle suggests a way to fix a tuple of partial sections. We create these sections stepwise, starting from the $0$-skeleton and extending sections further. Our toolbox contains harmonic chains and winding numbers. Harmonic chains appear as an answer to the following question: given a regular cell complex $\Gamma$ which is a combinatorial sphere $S^n$ and a closed chain $a\in C_k(\Gamma, \mathbb{Q})$, describe a canonical way to find $b\in C_{k+1}(\Gamma, \mathbb{Q})$ such that $\partial b=a$. Reader’s guide. For a shortcut to the first main result (§ 3) which coincides with Mnëv’s formula, it suffices to read § 2.1, and Definition 2. For a shortcut to the second main result (§ 4) one also needs Definitions 3 and 4. AcknowledgementsThe author is grateful to Alexander Gaifullin, Maxim Kazarian, Nikolai Mnëv, Ivan Panin and an anonymous referee for their useful comments. § 2. ToolboxWe denote the associated simplicial complexes of the base and total space by $K_B$ and $K_E$, respectively. 2.1. The combinatorics of the preimage of a simplexFix an oriented simplex $\sigma^{n+1} \in K_B$ in the base. We denote its vertices by $v_0,\dots,v_{n+1}$ so that their order agrees with the orientation of $\sigma^{n+1}$. It is instructive to think that the vertices are coloured, say, $v_0$ is red, $v_1$ is blue, $v_2$ is green and so on. Let us analyse $(2n+1)$-dimensional simplices lying in the preimage $\pi^{-1}(\sigma^{n+1})$. Each such simplex $\Delta^{2n+1}\in \pi^{-1}(\sigma^{n+1})$ has $2n+2$ vertices coloured in accordance with the colours of their projections. Clearly, for each $i$ there is a vertex of $\Delta^{2n+1}$ coloured with colour $i$. We say that $\Delta^{2n+1} \in \mathcal{A}_i(\sigma^{n+1})$ if
$$
\begin{equation*}
|{\operatorname{Vert}(\Delta^{2n+1})\cap \pi^{-1}(v_i)}|=n+1,
\end{equation*}
\notag
$$
that is, $\Delta^{2n+1}$ has $n+1$ vertices of colour $i$, whereas other colours appear only once each. Note that the classes $\mathcal{A}_i$ do not exhaust all simplices in the preimage. Now we analyse $\pi^{-1}(v_i)$. It is a triangulated oriented sphere $S^n$. Maximal simplices of the triangulation correspond to simplices in $\mathcal{A}_i(\sigma^{n+1})$. We denote the dual of the triangulation by $\Gamma_i$. It is a regular cell complex, and we imagine it to be coloured with colour $i$. Now take a point $x$ lying strictly inside an edge, for instance, $v_0v_1$. Its preimage $\pi^{-1}(x)$ (independently on the choice of $x$) is a tiling of $S^n$, whose dual complex is the superposition of $\Gamma_0$ and $\Gamma_1$, that is, the subtiling generated by $\Gamma_0$ and $\Gamma_1$ (see Figure 1). Generalizing, we conclude the following. Proposition 2. 1. For any point $x\in \sigma^{n+1}$ the combinatorics of the preimage $\pi^{-1}(x)$ depends only on the preimage of the face $F$ of $\sigma^{n+1}$ that contains $x$ as an interior point. 2. Let $\{v_{i_1},\dots,v_{i_k}\}$ be the set of vertices of $F$. Then the dual complex $\Gamma_{i_1,\dots,i_k}$ of the preimage $\pi^{-1}(x)$ is the superposition of the $\Gamma_i$ for $i$ ranging over the set $\{i_1,\dots,i_k\}$. 3. In particular, an interior point of $\sigma^{n+1}$, corresponds to the superposition of the $ n+2 $ coloured complexes $\Gamma_0,\dots,\Gamma_{n+2}$. Eliminating one (or several) of these complexes corresponds to moving $x$ to some face. Some vertices of $\Gamma_{i_1,\dots,i_k}$ are vertices of $\Gamma_i$ for some $i$. We say that these vertices are coloured with colour $i$. Other (uncoloured vertices) arise as the intersections of cells in $\Gamma_i$ for different $i$ (see Figure 1 for an illustration). 2.2. $s$-chains, patches and $s$-patchesAssume that a regular cell complex $\Gamma$ is a combinatorial sphere $S^n$. Let $C_{k+1}(\Gamma, \mathbb{Q})$ denote the rational chain group of $\Gamma$. Let $k<n$, and let $a\in C_k(\Gamma, \mathbb{Q})$ be a closed $k$-dimensional chain. A chain $c\in C_{k}(\Gamma, \mathbb{Q})$ is called a patch of $a$ if $\partial c=a$. Assume that there is a map from the oriented $k$-sphere to the $k$-skeleton It yields a map
$$
\begin{equation*}
S^k \to \operatorname{Skel}_k(\Gamma)/\operatorname{Skel}_{k-1}(\Gamma).
\end{equation*}
\notag
$$
The quotient space $\operatorname{Skel}_k(\Gamma)/\operatorname{Skel}_{k-1}(\Gamma)$ is a wedge of $k$-spheres, where the spheres correspond to $k$-cells of $\Gamma$. We conclude that $s$ gives rise to a closed chain $|s|\in C_k(\Gamma, \mathbb{Z})$ which represents the pushforward of the fundamental class of $S^k$. Now assume that there is a map
$$
\begin{equation*}
\widetilde{s}\colon D^{k+1}\to \operatorname{Skel}_{k+1}(\Gamma)
\end{equation*}
\notag
$$
such that the boundary of $D^{k+1}$ is mapped to $\operatorname{Skel}_k(\Gamma)$. In a similar way, the map $\widetilde{s}$ gives rise to a $(k+1)$-chain $|\widetilde{s}|\in C_{k+1}(\Gamma, \mathbb{Z}).$ Indeed, in this case $s$ yields a map
$$
\begin{equation*}
D^{k+1}/\partial D^{k+1}\to \operatorname{Skel}_{k+1}(\Gamma)/\operatorname{Skel}_{k}(\Gamma).
\end{equation*}
\notag
$$
If a chain $a$ equals $|s|$ for some $s$, then we say that $s$ supports $a$. In this case we call $a$ an $s$-chain. Let $a=|s|$ for some $s\colon S^k\to\operatorname{Skel}_k(\Gamma)$. Then a patch $c$ is called an $s$-patch of $a$ if $c=|\widetilde{s}|$ for an extension $\widetilde{s}\colon D^{k+1}\to \operatorname{Skel}_{k+1}(\Gamma)$ of the map $s$. Example 1. For $k=0$ let $a$ be an ordered pair of vertices $V_0$, $V_1$ interpreted as an $s$-chain. Each path on $\operatorname{Skel}_1$ connecting these vertices is an $s$-patch of $a$. However, not every patch of $a$ is an $s$-patch. For instance, the chain that averages two different paths is not an $s$-patch. Lemma 2. 1. If $c=|\widetilde{s}|$ is a chain supported by $\widetilde{s}\colon D^{k+1} \to \operatorname{Skel}_{k+1}(\Gamma)$ and $a$ is a chain supported by then $\partial c =a$. 2. Each rational closed $k$-chain is a rational linear combination of $s$-chains supported by some $s_i\colon S^{k} \to \operatorname{Skel}_{k}(\Gamma)$. 3. Let $k<n$. Let $a=|s|$ be a chain supported by some map $s\colon S^{k} \to \operatorname{Skel}_{k}$. Let $b$ be a rational chain such that $\partial b=a$. Then $b$ is an average of chains supported by some extensions of $s$. More precisely, there exist
$$
\begin{equation*}
\widetilde{s}_i\colon D^{k+1} \to \operatorname{Skel}_{k+1}(\Gamma), \qquad i=1,\dots,M,
\end{equation*}
\notag
$$
and such that the restrictions of all $\widetilde{s}_i$ and $\widetilde{s}^{\,\prime}_j$ to $\partial D^{k+1}$ equal $s$, and Proof. 1. This is trivial.
2. Let $k<n$. Let $\tau$ range over all the $(k+1)$-cells of the complex $\Gamma$. Then $\{\partial \tau \}$ generate the space of closed $k$-chains. Indeed, for any closed chain $p \in C_k(\Gamma)$, for some $r\in C_{k+1}(\Gamma)$ we have $p=\partial r=\partial (\sum a_i \tau_i)= \sum a_i\, \partial \tau_i$. For $k=n$ the proof is even simpler: all closed chains differ by a multiple. 3. Let $k<n-1$. Take any extension of $s$ to a map $\widetilde{s}$ from the disc $D^{k+1}$ to the $k$-skeleton, so that $\widetilde{s}_{|_{\partial D}}=s$. Then $b-|\widetilde{s}|$ is a closed rational $(k+1)$-chain and therefore a rational linear combination of the $ \partial \tau_i$ where $\tau_i$ ranges over $(k+2)$-cells. Let us prove that for each $(k+2)$-cell $\tau$ the chain $\partial (\tau)$ is a difference of some $|\widetilde{s}|$ and $|\widetilde{s}^{\,\prime}|$ such that the restrictions of $\widetilde{s}$ and $\widetilde{s}^{\,\prime}$ to the boundary of the disc equal $s$. Fix any subcomplex of $\partial \tau$ homeomorphic to $S^{k-1}$ and call it the Equator. Then the Equator partitions $\partial \tau$ into two hemispheres $B_1$ and $B_2$. More precisely, denote by $B_1$ and $B_2$ the chains such that $B_1+B_2=\partial \tau$. Consider a map of the cylinder $S^k\times[0,1]$ to the $k$-skeleton such that the restriction of this map to $S^k \times 0$ equals $s$, and its restriction to $S^k \times 1$ is a homeomorphism onto the Equator. Such a map exists since any two maps from a $k$-sphere to the $(k+1)$-skeleton are homotopic. We treat the cylinder as a disc $D^{k+1}$ from which a smaller $(k+1)$-disc is deleted. Extend the map to the whole of $D^{k+1}$ in two ways, using $B_1$ and $B_2$. Denote the extensions by $\widetilde{s}$ and $\widetilde{s}^{\,\prime}$. Now we have $\partial \tau= |\widetilde{s}|-|\widetilde{s}^{\,\prime}|$. For $k=n-1$ the proof is simpler again. In this case two rational patches of $a$ differ by a rational multiple of the chain representing the fundamental class of $S^n$. Each integer patch of $a$ is an $s$-patch. Lemma 2 is proved. In Lemma 2 we call sections $s_i$ ($s'_j$) positive (respectively, negative) entries of the averaging. 2.3. Harmonic extensionHarmonic extension provides a way to create a canonical inverse of the boundary operator $\partial$. We borrow the following basic knowledge about harmonic chains from [8]. Given a regular cell complex $K$, we assume that $C_{k}(K, \mathbb{Q})$ is endowed with a scalar product such that the set of all $k$-cells is an orthonormal basis. Let $\partial$ denote the standard boundary operator, and let (the coboundary) $\partial^*$ be its adjoint operator with respect to the scalar product. Definition 1. A chain $a\in C_{k}(K, \mathbb{Q})$ is harmonic if its boundary and coboundary vanish, that is, $\partial a=0$ and $\partial^* a=0$. Equivalently, $a$ is harmonic if it belongs to the kernel of the discrete Laplacian It is known (and is easy to prove) that each homology class is uniquely representable by a harmonic chain. If $\Delta\colon C_k(K, \mathbb{Q})\to C_k(K,\mathbb{Q})$ is invertible (that is, $H_k(K,\mathbb{Q})$ vanishes), then there exists an inverse operator $\Delta^{-1}\colon C_k(K,\mathbb{Q})\to C_k(K,\mathbb{Q})$, which is called the Greens operator. Assume that we have a regular cell complex $\Gamma$ which is a combinatorial sphere $S^n$. Take the cone over $\operatorname{Skel}_k(\Gamma)$. It is the cell complex defined by $\operatorname{Cone}(\operatorname{Skel}_k(\Gamma)) =\operatorname{Skel}_k(\Gamma)* O$, where $*$ denotes the join operation, and $O$ is the apex of the cone. We paste the cone to the combinatorial sphere $\Gamma$ via the natural inclusion map $\operatorname{Skel}_k \subset \operatorname{Cone}(\operatorname{Skel}_k)$. We also paste a disc $D^{n+1}$ to $\Gamma$ along a homeomorphism between $\Gamma$ and $\partial D^{n+1}$. We obtain a cell complex and denote it by $\operatorname{PC}_k(\Gamma)$. Let $k<n-1$, and let $a\in C_k(\Gamma, \mathbb{Q})$ be a closed chain. Consider the (uniquely defined) $(k+1)$-chain $\overline{a}\in C_{k+1}(\operatorname{Cone}(\operatorname{Skel}_k(\Gamma))$ such that $\partial \overline{a}=-a$. In simple words, $\overline{a}$ assigns the coefficient $-a(\sigma^k)$ to the cell $\sigma^k*O$. Now we extend the chain by keeping its coefficients at cells of type $\sigma^k*O$ and assigning some coefficients to $(k+1)$-cells of $\Gamma$. The chain $\overline{a}$ extends to a closed chain on $\operatorname{PC}_k(\Gamma)$ in many ways. All closed extensions represent one and the same homology class of $\operatorname{PC}_k(\Gamma)$. We denote the unique harmonic (with respect to $\operatorname{PC}_k(\Gamma)$) representative of this homology class by $h(a)$. We decompose $h(a)$ as $h(a)=\overline{a}+\mathcal{H}(a)$. We have necessarily $\mathcal{H}(a)\in C_{k+1}(\Gamma)$. Since $\Delta h(a)=0$, we have $\Delta \overline{a} +\Delta \mathcal{H}(a)=0$, and therefore $\Delta \mathcal{H}(a) =\partial^* a$ (here $\partial^*$ relates to the complex $\Gamma$, rather than to $\operatorname{PC}_k(\Gamma)$). Since $h(a)$ is a closed chain, $\partial h(a)= 0$, we have $\partial \overline{a}+\partial \mathcal{H}(a)=0$. Hence $\partial \mathcal{H}(a) =a$. Thus we arrive at the following definition. Definition 2. Let $k\leqslant n-1$, and let $a\in C_k(\Gamma, \mathbb{Q})$ be a closed $k$-dimensional chain. Then the harmonic extension of $a$ is the chain We have explained the (already known: see [8], for instance) construction of a harmonic chain with prescribed boundary. Lemma 3. 1. The harmonic extension is uniquely defined by the conditions 2. The harmonic extension of an $s$-chain $a$ can be represented as where $M \neq N$ are some natural numbers, $\partial c_i=\partial c_j' =a$, and each of the chains $c_i$ and $c_j'$ is an $s$-patch of $a$.Here are some important observations. Remark 1. The harmonic extension depends on the complex $\Gamma$, rather than on the chain $a$ alone. For instance, if one subdivides $\Gamma$ but preserves $a$, then the harmonic extension changes. Remark 2. Although the harmonic extension makes sense for all $k\leqslant n-1$, for $(n-1)$-dimensional chains we will also use simpler technique to invert the boundary operator (see below). Remark 3. Producing a canonical inversion of the boundary operator $\partial$ is a useful trick. For example, let us mention so-called Euclidean minimal chains used in [9] In short, the idea is that among all chains with prescribed boundary one takes the chain with the least Euclidean norm. 2.4. The winding number. A dimension $n$ extensionThroughout this section $\Gamma$ is a regular cell complex which is a combinatorial sphere $S^n$. Let $\Sigma \in C_{n-1}(\Gamma, \mathbb{Q})$ be a closed chain, and let $x,y\in S^n\setminus \operatorname{Skel}_{n-1}(\Gamma)$ be some points. Definition 3. The winding number $\mathcal{W}(x,y,\Sigma)$ is defined as the algebraic number of intersections of $\Sigma$ and $[x,y]$, where $[x,y]$ is a smooth (or a piecewise-linear) oriented path from $x$ to $y$ which intersects the $(n-1)$-skeleton of $\Gamma$ transversally. If the endpoints $x$ and $y$ are fixed, then the winding number is independent on the choice of the path $[x,y]$. Borrowing notation from intersection theory and treating $[x,y]$ as a $1$-chain, we write Example 2. 1. If $\Sigma$ arises as a pushforward of the fundamental class of some $\psi$: ${S^1\to S^{2}}$, and one regards $S^2$ as $\mathbb{R}^2$ compactified by the point $\infty=y$, then $\mathcal{W}(x,y,\Sigma)$ is exactly the winding number of $\Sigma$ around the point $x$ (see Figure 2 for an illustration). 2. If $\Sigma$ is an averaging of cycles represented by submanifolds, then $\mathcal{W}(x,y,\Sigma)$ is the average of the winding numbers. Lemma 4. Let $\Sigma\in C_{n-1}(\Gamma, \mathbb{Q})$ be a closed chain. 1. For each cell $\sigma_1^n\in \Gamma$ there exists a unique chain $\mathcal{C}=\mathcal{C}_{\Sigma,\sigma^n_1}\in C_{n}(\Gamma, \mathbb{Q})$ whose coefficient $\mathcal{C}(\sigma_1^n)$ at $\sigma_1^n$ vanishes, and $\partial \mathcal{C}=\Sigma$. 2. The coefficients of other cells in $\mathcal{C}$ are the winding numbers, that is, for all $\sigma_2^n \in \Gamma$, where $x\in \sigma_1$ and $y \in \sigma_2$.3. If $\Sigma$ is an $s$-chain, then $\mathcal{C}$ is an $s$-patch. Remark 4. Let $\Sigma \in C_{n-1}(\Gamma, \mathbb{Q})$. To compute its harmonic extension $\mathcal{H}(\Sigma)$ do the following: using Lemma 4, compute any $b$ with $\partial b=\Sigma$. The harmonic extension equals $b+c$, where $c$ is a constant chain defined by the condition that the sum of the coefficients of all $n$-cells in $b+c$ vanishes. Indeed, in this case $\partial^*(b+c)=0$. Since $\partial (b+c)=\partial b=\Sigma$, by Lemma 3 we have the harmonic extension. Definition 4. Let $\Sigma \in C_{n-1}(\Gamma, \mathbb{Q})$ be a closed chain, let $y\in S^n\setminus \operatorname{Skel}_{n-1}(\Gamma)$ be a point and $V$ be a vertex of $\Gamma$. Also assume that $x$ is incident to exactly $n+1$ top-dimensional cells of $\Gamma$. The winding number, or the dimension $n$ extension is defined as the average where the sum ranges over points $x_i$, one point from each $n$-cell incident to $V$. From now on, for fixed $V$ and $\Sigma$ we treat $\mathcal{W}(V,\Sigma)$ as an $n$-chain, the linear combination of $n$-cells such that the coefficient of a cell equals the winding number of an inner point of the cell.Lemma 5. 1. $\partial \mathcal{W}(V,y,\Sigma)=\Sigma$. 2. The operator $ \mathcal{W}(V,y,\Sigma)$ is linear in $\Sigma$. 3. If $\Sigma$ is an $s$-chain, then $ \mathcal{W}(V,\Sigma)$ is the average of its $s$-patches. § 3. The first formulaConsider a fibrewise oriented triangulated piecewise linear fibre bundle. 1. Fix a simplex $\sigma^{n+1}$ in the base $B$. Denote its vertices by $v_0,\dots,v_{n+1}$ and consider the complex $\Gamma_{01\dots n+1}=\Gamma_{01\dots n+1}(\sigma^{n+1})$. Take all existing $(n+2)$-tuples $V_0,\dots,V_{n+1}$ of its vertices, one vertex of each colour, so that $V_i\in \pi^{-1}(v_i)$ is a vertex of $\Gamma_i$. For each of tuple do the following. 2. For each oriented edge $V_iV_j$ take the $0$-chain $-V_i+V_j\in C_0(\Gamma_{ij}, \mathbb{Q})$ and compute its harmonic extension relative to the complex $\Gamma_{ij}$: 3. For each oriented triple of vertices $V_iV_jV_k$ compute the harmonic extension relative to the complex $\Gamma_{ijk}$:
$$
\begin{equation*}
\Sigma_{ijk}=\mathcal{H}( \Sigma_{jk}-\Sigma_{ik}+\Sigma_{ij}).
\end{equation*}
\notag
$$
4. Proceed in the same way for all $k\leqslant n$. Consider a $(k+1)$-tuple of vertices of the simplex $\sigma^{n+1}$, say, $v_0,\dots,v_{k}$. Assuming that the chains $\Sigma_{0\dots \widehat{i}\dots k}$ were defined at the previous step, compute the harmonic extension relative to the complex $\Gamma_{0\dots k}$:
$$
\begin{equation*}
\Sigma_{0\dots k}=\mathcal{H}\biggl(\sum_{i=0}^{k}(-1)^i\Sigma_{0\dots \widehat{i}\dots k}\biggr)\in C_k(\Gamma_{0\dots k}, \mathbb{Q}).
\end{equation*}
\notag
$$
5. Eventually, we arrive at a set of chains $\Sigma_{012\dots \widehat{i}\dots n+1}$. Then
$$
\begin{equation*}
\sum_{i=0}^{n+1}(-1)^i\Sigma_{012\dots \widehat{i}\dots n+1}\in C_{n}(\Gamma_{01\dots n+1}, \mathbb{Q})
\end{equation*}
\notag
$$
is a closed $n$-chain, and thus it represents a rational number $\mathfrak{e}=\mathfrak{e}(\sigma^{n+1}).$ Theorem 1. 1. The number $\mathfrak{e}(\sigma^{n+1})$ depends on the combinatorics of $\pi^{-1}(\sigma^{n+1})$ and the choice of $V_1,\dots,V_n$ only. 2. A change of orientation of $\sigma^{n+1}$ changes the sign of $\mathfrak{e}(V_0,\dots,V_{n+1};\Gamma_{01\dots n+1})$. 3. The function assigning to each $(n+1)$-dimensional simplex $\sigma^{n+1}$ the average of the numbers $\mathfrak{e}(V_0,\dots,V_{n+1};\Gamma_{01\dots n+1})$ over all $(n+2)$-tuples of vertices $(V_0,\dots,V_{n+1})$, one vertex from each colour, is a closed cochain that represents the rational Euler class of the fibre bundle.Remark 5. A detailed analysis (which is however omitted here for brevity and to keep the elementary level of our presentation) shows that the cochain in Theorem 1 coincides with the cochain obtained by Mnëv in [3], Theorem 8. § 4. The second formula: using a shortcut at the last step1. Fix a simplex $\sigma^{n+1}$ in the base $B$. Denote its vertices by $v_0,\dots,v_{n+1}$ and consider the complex $\Gamma_{01\dots n+1}=\Gamma_{01\dots n+1}(\sigma^{n+1})$. Take all the existing $(n+2)$-tuples $V_0,\dots,V_{n+1}$ of its vertices, one vertex of each colour, so that $V_i\in \pi^{-1}(v_i)$ is a vertex of $\Gamma_i$. Do the following for each tuple. 2. For each oriented edge $V_iV_j$ consider the $0$-chain $-V_i+V_j\in C_0(\Gamma_{ij}, \mathbb{Q})$ and compute its harmonic extension relative to the complex $\Gamma_{ij}$: 3. For each oriented triple of vertices $V_iV_jV_k$ compute the harmonic extension relative to the complex $\Gamma_{ijk}$:
$$
\begin{equation*}
\Sigma_{ijk}=\mathcal{H}( \Sigma_{jk}-\Sigma_{ik}+\Sigma_{ij}).
\end{equation*}
\notag
$$
4. Proceed in the same way for all $k<n$ as in § 3. Consider a $(k+1)$-tuple of vertices of the simplex $\sigma^{n+1}$, say, $v_0,\dots,v_{k}$. Assuming that the chains $\Sigma_{0\dots \widehat{i}\dots k}$ were defined at the previous step, compute the harmonic extension relative to the complex $\Gamma_{0\dots k}$:
$$
\begin{equation*}
\Sigma_{0\dots k}=\mathcal{H}\Big(\sum_{i=0}^{k}(-1)^i\Sigma_{0\dots \widehat{i}\dots k}\Big)\in C_k(\Gamma_{0\dots k}, \mathbb{Q}).
\end{equation*}
\notag
$$
5. The last step is different: assuming that the $\Sigma_{012\dots \widehat{i}\dots \widehat{j}\dots n+1}$ are already known, set
$$
\begin{equation*}
\Sigma_{\,\widehat{i}}=\Sigma_{012\dots \widehat{i}\dots n+1}=\sum_{j\neq i} (-1)^{j+(\operatorname{sign}(j-i)+1)/2} \Sigma_{012\dots \widehat{i}\dots \widehat{j}\dots n+1}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathfrak{E}_i=\frac{1}{n+1}\sum_{k\neq i}\mathcal{W}(V_i,V_k,\Sigma_{\,\widehat{i}})\in C_n(\Gamma_{0\dots n+1}, \mathbb{Q}).
\end{equation*}
\notag
$$
Theorem 2. 1. The chain $\mathfrak{E}_i$ does not depend on $i$. 2. The chain $\mathfrak{E}_i$ is closed, and thus it equals the fundamental class multiplied by a rational number $\mathfrak{e}=\mathfrak{e}(V_0, \dots,V_{n+1};\Gamma_{01\dots n+1})$, which depends on the combinatorics of $\pi^{-1}(\sigma^{n+1})$ and the choice of $V_1,\dots,V_n$ only. 3. A change of orientation of $\sigma^{n+1}$ changes the sign of $\mathfrak{e}(V_0,\dots,V_{n+1};\Gamma_{01\dots n+1})$. 4. The function assigning to each $(n+1)$-dimensional simplex the $\sigma^{n+1}$ the average of the numbers $\mathfrak{e}(V_0,\dots,V_{n+1};\Gamma_{01\dots n+1})$ over all $(n+2)$-tuples of vertices $(V_0,\dots,V_{n+1})$, one vertex of each colour, is a closed cochain representing the rational Euler class of the fibre bundle.§ 5. The proof of Theorems 1 and 2. Constructing partial sectionsWe construct a tuple of partial sections step by step, guided by the combinatorics of the triangulated fibre bundle. We start from partial sections over the zero skeleton $\operatorname{Skel}_0(K_B)$ and extend them stepwise to the skeleta of higher dimension up to $\operatorname{Skel}_n(K_B)$. We assume that an orientation is fixed for all the simplices of the complex $K_B$. Take an oriented simplex $\sigma^{n+1} \in K_B$, enumerate its vertices consistent with the orientation, and fix a local trivialization of the bundle over $\sigma^{n+1}$. Step 0. Take a vertex of $\sigma^{n+1}$, say, $v_0$. Its preimage $\pi^{-1}(v_0)$ is a triangulated sphere. Set a partial section $s$ over $v_0$ to be equal to a vertex $V_0$ of the dual cell complex $\Gamma_0$ of the triangulation. Repeat this for all the vertices of $K_B$. Then we obtain a system of partial sections over $\operatorname{Skel}_0(K_B)$. In other words, for the simplex $\sigma^{n+1}$ a partial section fixes a system $V_0,\dots,V_{n+1}$ of $n+2$ coloured vertices in the complex $\Gamma_{012\dots n+1}$, one vertex of each colour. Step 1. Take an edge of $\sigma^{n+1}$, say, $v_iv_j$, and consider the complex $\Gamma_{ij}$. Recall that it equals the superposition of $\Gamma_{i}$ and $\Gamma_{j}$. We have already defined a partial section over the vertices $v_i$ and $v_j$ by fixing $V_i$ and $V_j$. These are vertices of the complex $\Gamma_{ij}$. The chain
$$
\begin{equation*}
\Sigma_{ij} =\mathcal{H}(-V_i +V_j)\in C_1(\Gamma_{ij},\mathbb{Q})
\end{equation*}
\notag
$$
is the (uniquely defined) $1$-chain whose boundary is $-V_i+V_j$. By Lemma 3, $\Sigma_{ij}$ is the average of a system of chains supported by paths from $V_i$ to $V_j$, that is, an average of chains supported by partial sections over the edge $v_iv_j$. Treating the remaining edges and simplices in a similar way we arrive at a system of partial sections over $\operatorname{Skel}_1(K_B)$. Step 2. Now our goal is to extend each partial section to $\operatorname{Skel}_2(K_B)$. For each triple of vertices $v_i$, $v_j$, $v_k$ of $\sigma^{n+1}$ consider the closed $1$-chain ${\Sigma_{jk}-\Sigma_{ik}+\Sigma_{ij}}$. Here we assume tacitly that the simplex $v_iv_jv_k$ is oriented, and the order $ijk$ is consistent with the orientation. Recall that
$$
\begin{equation*}
\Sigma_{ijk}=\mathcal{H}( \Sigma_{jk}-\Sigma_{ik}+\Sigma_{ij}) \in C_2(\Gamma_{ijk}, \mathbb{Q})
\end{equation*}
\notag
$$
is the harmonic extension taken with respect to $\Gamma_{ijk}$. Lemma 6. The chain $\Sigma_{ijk}$ is an average of chains supported by partial sections of the bundle defined on the face $(v_iv_jv_k)$ of the simplex $\sigma^{n+1}$. Proof. We know already that $\Sigma_{ij}$, $\Sigma_{jk}$ and $\Sigma_{ki}$ are averages of chains supported by some paths. We may assume that in the averaging expressions
$$
\begin{equation*}
\frac{1}{M-N}\biggl(\sum_{p=1}^M |\widetilde{s}_p|-\sum_{q=1}^N |\widetilde{s}^{\,\prime}_q|\biggr)
\end{equation*}
\notag
$$
in Lemma 2 the numbers $M$ (respectively, $N$) are the same for each of the three chains. If not, the following tricks help:
Moreover, for the sake of consistency we make these numbers the same for all the edges of the complex $K_B$, which is possible for the same reason. We collect the terms appearing in the decompositions of $\Sigma_{ij}$, $\Sigma_{jk}$ and $-\Sigma_{ik}$, positive ones with positive ones, and negative ones with negative ones, to form closed triples of paths. We apply Lemma 3 to each triple. Treating each $2$-face in a similar way we arrive at an average of partial sections over $\operatorname{Skel}_2(K_B)$. We proceed stepwise in a similar manner. Here is how the general step looks like. Step $k$. Extend each partial section to $\operatorname{Skel}_k(K_B)$. For each $(k+1)$-tuple of vertices, say, $v_0,\dots,v_{k}$ of $\sigma^{n+1}$ consider the closed $k$-chain
$$
\begin{equation*}
\sum_{i=0}^{k}(-1)^i\Sigma_{0\dots \widehat{i}\dots k} \in C_{k-1}(\Gamma_{0\dots k}).
\end{equation*}
\notag
$$
In accordance with the previous section, set
$$
\begin{equation*}
\Sigma_{0\dots k}=\mathcal{H}\biggl(\sum_{i=0}^{k}(-1)^i\Sigma_{0\dots \widehat{i}\dots k}\biggr)\in C_k(\Gamma_{0\dots k}).
\end{equation*}
\notag
$$
The harmonic extension is taken with respect to the complex $\Gamma_{0\dots k}$. The chain $\Sigma_{0\dots k}$ is an average of chains supported by partial sections over the simplex $[v_0\dots v_k]$. This is proved using arguments similar to the ones in Lemma 6. Remark 6. The same $k$-simplex $v_0,\dots,v_{k}$ is a face of several $(n+1)$-simplices. However, our construction relies on the complex $\Gamma_{0\dots k}$ only, so it is independent of the choice of $\sigma^{n+1}$ containing $v_0,\dots,v_{k}$. $\bullet$ For the proof of Theorem 1 we proceed in this way for all $k=1,2,\dots,n$, including $k=n$. At step $n$ we arrive at a closed $n$-chain which is an average of chains supported by partial sections over the boundary of $\sigma^{n+1}$. Proposition 1 and the idea of averaging complete the proof of Theorem 1. $\bullet$ In the proof of Theorem 2 the final step $n$ is different. Step $n$. Our goal is to extend averaged partial sections to $\operatorname{Skel}_n(K_B)$. Now we use the dimension $n$ extension. At this step we average some extended sections further, but some terms luckily get cancelled. For each $i$ set
$$
\begin{equation*}
\Sigma_{\,\widehat{i}}= \sum_{j\in \{0,1,\dots,\widehat{i},\dots,n+1\}} \ (-1)^{j+(\operatorname{sign}(j-i)+1)/{2}} \Sigma_{012\dots \widehat{i}\dots \widehat{j}\dots n+1}.
\end{equation*}
\notag
$$
By construction $\Sigma_{\,\widehat{i}}$ is a closed chain related to $\Gamma_{012\dots \widehat{i}\dots n+1}$. Fix $\Sigma_{\,\widehat{i}}$. For each $V_j$, $j\neq i$, we consider the associated dimension $n$ extensions $\mathcal{W}(V_j,\Sigma_{\,\widehat{i}})$. Altogether we have $n+1$ extensions of $\Sigma_{\,\widehat{i}}$, each of which is the average of some $s$-patches of the chain $\Sigma_{\,\widehat{i}}$ by Lemma 2. Therefore, the chain
$$
\begin{equation*}
\frac{1}{n+1} \sum_{j \neq i} \mathcal{W}(V_j,\Sigma_{\,\widehat{i}})
\end{equation*}
\notag
$$
is also an average of several $s$-patches of the chain $\Sigma_{\,\widehat{i}}$. The further summation over $i$ yields a closed chain which is the average of some partial sections defined on $\partial \sigma^{n+1}$:
$$
\begin{equation*}
\mathfrak{E}= \frac{1}{n+1}\sum_i \sum_{j \neq i} \mathcal{W}(V_j,\Sigma_{\,\widehat{i}}).
\end{equation*}
\notag
$$
We evaluate this chain at a point $x$:
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{E}(x) &= \frac{1}{n+1}\sum_i \sum_{j \neq i} \mathcal{W}(V_j,x,\Sigma_{\,\widehat{i}}) \\ &=\frac{1}{n+1}\sum_i \sum_{j} \mathcal{W}(V_j,x,\Sigma_{\,\widehat{i}})- \frac{1}{n+1}\sum_i \mathcal{W}(V_i,x,\Sigma_{\,\widehat{i}}). \end{aligned}
\end{equation*}
\notag
$$
We substitute in $x=V_k$ and observe that the first sum vanishes by Lemma 7.
$$
\begin{equation*}
\mathfrak{E}(V_k)=- \frac{1}{n+1}\sum_i \mathcal{W}(V_i,V_k,\Sigma_{\,\widehat{i}})=\frac{1}{n+1}\sum_{i} \mathcal{W}(V_k,V_i,\Sigma_{\,\widehat{i}}).
\end{equation*}
\notag
$$
So $\mathfrak{E}$ is a closed $n$-chain, which is, by construction and by Lemmas 5 and 2, the average of chains supported by some partial sections defined on $\partial \sigma^{n+1}$ such that its restrictions to the $v_i$ are $V_i$. One more averaging over all possible choices of $V_0,\dots,V_{n+1}$, $V_i\in \operatorname{Vert}(\Gamma_i)$, yields a cochain $\mathcal{E}(\sigma^{n+1})$ representing the Euler class. The proof of Theorem 2 is complete. § 6. Concluding remarks6.1. Circle bundlesLet us show that for $n=1$ Theorem 2 leads to the formula from [2] and [1]. For a simplex $\sigma^{n+1}=\sigma^2$ in the base the complex $\Gamma_{012}$ is a triangulated circle, which is viewed as a necklace with beads coloured with three colours, say, red, blue and green. The algorithm consists of two steps. Step $0$ fixes a multicoloured triple of beads. Combinatorially, there are two types of red-blue-green triples, positive and negative oriented ones. Step $1$ is incidentally Step $n$, so we need not use harmonic extensions in this case.1 We only use dimension $n$ extensions. The chain $\Sigma_{\widehat{0}}$ is a $0$-chain represented by a pair of beads $V_1$ and $V_2$. It can be glued in by one of the arcs of a circle which connect $V_1$ and $V_2$ (surely, there also exist other patches that wind several times around the circle, but according to our formula we use only the ones we have mentioned). Its dimension $n$ extension equals the average of these two patches. One concludes that for each triple we average over eight sections; Figure 3 shows one of them. A simple analysis shows that each positive oriented triple of beads contributes ${-1}/{2}$ to the cochain, whereas a negative orientated one contributes ${1}/{2}$. Thus, the cochain representing the Euler class assigns the number
$$
\begin{equation*}
\frac{\sharp(\mathrm{neg})-\sharp(\mathrm{pos})}{2\cdot \sharp(\mathrm{red})\cdot \sharp(\mathrm{blue})\cdot \sharp(\mathrm{green})}
\end{equation*}
\notag
$$
to each simplex $\sigma^2$ in the base. Here $\sharp(\mathrm{pos})$ ($\sharp(\mathrm{neg})$) is the number of positive (respectively, negative) oriented multicoloured triples of beads; $\sharp(\mathrm{red})$ denotes the number of red beads, and so on. 6.2. Other local combinatorial formulaeThis approach leaves much room for deducing other local combinatorial formulae. For instance, instead of harmonic extension we can use minimal extension: at the first step we connect $V_i$ and $V_j$ by the shortest path in the $1$-skeleton of the complex $\Gamma_{ij}$. By the length of a path in a graph we mean the number of edges. If there are several shortest paths, then we must take the average. At the second step we take a patch of ‘minimum area’, that is, the subcomplex of the $2$-skeleton of $\Gamma_{ijk}$ with the least number of $2$-cells, and so on. Another way to obtain a local combinatorial formula is to use a deformed Laplacian instead of the usual one. That is, we can fix another scalar product on $C_k(\Gamma)$, for instance, to set the length of a vector associated with a cell to depend on the combinatorics of the cell (say, on the number of vertices of the cell). As a result, one expects a formula for some other cochain (representing the same cohomology class).
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\Bibitem{Pan23} |
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