Spin-structures on real Bott manifolds with K\"ahler structures

TL;DR

This paper characterizes when real Bott manifolds with Kähler structures admit Spin-structures, using cohomological techniques and Ishida's characterization, contributing to the understanding of their topological properties.

Contribution

It provides necessary and sufficient conditions for the existence of Spin-structures on real Bott manifolds with Kähler structures, extending previous cohomological methods.

Findings

Identifies conditions for Spin-structure existence

Uses Ishida's characterization and cohomological techniques

Builds on methods from Hantzsche-Wendt manifolds

Abstract

Let M be a real Bott manifold with K\"{a}hler structure. Using Ishida characterization we give necessary and sufficient condition for the existence of the Spin-structure on M. In proof we use the technic developed in Popko, Szczepa\'{n}ski "Cohomological rigity of oriented Hantzsche-Wendt manifolds" (Adv. Math. 302 (2016), 1044 - 1068) and characteristic classes.

Equations

$$0\to {\mathbb{Z}}^n\stackrel{\iota }{\to }\Gamma \stackrel{\pi }{\to }{\mathbb{Z}}_2^k\to 1.$$

$$\varrho (g)(z)=\iota (\gamma \iota (z){\gamma }^{-1})$$

$$g_0(z)=z,g_1(z)=-z,g_2(z)=\bar{z},g_3(z)=-\bar{z},$$

$$g_0([t])=[t],g_1([t])=\left[t+\frac{1}{2}\right],g_2([t])=[-t],g_3([t])=\left[-t+\frac{1}{2}\right].$$

$$(t_1,\dots ,t_n)(z_1,\dots ,z_n)=(t_1{z}_1,\dots ,t_n{z}_n)$$

$${\alpha }_j:{\mathbb{Z}}_2^d\subseteq D^n\stackrel{p{r}_j}{\to }D\stackrel{\alpha }{\to }{\mathbb{F}}_2,{\beta }_j:{\mathbb{Z}}_2^d\subseteq D^n\stackrel{p{r}_j}{\to }D\stackrel{\beta }{\to }{\mathbb{F}}_2$$

$$\phi (x)=\mathrm{diag}\left[(-1{)}^{({\alpha }_1+{\beta }_1)(x)},\dots ,(-1{)}^{({\alpha }_n+{\beta }_n)(x)}\right],$$

$${\theta }_j={\alpha }_j\cup {\beta }_j\in H^2({\mathbb{Z}}_2^d,{\mathbb{F}}_2),$$

$${\alpha }_j=\sum _{i=1}^d\alpha (p{r}_j(b_i))x_i,{\beta }_j=\sum _{i=1}^d\beta (p{r}_j(b_i))x_i\in {\mathbb{Z}}_2^d[x_1,\dots ,x_d],$$

$${\alpha }_j=\sum _{i=1}^d\alpha (P_{ij})x_i,{\beta }_j=\sum _{i=1}^d\beta (P_{ij})x_i,{\theta }_j^P={\alpha }_j{\beta }_j.$$

$$0\to H^1({\mathbb{Z}}_2^d,{\mathbb{F}}_2)\stackrel{{\pi }^{\ast }}{\to }{H}^1(\Gamma ,{\mathbb{F}}_2)\stackrel{{\iota }^{\ast }}{\to }{H}^1({\mathbb{Z}}^n,{\mathbb{F}}_2)\stackrel{d_2}{\to }{H}^2({\mathbb{Z}}_2^d,{\mathbb{F}}_2)\stackrel{{\pi }^{\ast }}{\to }{H}^2(\Gamma ,{\mathbb{F}}_2)$$

$$w(M)={\pi }^{\ast }\left(\prod _{j=1}^n(1+{\alpha }_j+{\beta }_j)\right)\in H^{\ast }(\Gamma ,{\mathbb{F}}_2)=H^{\ast }(M,{\mathbb{F}}_2).$$

$$⟨\mathrm{Im}(d_2)⟩=⟨{\theta }_1,\dots ,{\theta }_n⟩\subseteq {\mathbb{F}}_2[x_1,x_2,\dots ,x_n.]$$

$$w=\prod _{j=1}^n\left(1+{\alpha }_j+{\beta }_j\right)\in {\mathbb{F}}_2[x_1,\dots ,x_d].$$

$$s_i=\left(\mathrm{diag}\left[1,\dots ,1,(-1{)}^{a_{i,i+1}},\dots ,(-1{)}^{a_{i,n}}\right],{\left(0,\dots ,0,\frac{1}{2},0\dots ,0\right)}^T\right)$$

$${\beta }_j=\sum _{i=1}^d\beta \left(p_{ij}\right)x_i=\beta \left(p_{jj}\right)x_j=x_j\text{ for all }1\le i\le n.$$

$${\alpha }_j+{\beta }_j$$

$$=\sum _{\begin{array}{c}i=1,i\ne j\end{array}}^{j-1}\alpha (p_{ij})x_i+\left(\alpha (p_{jj})+\beta (p_{jj})\right)x_j=\sum _{i\ne j}\alpha (p_{ij})x_i=\sum _{i=1}^{j-1}{b}_{ji}{x}_i,$$

$${\alpha }_i+{\beta }_i=\sum _{j=1}^{i-1}{b}_{ij}{x}_j,$$

$${\alpha }_k=x_k={\beta }_k,$$

$$w_2(M(A))=\sum _{i=1}^n\left(b_{j_{1}i}+b_{j_{2}i}+\dots +b_{j_{k}i}\right)x_i^2\in {\mathbb{F}}_2[x_1,x_2,\dots ,x_n]$$

$${\alpha }_{j_{k+n}}+{\beta }_{j_{k+n}}={\alpha }_{j_k}+{\beta }_{j_k}=\sum _{i=1}^n{b}_{j_{k}i}{x}_i,$$

$$w(M(A))$$

$$=\prod _{k=1}^n{\left(1+\sum _{i=1}^{j_k-1}{b}_{j_{k}i}{x}_i\right)}^2=\prod _{k=1}^n\left(1+\sum _{i=1}^{j_k-1}{b}_{j_{k}i}{x}_i^2\right)$$

$$=1+\sum _{i=1}^n\left(b_{j_{1}i}+b_{j_{2}i}+\dots +b_{j_{n}i}\right)x_i^2+\dots .$$

$$w_2(M(A))=\sum _{i=1}^n\left(b_{j_{1}i}+b_{j_{2}i}+\dots +b_{j_{n}i}\right)x_i^2.$$

$$w_2(M(A))=\sum _{i=1}^n\left(b_{j_{1}i}+b_{j_{2}i}+\dots +b_{j_{n}i}\right)x_i^2=0$$

$${\alpha }_{i_1}+{\beta }_{i_1}=\dots ={\alpha }_{i_{2k}}+{\beta }_{i_{2k}}=\sum _{k=1}^{i_1}{b}_{i_{1}k}{x}_k,$$

$$w(M(A))$$

$$={\left(1+\sum _{r=1}^{j_l-1}{b}_{j_{l}r}{x}_r^2\right)}^k={\left(1+\sum _{r=1}^{j_l-1}{b}_{j_{l}r}{x}_r^2\right)}^{2s}={\left(1+\sum _{r=1}^{j_l-1}{b}_{j_{l}r}{x}_r^4\right)}^s.$$

$$A=\left[\begin{array}{cccccc}1& 0& 2& 2& 2& 2\\ 0& 1& 2& 2& 2& 2\\ 0& 0& 1& 0& 2& 2\\ 0& 0& 0& 1& 2& 2\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$$

$${\mathrm{Id}}_A$$

Full text

Author is supported by the Polish National Science Center grant DEC-2013/09/B/ST1/04125

1. Introduction

Let $\Gamma$ be a fundamental group of a real Bott manifold $M$. From [10] we know that $\Gamma$ defines a short exact sequence

[TABLE]

However above we have a induced holonomy representation $\varrho:\mathbb{Z}_{2}^{k}\to GL(n,\mathbb{Z})$

[TABLE]

for all $g\in\mathbb{Z}_{2}^{k}$, $\pi(\gamma)=g$, $\gamma\in\Gamma$, $z\in\mathbb{Z}^{n}$, where $\varrho(\mathbb{Z}_{2})^{k}\subset D=\{A=(a_{ij})\in GL(n,\mathbb{Z})|a_{ij}=0,i\neq j,a_{ii}=\pm 1,1\leq i,j\leq n\}.$

It is well known, ([10]), that $M$ is determined by a certain matrix $A=[a_{ij}]$, where $a_{ij}\in\mathbb{F}_{2}$. We call $A$ a Bott matrix and we shall denote the manifold $M$ by $M(A)$. In [9] Ishida gave the following necessary and sufficient condition for existence of the Kähler structure on $M(A)$.

Theorem 1.1**.**

([9] Theorem 3.1)* Let $A$ be $2n-$dimensional matrix of real Bott manifold $M(A)$. Then the following conditions are equivalent:*

1. (1)

there exist $n$ subsets $\{j_{1},j_{n+1}\}$, $\ldots$, $\{j_{n},j_{2n}\}$ of $\{1,2,\ldots,2n\}$ such that

1. (a)

$\coprod_{k=1}^{n}\{j_{k},j_{k+n}\}=\{1,2,\ldots,2n\}$, 2. (b)

$A^{j_{k}}=A^{j_{k+n}}$* for all $1\leq i<j_{k}$, where $A^{k}$ is the *$k-$th column of the matrix $A$. 2. (2)

there exist a Kähler structure on $M(A)$.

In this note we are going to examine the existence of the Spin structure on a real Bott manifold with a Kähler structure. We would like to mention that the general condition for existence of Spin structure is considered in ([6], [4]). However we use different new methods which we find interesting. We use different definition of the real Bott manifold which we introduce at the section 2. In the section 3 we prove our main result.

2. New definition of real Bott manifold

In this section we recall methods introduced in [12] and developed in [11]. Let $S^{1}$ be a unit circle in $\mathbb{C}$ and we consider authomorphisms $g_{i}:S^{1}\to S^{1}$ given by

[TABLE]

for all $z\in S^{1}$. We can identify $S^{1}$ with $\mathbb{R}/\mathbb{Z}$ and for each ${[t]\in\mathbb{R}/\mathbb{Z}}$ we have

[TABLE]

Let ${\it{D}}=\langle g_{i}:i=0,1,2,3\rangle$. Then ${\it{D}}\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ and $g_{3}=g_{1}g_{2}$. We define an action ${\it{D}}^{n}$ on $T^{n}$ by

[TABLE]

for $(t_{1},\ldots,t_{n})\in{\it{D}}^{n}$ and $(z_{1},\ldots,z_{n})\in T^{n}=\underbrace{S^{1}\times\ldots\times S^{1}}_{n}$.

Any subgroup $\mathbb{Z}_{2}^{d}\sqsubseteq{\it{D}}^{n}$ defines $(d\times n)-$matrix with entries in ${\it{D}}$ which defines a matrix $P$ with entries in the set $S=\{0,1,2,3\}$ under the identification $i\leftrightarrow g_{i}$, $i=0,1,2,3$.

We have the following characterisation of the action of $\mathbb{Z}_{2}^{d}$ on $T^{n}$ and the associated orbit space $T^{n}/\mathbb{Z}_{2}^{d}$ via the matrix $P$. Let $\mathbb{Z}_{2}^{d}\subseteq{\it{D}}^{n}$ and $P\in S^{d\times n}$. Then the action of $\mathbb{Z}_{2}^{d}$ on $T^{n}$ is free if and only if there is 1 in the sum of any distinct collection of rows of $P$. Group $\mathbb{Z}_{2}^{d}$ is the holonomy group of $T^{n}/\mathbb{Z}_{2}^{d}$ if and only if there is either 2 or 3 in each row of $P$.

Let us consider the epimorphisms $\alpha,\beta:{\it{D}}\to\mathbb{F}_{2}=\{0,1\}$ where values of $\alpha$ and $\beta$ on ${\it{D}}$ are given by

[TABLE]

.

For $j=1,2,\ldots,n$ and $\mathbb{Z}_{2}^{d}\subseteq{\it{D}}^{n}$ we define epimorphisms

[TABLE]

by $\alpha_{j}(t_{1},\ldots,t_{n})=\alpha(t_{j}),\;\;\beta_{j}(t_{1},\ldots,t_{n})=\beta(t_{j}).$ Using definitions of $\alpha$ and $\beta$ and the translations given by (3), we obtain the following lemma.

Lemma 2.1**.**

([11])* Suppose a subgroup $\mathbb{Z}_{2}^{d}\subseteq{\it{D}}$ acts free on $T^{n}$. Then a holonomy representation $\varphi:\mathbb{Z}_{2}^{d}\to\operatorname{GL}(n,\mathbb{Z})$ of the flat manifold $T^{n}/\mathbb{Z}_{2}^{d}$ is given by*

[TABLE]

for all $x\in\mathbb{Z}_{2}^{d}$.

Since $H^{1}(\mathbb{Z}_{2}^{d},\mathbb{F}_{2})=\operatorname{Hom}(\mathbb{Z}_{2}^{d},\mathbb{Z}_{2})$ we can view $\alpha_{i}$ and $\beta_{i}$ as 1-cocycles and define

[TABLE]

where $\cup$ denotes the cup product. It is well known that $H^{*}(\mathbb{Z}_{2}^{d},\mathbb{F}_{2})\cong\mathbb{F}_{2}[x_{1},\ldots,x_{d}]$ where $\{x_{1},\ldots,x_{d}\}$ is a basis of $H^{1}(\mathbb{Z}_{2}^{d},\mathbb{F}_{2})$. Hence, elements $\alpha_{j}$ and $\beta_{j}$ correspond to

[TABLE]

where $\{x_{1},\ldots,x_{d}\}$ is the standard basis of $\mathbb{Z}_{2}^{d}$ and $j=1,\ldots,n$ ([2], Proposition 1.3). Moreover, from the definition of matrix $P\in S^{d\times n}$ we can write equations (6) and (7) as follows

[TABLE]

There is an exact sequence

[TABLE]

where $d_{2}$ is the transgression and $\pi^{*}$ is induced by the quotient map $\pi:\Gamma\to\mathbb{Z}_{2}^{d}$, [3].

Proposition 2.1**.**

([11])* Suppose $\mathbb{Z}_{2}^{d}$ acts freely and diagonally on $T^{n}$. Let $M=T^{n}/\mathbb{Z}_{2}^{n}$, $\Gamma=\pi_{1}(M)$ and consider the associated to the group extension of (1). Then*

1. (1)

$\forall_{1\leq l\leq n\;\;}\theta_{l}=d_{2}(\varepsilon_{l})$*, where *$\{\varepsilon_{1},\ldots,\varepsilon_{n}\}$is the basis of $H^{1}(\mathbb{Z}^{n},\mathbb{F}_{2})$ dual to the standard basis of $\mathbb{Z}^{n}\otimes\mathbb{F}_{2}$, 2. (2)

the total Stiefel-Whitney class of $M$ is

[TABLE]

One can see by part (1) of Proposition 2.1 that the image of differential $d_{2}$ is an ideal generated by $\theta_{j}$ and

[TABLE]

For matrix $P\in S^{d\times n}$, using (8), we set $\operatorname{Id}_{P}=\langle\theta_{1}^{P},\ldots,\theta_{n}^{P}\rangle$ and we call this the characteristic ideal of $P$. The quotient $C_{P}=\mathbb{F}_{2}[x_{1},\ldots,x_{d}]/\operatorname{{Id}}_{P}$ we call characteristic algebra of $P$.

Corollary 2.1**.**

([11])* Suppose $\mathbb{Z}_{2}^{d}$ acts freely and diagonally on $T^{n}$. There is a canonical homomorphism of graded algebras $\Phi:C\to H^{*}(T^{n},\mathbb{Z}_{2}^{d},\mathbb{F}_{2})$ such that $\Phi([w])=w(T^{n}/\mathbb{Z}_{2}^{d})$ where $[w]$ is the class of*

[TABLE]

Moreover, $\Phi$ is a monomorphism in degree less that or equal to two.

Definition 2.1.

Given a matrix $P\in S^{d\times n}$, we define the Stiefel-Whitney class of $P$, to be the class $[w]\in C_{P}$ defined by (9).

Corollary 2.2**.**

([11])* Suppose $P\in S^{d\times n}$ is free and $T^{n}/\mathbb{Z}_{2}^{d}$ is the corresponding flat manifold. Then $\Phi(w(P))=w(T^{n}/\mathbb{Z}_{2}^{d})$.*

Now, we describe a real Bott manifold $M(A)$. Let $A=[a_{ij}]$ be an strictly upper triangular matrix with entries 0 or 1 and let $s_{i}$, $i=1,\ldots,n$ be Euclidean motions on $\mathbb{R}^{n}$ defined by

[TABLE]

where $(-1)^{a_{i,i+1}}$ is at the $(i+1,i+1)$ position and $\frac{1}{2}$ is the $i-$th coordinate of the column, $i=1,2,...,n-1.$ $s_{n}=\left(I,\left(0,0,...,0,\frac{1}{2}\right)\right)\in E(n).$ The group $\Gamma(A)$ generated by $s_{1},\ldots,s_{n}$ is crystallographic group. The subgroup generated by $s_{1}^{2},s_{2}^{2},...,s_{n}^{2}$ consists of all transitions by $\mathbb{Z}^{n}$. The action of $\Gamma(A)$ on $\mathbb{R}^{n}$ is free and the orbit space $\mathbb{R}^{n}/\Gamma(A)$ is compact.

3. Main results

We know that $s_{i}$ ([10]) are generators of the crystallographic group $\Gamma(A)$. Using the same methods as in [12], [11], for each strictly upper triangular matrix $A=[a_{ij}]$ which generates the fundamental group of real Bott manifold $M(A)$ we have $P-$matrix $P_{A}=[p_{ij}]$ with diagonal entries 1, entries 0 or 2 in the upper triangular part and entries 0 in the lower triangular part. Since $\beta(0)=\beta(2)=1$ and $\beta(1)=1$ we get

[TABLE]

Now, $\alpha(0)=0$, $\alpha(1)=\alpha(2)=1$ and

[TABLE]

where $b_{ji}\in\{0,1\}$ for all $i=1,2,\ldots,j-1$. So, we get

[TABLE]

where $b_{ij}\in\{0,1\}$ for all $j=1,2,\ldots,i-1$.

Now, let us consider a real Bott manifold with a Kähler structure. We will denote this manifold by RBK manifold. Let the column $P^{k}$ of matrix $P_{A}$ has $p_{kk}=1$ and all others entries equal to 0. Then

[TABLE]

so $x_{k}^{2}\in\operatorname{Id}_{P_{A}}$.

Theorem 3.1**.**

Let $P_{A}$ be a $P-$matrix of $2n-$dimensional RBK manifold $M(A)$. Then

[TABLE]

and $M(A)$ has a Spin-structure if and only if either $\left(b_{j_{1}i}+b_{j_{2}i}+\ldots+b_{j_{k}i}\right)=0\mod 2$ or $x_{i}^{2}\in\operatorname{Id}_{P_{A}}$ for all $i=1,\ldots,n$.

Proof. $M(A)$ is $2n-$dimensional RBK manifold, so from (11) we get

[TABLE]

where $b_{j_{k}i}\in\{0,1\}$ for all $i=1,2,\ldots,j_{k}-1$ and

[TABLE]

From the above consideration we have

[TABLE]

It is well known that the manifold $M(A)$ has the Spin-structure if $w_{2}(M(A))=0$. In our case

[TABLE]

if either $\left(b_{j_{1}i}+b_{j_{2}i}+\ldots+b_{j_{n}i}\right)=0\mod 2$ or $x_{i}^{2}\in\operatorname{Id}$ for all $i=1,\ldots,n$.

$\square$

At the end we consider a special case of RBK manifold.

Lemma 3.1**.**

Let $M(A)$ be a RBK manifold with matrix $P$. Let $k$ be an even number and let $P^{i_{1}}=P^{i_{2}}=\ldots=P^{i_{2k}}$ be columns with nonzero entries and all others columns of matrix $P$ have only 0 entries. Then the manifold $M(A)$ has the Spin-structure.

Proof. Since $P^{i_{1}}=P^{i_{2}}=\ldots=P^{i_{2k}}$ we get

[TABLE]

where $b_{i_{1}k}\in\{0,1\}$ for all $k=1,2,\ldots,i_{1}$ and from the proof of Theorem 3.1 and since $k=2s$ we get

[TABLE]

So, $w_{2}(M(A))=0$ and RBK manifold has the Spin-structure.

$\square$

Example 3.1.

Let

[TABLE]

be a matrix of the manifold $M(A)$. Then

[TABLE]

and

[TABLE]

Since $x_{3}^{2},x_{4}^{2}\not\in\operatorname{Id}_{A}$, so $w_{2}(M(A))=x_{3}^{2}+x_{4}^{2}\neq 0$ and $M(A)$ has no Spin-structure.

Let $A$ be the matrix of $2n-$dimensional RBK manifold $M(A)$ and let $S_{k}=\sum_{i=1}^{n}a_{kj_{i}}$ for all $1\leq k\leq 2n$. Then from Theorem 1.1 and following Theorem 3.1 we get

Corollary 3.1**.**

Let $A$ be a matrix of a $2n-$dimensional RBK manifold $M(A)$. The manifold $M(A)$ has the Spin-structure if or $S_{k}=0\mod 2$ or $S_{k}=1\mod 2$ and the column $A^{k}$ has only entries 0, for all $1\leq k\leq 2n$.

Example 3.2.

Let

[TABLE]

be a matrix of a manifold $M(A)$. Then $S_{1}=S_{2}=2\mod 2=0,S_{3}=S_{4}=1\mod 2=1,S_{5}=S_{6}=0\mod 2=0$ and there are entries equal to 1 in columns $A^{3}$ and $A^{4}$, so $M(A)$ has no Spin-structure.