Geometric structures and configurations of flags in orbits of real forms

Abstract

This is an introduction and a survey on geometric structures modelled on closed orbits of real forms acting on spaces of flags. We focus on 3-manifolds and the flag space of all pairs of a point and a line containing it in ${\mathbb{P}}({\mathbb{C}}^3)$. It includes a description of general flag structures which are not necessarily flat and a combinatorial description of flat structures through configurations of flags in closed orbits of real forms. We also review volume and Chern–Simons invariants for those structures.

Appendix

In this “Appendix”, we will discuss the relations between Bloch group and algebraic K-theory. We will use the notations introduced in Sect. 5.

1.1 Basic definitions and properties

Let F be a field. We first recall the definition of Milnor K-groups of F which was introduced in [39]. Let $F^{*}=F{\setminus } \{0\}$ be the multiplicative group of F. Let $T(F^{*})$ be its graded tensor algebra over ${\mathbb {Z}}$. That is,

$$\begin{aligned} T(F^{*})=\bigoplus _{n\ge 0}(F^{*})^{\otimes n}, \end{aligned}$$

where $(F^{*})^{\otimes 0}={\mathbb {Z}}$, $(F^{*})^{\otimes 1}=F^{*}$, for $n\ge 2$, $(F^{*})^{\otimes n}=F^{*}\otimes _{_{\mathbb {Z}}}\cdots \otimes _{_{\mathbb {Z}}} F^{*}$ (n factors).

Let I be the homogeneous ideal of $T(F^{*})$ generated by the elements of the form $a\otimes (1-a)$, $a\in F{\setminus } \{0,1\}$.

Definition 8.1

The Milnor K ring of F is the graded ring $K^{M}_{*}=T(F^{*})/I$. Its degree n part is called the n-th Milnor K-group of F, denoted by $K_n^{M}(F)$.

By definition, since the generators of I are of degree 2, we have

$$\begin{aligned} K_0^{M}(F)={\mathbb {Z}}, K_1^{M}(F)=F^{*}. \end{aligned}$$

For $n\ge 2$, as abelian group,

$$\begin{aligned} K_n^{M}(F)=(F^{*})^{\otimes n}/I_n, \end{aligned}$$

where $I_n$ is the subgroup generated by the tensors of the form $a_1\otimes \cdots \otimes a_n$ with $a_i+a_{i+1}=0$ for some i, $1\le i\le n-1$.

The definition of Quillen’s algebraic K-groups $K_{*}(F)$ of F is more involved (See [42]). They are defined as the homotopy groups of a certain topological space which is closely related to the classifying space of the infinite general linear group GL(F) of F. Here GL(F) is the direct limit of GL(n, F), $n\ge 1$ with respect to the homomorphisms $d_n:GL(n,F)\rightarrow GL(n+1,F)$, defined by

$$\begin{aligned}d_n(A)= \left( \begin{array}{cc} A &{} \quad 0 \\ 0 &{} \quad 1 \\ \end{array} \right) . \end{aligned}$$

It is well-known that $K_{1}(F)\cong K^{M}_1(F) =F^{*}$ and $K^{M}_2(F)\cong K_{2}(F)$. Moreover, there is a natural homomorphism from $K^{M}_{n}(F)$ to $K_{n}(F)$ for each n. The cokernel of this map is called the group of indecomposable elements, denoted by $K_{n}^{{\text {ind}}}(F)$. The first non-trivial case is when $n=3$.

For more information and details about the algebraic K-theory, besides the original papers [39, 42], we recommend the books [45, 52] and the references therein. In this paper, we will focus on $K_3$.

Now for $n=3$ and F an infinite field, we have the following fundamental exact sequence, due to Suslin [46, Theorem 5.2]:

(8.1)

where $\mu (F)$ is the group of roots of unity in F. ${\text {Tor}}(\mu (F),\mu (F))^{\sim }$ is the unique nontrivial extension of ${\text {Tor}}(\mu (F),\mu (F))$ by ${\mathbb {Z}}/2$.

When $F=\mathbb {C}$, ${\text {Tor}}(\mu (\mathbb {C}),\mu (\mathbb {C}))^{\sim }={\mathbb {Q}}/{\mathbb {Z}}$, we have the exact sequence:

(8.2)

When $F={\mathbb {R}}$, ${\text {Tor}}(\mu ({\mathbb {R}}),\mu ({\mathbb {R}}))^{\sim }={\mathbb {Z}}/4{\mathbb {Z}}$, we have the exact sequence

(8.3)

We also know that (See [44])

$$\begin{aligned} K_{3}^{{\text {ind}}}(\mathbb {C})\simeq H_3(SL(2,\mathbb {C}),{\mathbb {Z}}),\; K_{3}^{{\text {ind}}}({\mathbb {R}})\simeq H_3(SL(2,{\mathbb {R}}),{\mathbb {Z}}), \end{aligned}$$

where the group homologies are the homologies of $SL(2,\mathbb {C})$, and $SL(2,{\mathbb {R}})$ viewed as discrete groups.

Complex conjugation induces an involution on $H_3(SL(2,\mathbb {C}),{\mathbb {Z}})$, which will be denoted by c. Set

$$\begin{aligned} H_3(SL(2,\mathbb {C}),{\mathbb {Z}})^{+}=\{u\in H_3(SL(2,\mathbb {C}),{\mathbb {Z}})|c(u)=u\}; \end{aligned}$$

and

$$\begin{aligned} H_3(SL(2,\mathbb {C}),{\mathbb {Z}})^{-}=\{u\in H_3(SL(2,\mathbb {C}),{\mathbb {Z}})|c(u)=-u\}. \end{aligned}$$

By [19, 44], the natural homomorphism $g:H_3(SL(2,{\mathbb {R}}),{\mathbb {Z}})\rightarrow H_3(SL(2,\mathbb {C}),{\mathbb {Z}})$ is injective and its image is equal to $H_3(SL(2,\mathbb {C}),{\mathbb {Z}})^{+}$.

Let $f:{\mathcal {B}}({\mathbb {R}})\rightarrow {\mathcal {B}}(\mathbb {C})$ be the natural homomorphism induced by the inclusion of ${\mathbb {R}}$ into $\mathbb {C}$. The following proposition is probably known to the experts, but we can’t find a proof in the literature. We include a proof here.

Proposition 8.2

The image of f is ${\mathcal {B}}(\mathbb {C})^{+}$ and the kernel of f is the torsion subgroup of ${\mathcal {B}}({\mathbb {R}})$. Hence ${\mathcal {B}}({\mathbb {R}})/\{{\text {torsion}}\}\cong {\mathcal {B}}(\mathbb {C})^{+}$, equivalently, ${\mathcal {B}}({\mathbb {R}})\otimes {\mathbb {Q}}\cong {\mathcal {B}}(\mathbb {C})^{+}$.

Proof

By [18, 44, 46], we have the following commutative diagram:

Both kernels of $t_{\mathbb {C}}$ and $t_{{\mathbb {R}}}$ consist of torsion elements. g is the natural homomorphism induced by the inclusion of $SL(2,{\mathbb {R}})$ into $SL(2,\mathbb {C})$. The complex conjugation acts both on $H_3(SL(2,\mathbb {C}),{\mathbb {Z}})$ and ${\mathcal {B}}(\mathbb {C})$. Let $H_3(SL(2,\mathbb {C}),{\mathbb {Z}})^{+}$ and ${\mathcal {B}}(\mathbb {C})^{+}$ be the corresponding subgroups which are invariant under the action. For $a\in H_3(SL(2,\mathbb {C}),{\mathbb {Z}})$ or $a\in {\mathcal {B}}(\mathbb {C})$, we will denote by ${\overline{a}}$ its image under the complex conjugation. It is known that $t_{\mathbb {C}}$ is compatible with the complex conjugation, i.e. for $a\in H_3(SL(2,\mathbb {C}),{\mathbb {Z}})$, we have

$$\begin{aligned} t_{\mathbb {C}}({\overline{a}})=\overline{t_{\mathbb {C}}(a)}. \end{aligned}$$

Let’s first show that the image of f is ${\mathcal {B}}(\mathbb {C})^{+}$. By the definition, the image of f is contained in ${\mathcal {B}}(\mathbb {C})^{+}$. Suppose $a\in {\mathcal {B}}(\mathbb {C})^{+}$,i.e. $a={\overline{a}}$. Since ${\mathcal {B}}(\mathbb {C})$ is a ${\mathbb {Q}}$-vector space, there is a unique $b\in {\mathcal {B}}(\mathbb {C})$ such that $b=\frac{1}{2}a$, i.e. $2b=a$. Since $a\in {\mathcal {B}}(\mathbb {C})^{+}$, so is b. Since $t_{\mathbb {C}}$ is surjective, there is a $c\in H_3(SL(2,\mathbb {C}),{\mathbb {Z}})$ such that $t_{\mathbb {C}}(c)=b$. Let $x=c+{\overline{c}}$, then $x\in H_3(SL(2,\mathbb {C}),{\mathbb {Z}})^{+}$ and

$$\begin{aligned} t_{\mathbb {C}}(x)=t_{\mathbb {C}}(c)+t_{\mathbb {C}}({\overline{c}})=b+{\overline{b}}=2b=a. \end{aligned}$$

Since g maps $H_3(SL(2,{\mathbb {R}}),{\mathbb {Z}})$ onto $H_3(SL(2,\mathbb {C}),{\mathbb {Z}})^{+}$, $\exists y\in H_3(SL(2,{\mathbb {R}}),{\mathbb {Z}})$ such that $g(y)=x$. By the diagram chasing, we see that $f(t_{{\mathbb {R}}}(y))=a$. Therefore the image of f is ${\mathcal {B}}(\mathbb {C})^{+}$.

If a is a torsion of ${\mathcal {B}}({\mathbb {R}})$, since ${\mathcal {B}}(\mathbb {C})$ is torsion-free, we see $f(a)=0$. Conversely suppose $a\in \ker {f}$. Since $t_{{\mathbb {R}}}$ is surjective, $\exists b\in H_3(SL(2,{\mathbb {R}}),{\mathbb {Z}})$ such that $a=t_{{\mathbb {R}}}(b)$. By the commutative diagram, we obtain $t_{\mathbb {C}}(g(b))=0$. Since $\ker {t_{\mathbb {C}}}$ is torsion, g(b) is a torsion. Since g is injective, b is a torsion. Hence $a=t_{{\mathbb {R}}}(b)$ is a torsion. □

1.2 $H_3({{\,\mathrm{SL}\,}}(n,\mathbb {C}),{\mathbb {Z}})$ and $K_3(\mathbb {C})$

Let F be a field. The natural imbedding $a_n:{{\,\mathrm{SL}\,}}(n,F)\rightarrow {{\,\mathrm{SL}\,}}(n+1,F)$ defined by

$$\begin{aligned} a_n(A)= \left( \begin{array}{cc} A &{} \quad 0 \\ 0 &{} \quad 1 \\ \end{array} \right) \end{aligned}$$

induces the natural homomorphism

$$\begin{aligned} a(i,n):H_i({{\,\mathrm{SL}\,}}(n,F),{\mathbb {Z}})\rightarrow H_i({{\,\mathrm{SL}\,}}(n+1,F),{\mathbb {Z}}) \end{aligned}$$

for each i and n.

The following proposition was proved by Sah.

Proposition 8.3

(Sah [44]) Let $F=\mathbb {C}$ or ${\mathbb {R}}$.

• 1. $a(3,n):H_3({{\,\mathrm{SL}\,}}(n,F),{\mathbb {Z}})\rightarrow H_3({{\,\mathrm{SL}\,}}(n+1,F),{\mathbb {Z}})$ are isomorphisms for $n\ge 3$ ;

• 2. $a(3,2):H_3({{\,\mathrm{SL}\,}}(2,F),{\mathbb {Z}})\rightarrow H_3({{\,\mathrm{SL}\,}}(3,F),{\mathbb {Z}})$ is injective.

Moreover, for $F=\mathbb {C}$, we have for each $n\ge 3$,

$$\begin{aligned} K_3(\mathbb {C})\simeq H_3({{\,\mathrm{SL}\,}}(n,\mathbb {C}),{\mathbb {Z}}), \end{aligned}$$

and

$$\begin{aligned} K_3(\mathbb {C})\simeq H_3({{\,\mathrm{SL}\,}}(2,\mathbb {C}),{\mathbb {Z}})\bigoplus K_3^{M}(\mathbb {C})\simeq K_3^{ind}(\mathbb {C}) \bigoplus K_3^{M}(\mathbb {C}). \end{aligned}$$