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''solvmanifold, solvable manifold''
 
''solvmanifold, solvable manifold''
  
A homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s0861001.png" /> of a connected solvable Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s0861002.png" /> (cf. [[Lie group, solvable|Lie group, solvable]]). It can be identified with the coset space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s0861003.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s0861004.png" /> is the stabilizer subgroup of some point of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s0861005.png" />.
+
A homogeneous space $  M $
 
+
of a connected solvable Lie group $  G $(
Examples: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s0861006.png" />, the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s0861007.png" />, the Iwasawa manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s0861008.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s0861009.png" /> is the group of all upper-triangular matrices with 1's on the main diagonal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610011.png" /> is the subgroup of all integer points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610012.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610013.png" /> (the Klein bottle), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610014.png" /> (the Möbius band).
+
cf. [[Lie group, solvable|Lie group, solvable]]). It can be identified with the coset space $  G / H $,  
 +
where $  H $
 +
is the stabilizer subgroup of some point of the manifold $  M $.
  
The first solvmanifolds studied were those in the narrower class of nil manifolds (cf. [[Nil manifold|Nil manifold]]), that is, homogeneous spaces of nilpotent Lie groups (such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610017.png" />, but not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610019.png" />). The following results are due to A.I. Mal'tsev (see [[#References|[5]]]). 1) Every nil manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610020.png" /> is diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610022.png" /> is a compact nil manifold. 2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610023.png" /> is compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610024.png" /> acts effectively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610025.png" />, then the stabilizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610026.png" /> is a [[Discrete subgroup|discrete subgroup]]. 3) A nilpotent Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610027.png" /> (cf. [[Lie group, nilpotent|Lie group, nilpotent]]) acts transitively and locally effectively on some compact manifold if and only if its Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610028.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610029.png" />-form. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610030.png" /> is simply connected, then it is isomorphic to a unipotent algebraic group defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610032.png" /> is an arithmetic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610033.png" />. 4) The [[Fundamental group|fundamental group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610034.png" /> of a compact nil manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610035.png" /> (which is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610036.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610037.png" /> is simply connected and its action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610038.png" /> is locally effective) determines it up to a diffeomorphism. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610039.png" /> that can arise here are just the finitely-generated nilpotent torsion-free groups.
 
  
These results admit partial generalizations to arbitrary solvmanifolds. Thus, for any solvmanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610040.png" /> there is a solvmanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610041.png" /> which is a finitely-sheeted covering of it and is diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610043.png" /> is some compact solvmanifold. An arbitrary solvmanifold cannot always be decomposed into a direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610044.png" />, but it is diffeomorphic (see [[#References|[1]]], [[#References|[4]]]) to the space of a vector bundle over some compact solvmanifold (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610045.png" /> the corresponding bundle is a non-trivial line bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610046.png" />). The fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610047.png" /> of an arbitrary solvmanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610048.png" /> is polycyclic (cf. [[Polycyclic group|Polycyclic group]]), and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610049.png" /> is compact, it determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610050.png" /> uniquely up to a diffeomorphism. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610051.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610052.png" /> for some compact solvmanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610053.png" /> if and only if it is contained in an exact sequence of the form
+
Examples:  $  \mathbf R ^{n} $,
 +
the torus  $  T ^{n} $,  
 +
the Iwasawa manifold  $  N / I $(
 +
where  $  N $
 +
is the group of all upper-triangular matrices with 1's on the main diagonal in  $  \mathop{\rm GL}\nolimits ( 3 ,\  \mathbf R ) $
 +
and  $  I $
 +
is the subgroup of all integer points in  $  N $),
 +
$  K ^{2} $(
 +
the Klein bottle), and  $  \mathop{\rm Mb}\nolimits $(
 +
the Möbius band).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610054.png" /></td> </tr></table>
+
The first solvmanifolds studied were those in the narrower class of nil manifolds (cf. [[Nil manifold|Nil manifold]]), that is, homogeneous spaces of nilpotent Lie groups (such as  $  \mathbf R ^{n} $,
 +
$  T ^{n} $,
 +
$  N / I $,
 +
but not  $  K ^{2} $
 +
and  $  \mathop{\rm Mb}\nolimits $).
 +
The following results are due to A.I. Mal'tsev (see [[#References|[5]]]). 1) Every nil manifold  $  M = G / H $
 +
is diffeomorphic to  $  M ^{*} \times \mathbf R ^{n} $,
 +
where  $  M ^{*} $
 +
is a compact nil manifold. 2) If  $  M $
 +
is compact and  $  G $
 +
acts effectively on  $  M $,
 +
then the stabilizer  $  H $
 +
is a [[Discrete subgroup|discrete subgroup]]. 3) A nilpotent Lie group  $  G $(
 +
cf. [[Lie group, nilpotent|Lie group, nilpotent]]) acts transitively and locally effectively on some compact manifold if and only if its Lie algebra  $  \mathfrak G $
 +
has a  $  \mathbf Q $-
 +
form. If, in addition,  $  G $
 +
is simply connected, then it is isomorphic to a unipotent algebraic group defined over  $  \mathbf Q $
 +
and  $  H $
 +
is an arithmetic subgroup of  $  G $.  
 +
4) The [[Fundamental group|fundamental group]]  $  \pi _{1} (M) $
 +
of a compact nil manifold  $  M $(
 +
which is isomorphic to  $  H $
 +
when  $  G $
 +
is simply connected and its action on  $  M $
 +
is locally effective) determines it up to a diffeomorphism. The groups  $  \pi _{1} (M) $
 +
that can arise here are just the finitely-generated nilpotent torsion-free groups.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610055.png" /> is a finitely-generated nilpotent torsion-free group. Every polycyclic group has a subgroup of finite index that is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610056.png" /> for some compact solvmanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610057.png" />. If a solvable Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610058.png" /> acts transitively and locally effectively on a compact solvmanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610059.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610060.png" /> is fibred over a torus with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610061.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610062.png" /> is the nil radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610063.png" />. A solvmanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610064.png" /> is compact if and only if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610065.png" />-invariant measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610066.png" /> with respect to which the volume of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610067.png" /> is finite.
+
These results admit partial generalizations to arbitrary solvmanifolds. Thus, for any solvmanifold  $  M $
 +
there is a solvmanifold  $  M ^ \prime  $
 +
which is a finitely-sheeted covering of it and is diffeomorphic to  $  M ^{*} \times \mathbf R ^{n} $,
 +
where $  M ^{*} $
 +
is some compact solvmanifold. An arbitrary solvmanifold cannot always be decomposed into a direct product  $  M ^{*} \times \mathbf R ^{n} $,
 +
but it is diffeomorphic (see [[#References|[1]]], [[#References|[4]]]) to the space of a vector bundle over some compact solvmanifold (for  $  \mathop{\rm Mb}\nolimits $
 +
the corresponding bundle is a non-trivial line bundle over  $  S ^{1} $).
 +
The fundamental group  $  \pi _{1} (M) $
 +
of an arbitrary solvmanifold  $  M $
 +
is polycyclic (cf. [[Polycyclic group|Polycyclic group]]), and if  $  M $
 +
is compact, it determines  $  M $
 +
uniquely up to a diffeomorphism. A group  $  \pi $
 +
is isomorphic to  $  \pi _{1} (M) $
 +
for some compact solvmanifold  $  M $
 +
if and only if it is contained in an exact sequence of the form $$
 +
\{ e \}  \rightarrow  \Delta  \rightarrow  \pi  \rightarrow  \mathbf Z ^{s}  \rightarrow  \{ e \} ,
 +
$$
 +
where  $  \Delta $
 +
is a finitely-generated nilpotent torsion-free group. Every polycyclic group has a subgroup of finite index that is isomorphic to $  \pi _{1} (M) $
 +
for some compact solvmanifold $  M $.  
 +
If a solvable Lie group $  G $
 +
acts transitively and locally effectively on a compact solvmanifold $  M = G / H $,  
 +
then $  M $
 +
is fibred over a torus with fibre $  N / (H \cap N ) $,  
 +
where $  N $
 +
is the nil radical of $  G $.  
 +
A solvmanifold $  M = G / H $
 +
is compact if and only if there is a $  G $-
 +
invariant measure on $  M $
 +
with respect to which the volume of $  M $
 +
is finite.
  
Every solvmanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610068.png" /> is aspherical (that is, the homotopy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610069.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610070.png" />). Among all compact homogeneous spaces, compact solvmanifolds are characterized by asphericity and the solvability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086100/s08610071.png" /> (see [[#References|[3]]]).
+
Every solvmanifold $  M $
 +
is aspherical (that is, the homotopy group $  \pi _{i} (M) = 0 $
 +
for $  i \geq 2 $).  
 +
Among all compact homogeneous spaces, compact solvmanifolds are characterized by asphericity and the solvability of $  \pi _{1} (M) $(
 +
see [[#References|[3]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Auslander, "An exposition of the structure of solvmanifolds I, II" ''Bull. Amer. Math. Soc.'' , '''79''' : 2 (1973) pp. 227–261; 262–285 {{MR|486308}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L. Auslander, R. Szczarba, "Vector bundles over tori and noncompact solvmanifolds" ''Amer. J. Math.'' , '''97''' : 1 (1975) pp. 260–281 {{MR|0383443}} {{ZBL|0303.22006}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.V. Gorbatsevich, "On Lie groups, transitive on Solv manifolds" ''Math. USSR.-Izv.'' , '''11''' (1977) pp. 271–291 ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''41''' (1977) pp. 285–307 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> G. Mostow, "Some applications of representative functions to solvmanifolds" ''Amer. J. Math.'' , '''93''' : 1 (1971) pp. 11–32 {{MR|0283819}} {{ZBL|0228.22015}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M. Raghunatan, "Discrete subgroups of Lie groups" , Springer (1972) {{MR|}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Auslander, "An exposition of the structure of solvmanifolds I, II" ''Bull. Amer. Math. Soc.'' , '''79''' : 2 (1973) pp. 227–261; 262–285 {{MR|486308}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L. Auslander, R. Szczarba, "Vector bundles over tori and noncompact solvmanifolds" ''Amer. J. Math.'' , '''97''' : 1 (1975) pp. 260–281 {{MR|0383443}} {{ZBL|0303.22006}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.V. Gorbatsevich, "On Lie groups, transitive on Solv manifolds" ''Math. USSR.-Izv.'' , '''11''' (1977) pp. 271–291 ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''41''' (1977) pp. 285–307 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> G. Mostow, "Some applications of representative functions to solvmanifolds" ''Amer. J. Math.'' , '''93''' : 1 (1971) pp. 11–32 {{MR|0283819}} {{ZBL|0228.22015}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M. Raghunatan, "Discrete subgroups of Lie groups" , Springer (1972) {{MR|}} {{ZBL|}} </TD></TR></table>

Latest revision as of 00:30, 22 December 2019


solvmanifold, solvable manifold

A homogeneous space $ M $ of a connected solvable Lie group $ G $( cf. Lie group, solvable). It can be identified with the coset space $ G / H $, where $ H $ is the stabilizer subgroup of some point of the manifold $ M $.


Examples: $ \mathbf R ^{n} $, the torus $ T ^{n} $, the Iwasawa manifold $ N / I $( where $ N $ is the group of all upper-triangular matrices with 1's on the main diagonal in $ \mathop{\rm GL}\nolimits ( 3 ,\ \mathbf R ) $ and $ I $ is the subgroup of all integer points in $ N $), $ K ^{2} $( the Klein bottle), and $ \mathop{\rm Mb}\nolimits $( the Möbius band).

The first solvmanifolds studied were those in the narrower class of nil manifolds (cf. Nil manifold), that is, homogeneous spaces of nilpotent Lie groups (such as $ \mathbf R ^{n} $, $ T ^{n} $, $ N / I $, but not $ K ^{2} $ and $ \mathop{\rm Mb}\nolimits $). The following results are due to A.I. Mal'tsev (see [5]). 1) Every nil manifold $ M = G / H $ is diffeomorphic to $ M ^{*} \times \mathbf R ^{n} $, where $ M ^{*} $ is a compact nil manifold. 2) If $ M $ is compact and $ G $ acts effectively on $ M $, then the stabilizer $ H $ is a discrete subgroup. 3) A nilpotent Lie group $ G $( cf. Lie group, nilpotent) acts transitively and locally effectively on some compact manifold if and only if its Lie algebra $ \mathfrak G $ has a $ \mathbf Q $- form. If, in addition, $ G $ is simply connected, then it is isomorphic to a unipotent algebraic group defined over $ \mathbf Q $ and $ H $ is an arithmetic subgroup of $ G $. 4) The fundamental group $ \pi _{1} (M) $ of a compact nil manifold $ M $( which is isomorphic to $ H $ when $ G $ is simply connected and its action on $ M $ is locally effective) determines it up to a diffeomorphism. The groups $ \pi _{1} (M) $ that can arise here are just the finitely-generated nilpotent torsion-free groups.

These results admit partial generalizations to arbitrary solvmanifolds. Thus, for any solvmanifold $ M $ there is a solvmanifold $ M ^ \prime $ which is a finitely-sheeted covering of it and is diffeomorphic to $ M ^{*} \times \mathbf R ^{n} $, where $ M ^{*} $ is some compact solvmanifold. An arbitrary solvmanifold cannot always be decomposed into a direct product $ M ^{*} \times \mathbf R ^{n} $, but it is diffeomorphic (see [1], [4]) to the space of a vector bundle over some compact solvmanifold (for $ \mathop{\rm Mb}\nolimits $ the corresponding bundle is a non-trivial line bundle over $ S ^{1} $). The fundamental group $ \pi _{1} (M) $ of an arbitrary solvmanifold $ M $ is polycyclic (cf. Polycyclic group), and if $ M $ is compact, it determines $ M $ uniquely up to a diffeomorphism. A group $ \pi $ is isomorphic to $ \pi _{1} (M) $ for some compact solvmanifold $ M $ if and only if it is contained in an exact sequence of the form $$ \{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow \mathbf Z ^{s} \rightarrow \{ e \} , $$ where $ \Delta $ is a finitely-generated nilpotent torsion-free group. Every polycyclic group has a subgroup of finite index that is isomorphic to $ \pi _{1} (M) $ for some compact solvmanifold $ M $. If a solvable Lie group $ G $ acts transitively and locally effectively on a compact solvmanifold $ M = G / H $, then $ M $ is fibred over a torus with fibre $ N / (H \cap N ) $, where $ N $ is the nil radical of $ G $. A solvmanifold $ M = G / H $ is compact if and only if there is a $ G $- invariant measure on $ M $ with respect to which the volume of $ M $ is finite.

Every solvmanifold $ M $ is aspherical (that is, the homotopy group $ \pi _{i} (M) = 0 $ for $ i \geq 2 $). Among all compact homogeneous spaces, compact solvmanifolds are characterized by asphericity and the solvability of $ \pi _{1} (M) $( see [3]).

References

[1] L. Auslander, "An exposition of the structure of solvmanifolds I, II" Bull. Amer. Math. Soc. , 79 : 2 (1973) pp. 227–261; 262–285 MR486308
[2] L. Auslander, R. Szczarba, "Vector bundles over tori and noncompact solvmanifolds" Amer. J. Math. , 97 : 1 (1975) pp. 260–281 MR0383443 Zbl 0303.22006
[3] V.V. Gorbatsevich, "On Lie groups, transitive on Solv manifolds" Math. USSR.-Izv. , 11 (1977) pp. 271–291 Izv. Akad. Nauk. SSSR Ser. Mat. , 41 (1977) pp. 285–307
[4] G. Mostow, "Some applications of representative functions to solvmanifolds" Amer. J. Math. , 93 : 1 (1971) pp. 11–32 MR0283819 Zbl 0228.22015
[5] M. Raghunatan, "Discrete subgroups of Lie groups" , Springer (1972)
How to Cite This Entry:
Solv manifold. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Solv_manifold&oldid=44320
This article was adapted from an original article by V.V. Gorbatsevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article