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A smooth odd-dimensional manifold of a special type which is the boundary of an even-dimensional manifold constructed from fibrations over spheres by a glueing scheme specified by some graph (tree).
 
A smooth odd-dimensional manifold of a special type which is the boundary of an even-dimensional manifold constructed from fibrations over spheres by a glueing scheme specified by some graph (tree).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d0310101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d0310102.png" /> be a fibration over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d0310103.png" />-spheres with as fibre the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d0310104.png" />-ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d0310105.png" /> and as structure group the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d0310106.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d0310107.png" /> be the closed standard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d0310108.png" />-ball in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d0310109.png" />-sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101010.png" />; then
+
Let $  p _ {i} : E _ {i}  ^ {2n} \rightarrow S _ {i}  ^ {n} $,  
 +
$  i = 1, 2 \dots $
 +
be a fibration over $  n $-
 +
spheres with as fibre the $  n $-
 +
ball $  D  ^ {n} $
 +
and as structure group the group $  \mathop{\rm SO} _ {n} $,  
 +
and let $  B _ {i}  ^ {n} $
 +
be the closed standard $  n $-
 +
ball in the $  n $-
 +
sphere $  S _ {i}  ^ {n} $;  
 +
then
  
<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/d/d031/d031010/d03101011.png" /></td> </tr></table>
+
$$
 +
p _ {i}  ^ {-} 1 ( B _ {i}  ^ {n} )  \approx  B _ {i}  ^ {n} \times D _ {i}  ^ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101012.png" /> is the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101013.png" />. Let
+
where $  D _ {i}  ^ {n} $
 +
is the fibre $  p _ {i} $.  
 +
Let
  
<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/d/d031/d031010/d03101014.png" /></td> </tr></table>
+
$$
 +
\gamma _ {ij} : B _ {i}  ^ {n} \times D _ {i}  ^ {n}  \rightarrow  B _ {j}  ^ {n} \times D _ {j}  ^ {n} ,\  i = 1, 2 ,
 +
$$
  
be a homeomorphism realizing the glueing of two fibrations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101016.png" /> and mapping each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101017.png" />-ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101018.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101019.png" /> into some ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101020.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101021.png" /> (the glueing alters the factors of the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101022.png" />). The result of glueing two fibrations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101024.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101025.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101026.png" /> which, as a result of "angle smoothing" , is converted to a smooth manifold.
+
be a homeomorphism realizing the glueing of two fibrations $  p _ {i} $,  
 +
$  p _ {j} $
 +
and mapping each $  n $-
 +
ball $  B  ^ {n} \times x $
 +
from $  B _ {i}  ^ {n} \times D _ {i}  ^ {n} $
 +
into some ball $  y \times D  ^ {n} $
 +
from $  B _ {j}  ^ {n} \times D _ {j}  ^ {n} $(
 +
the glueing alters the factors of the direct product $  B  ^ {n} \times D  ^ {n} $).  
 +
The result of glueing two fibrations $  p _ {i} $,  
 +
$  p _ {j} $
 +
is the $  2n $-
 +
dimensional manifold $  E _ {i}  ^ {2n} \cup _ {\gamma _ {ij}  } E _ {j}  ^ {2n} $
 +
which, as a result of "angle smoothing" , is converted to a smooth manifold.
  
The fibrations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101027.png" /> are considered as "structural blocks" from which it is possible to construct, by pairwise glueing, the resulting smooth manifold as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101028.png" /> be a one-dimensional finite complex (a graph). Each vertex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101029.png" /> is brought into correspondence with a block <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101030.png" />; next, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101032.png" /> non-intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101033.png" />-balls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101034.png" /> are selected in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101036.png" /> is equal to the branching index of the respective vertex, and the glueing is performed according to the scheme indicated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101037.png" />. The manifold with boundary thus obtained is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101038.png" /> (neglecting the dependence on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101039.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101040.png" /> is a tree, and therefore the graph is without cycles, the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101041.png" /> is said to be a dendritic manifold.
+
The fibrations $  E _ {i}  ^ {2n} $
 +
are considered as "structural blocks" from which it is possible to construct, by pairwise glueing, the resulting smooth manifold as follows. Let $  T $
 +
be a one-dimensional finite complex (a graph). Each vertex of $  T $
 +
is brought into correspondence with a block $  E _ {i}  ^ {2n} $;  
 +
next, $  k $,
 +
$  k = 1 , 2 \dots $
 +
non-intersecting $  n $-
 +
balls $  B _ {i _ {k}  }  ^ {k} $
 +
are selected in $  S _ {i}  ^ {n} $,  
 +
where $  k $
 +
is equal to the branching index of the respective vertex, and the glueing is performed according to the scheme indicated by $  T $.  
 +
The manifold with boundary thus obtained is denoted by $  W  ^ {2n} ( T ) $(
 +
neglecting the dependence on the choice of $  E _ {i}  ^ {2n} $).  
 +
If $  T $
 +
is a tree, and therefore the graph is without cycles, the boundary $  \partial  W  ^ {2n} ( T ) = M  ^ {2n-} 1 $
 +
is said to be a dendritic manifold.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101042.png" /> is a tree, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101043.png" /> has the homotopy type of a bouquet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101044.png" /> spheres, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101045.png" /> is the number of vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101046.png" />.
+
If $  T $
 +
is a tree, $  W  ^ {2n} ( T ) $
 +
has the homotopy type of a bouquet of $  k $
 +
spheres, where $  k $
 +
is the number of vertices of $  T $.
  
The dendritic manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101047.png" /> is an integral homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101048.png" />-sphere if and only if the determinant of the matrix of the integral bilinear intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101049.png" />-form defined on the lattice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101050.png" />-dimensional homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101051.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101052.png" />. If this condition is met, the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101053.png" /> is called a plumbing.
+
The dendritic manifold $  M  ^ {2n-} 1 = \partial  W  ^ {2n} ( T ) $
 +
is an integral homology $  ( 2n - 1 ) $-
 +
sphere if and only if the determinant of the matrix of the integral bilinear intersection $  ( - 1 )  ^ {n} $-
 +
form defined on the lattice of $  n $-
 +
dimensional homology groups $  H _ {n} ( W  ^ {2n} , \mathbf Z ) $
 +
equals $  \pm  1 $.  
 +
If this condition is met, the manifold $  W  ^ {2n} ( T ) $
 +
is called a plumbing.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101054.png" /> is a tree and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101056.png" /> is simply connected; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101057.png" /> is a plumbing, the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101058.png" /> is a homotopy sphere if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101059.png" />.
+
If $  T $
 +
is a tree and $  n \geq  3 $,  
 +
$  \partial  W  ^ {2n} ( T ) $
 +
is simply connected; if $  W  ^ {2n} $
 +
is a plumbing, the boundary $  \partial  W  ^ {2n} $
 +
is a homotopy sphere if $  n \geq  3 $.
  
If the plumbing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101060.png" /> is parallelizable, the diagonal of the intersection matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101061.png" />-dimensional cycles is occupied by even numbers; in such a case the signature of the intersection matrix is divisible by 8. The plumbing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101062.png" /> is parallelizable if and only if all the fibrations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101063.png" /> used in constructing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101064.png" /> are stably trivial; e.g., if all fibrations used in constructing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101065.png" /> are tangent bundles on discs over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101066.png" />-dimensional spheres, the plumbing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101067.png" /> is parallelizable. The plumbing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101068.png" /> will be parallelizable if and only if any fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101069.png" /> used as a block in the construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101070.png" /> is either trivial or is a tubular neighbourhood of the diagonal in the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101071.png" />, i.e. is a tangent bundle on discs over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101072.png" />. If the plumbing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101073.png" /> is parallelizable, its intersection matrix can be reduced to the symplectic form consisting of blocks
+
If the plumbing $  W  ^ {4k} $
 +
is parallelizable, the diagonal of the intersection matrix of $  2k $-
 +
dimensional cycles is occupied by even numbers; in such a case the signature of the intersection matrix is divisible by 8. The plumbing $  W  ^ {4k} $
 +
is parallelizable if and only if all the fibrations over $  S  ^ {2k} $
 +
used in constructing $  W  ^ {4k} $
 +
are stably trivial; e.g., if all fibrations used in constructing $  W  ^ {4k} $
 +
are tangent bundles on discs over $  2k $-
 +
dimensional spheres, the plumbing $  W  ^ {4k} $
 +
is parallelizable. The plumbing $  W  ^ {4k+} 2 $
 +
will be parallelizable if and only if any fibration $  E _ {i}  ^ {4k+} 2 $
 +
used as a block in the construction of $  W  ^ {4k+} 2 $
 +
is either trivial or is a tubular neighbourhood of the diagonal in the product $  S  ^ {2k+} 1 \times S  ^ {2k+} 1 $,  
 +
i.e. is a tangent bundle on discs over $  S  ^ {2k+} 1 $.  
 +
If the plumbing $  W  ^ {4k+} 2 $
 +
is parallelizable, its intersection matrix can be reduced to the symplectic form consisting of blocks
  
<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/d/d031/d031010/d03101074.png" /></td> </tr></table>
+
$$
 +
\left \|
 +
\begin{array}{rr}
 +
0  & 1  \\
 +
- 1  & 0 \\
 +
\end{array}
 +
\right \|
 +
$$
  
 
situated along the main diagonal.
 
situated along the main diagonal.
  
Especially important plumbings are the Milnor manifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101076.png" />, and the Kervaire manifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101078.png" />. The Milnor manifolds are constructed as follows: A few copies of the tubular neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101079.png" /> of the diagonal in the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101080.png" /> are taken as blocks, while the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101081.png" /> is of the form
+
Especially important plumbings are the Milnor manifolds of dimension $  4k $,  
 +
$  k > 1 $,  
 +
and the Kervaire manifolds of dimension $  4k + 2 $,  
 +
$  k \geq  0 $.  
 +
The Milnor manifolds are constructed as follows: A few copies of the tubular neighbourhood $  E  ^ {4k} $
 +
of the diagonal in the product $  S  ^ {2k} \times S  ^ {2k} $
 +
are taken as blocks, while the graph $  T $
 +
is of the form
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031010a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031010a.gif" />
Line 31: Line 130:
 
Figure: d031010a
 
Figure: d031010a
  
Under these conditions the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101082.png" /> realizes a quadratic form of order eight, in which every element on the main diagonal equals 2, while the signature equals 8.
+
Under these conditions the manifold $  W  ^ {4k} ( T ) $
 +
realizes a quadratic form of order eight, in which every element on the main diagonal equals 2, while the signature equals 8.
  
In constructing the Kervaire manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101083.png" /> one takes two copies of the block obtained as the tubular neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101084.png" /> of the diagonal in the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101085.png" />. They are glued together so that the intersection matrix has the form
+
In constructing the Kervaire manifolds $  K  ^ {4k+} 2 $
 +
one takes two copies of the block obtained as the tubular neighbourhood $  E  ^ {4k+} 2 $
 +
of the diagonal in the product $  S  ^ {2k+} 1 \times S  ^ {2k+} 1 $.  
 +
They are glued together so that the intersection matrix has the form
  
<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/d/d031/d031010/d03101086.png" /></td> </tr></table>
+
$$
 +
\left \|
 +
\begin{array}{rr}
 +
0  & 1  \\
 +
- 1  & 0 \\
 +
\end{array}
 +
\right \| .
 +
$$
  
The boundary of a Milnor manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101087.png" /> (a Milnor sphere) is never diffeomorphic to the standard sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101088.png" />. As regards Kervaire manifolds, this problem has not yet (1987) been conclusively solved. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101089.png" />, then the boundary of a Kervaire manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101090.png" /> (a Kervaire sphere) is always non-standard; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101091.png" />, one obtains the standard sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101092.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101093.png" />, while for other <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101094.png" /> it remains unsolved (cf. [[Kervaire invariant|Kervaire invariant]]).
+
The boundary of a Milnor manifold $  \partial  M  ^ {4k} $(
 +
a Milnor sphere) is never diffeomorphic to the standard sphere $  S  ^ {4k-} 1 $.  
 +
As regards Kervaire manifolds, this problem has not yet (1987) been conclusively solved. If $  2k + 1 \neq 2  ^ {i} - 1 $,  
 +
then the boundary of a Kervaire manifold $  \partial  K  ^ {4k+} 2 $(
 +
a Kervaire sphere) is always non-standard; if $  2k + 1 = 2  ^ {i} - 1 $,  
 +
one obtains the standard sphere $  S  ^ {4k+} 1 $
 +
for $  1 \leq  i \leq  6 $,  
 +
while for other $  i $
 +
it remains unsolved (cf. [[Kervaire invariant|Kervaire invariant]]).
  
The Kervaire manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101095.png" /> of dimension 2, 6 or 14 are products of spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101097.png" /> respectively, after an open cell has been discarded, while all other Kervaire manifolds are not homeomorphic to the products of spheres with a discarded cell.
+
The Kervaire manifolds $  K  ^ {4k+} 2 $
 +
of dimension 2, 6 or 14 are products of spheres $  S  ^ {2k+} 1 \times S  ^ {2k+} 1 $,
 +
$  k = 0 , 1 , 3 $
 +
respectively, after an open cell has been discarded, while all other Kervaire manifolds are not homeomorphic to the products of spheres with a discarded cell.
  
The PL-manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d03101099.png" /> are often used in the topology of manifolds. These manifolds are obtained by adding a cone over the boundary of, respectively, the Milnor manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d031010100.png" /> and the Kervaire manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d031010101.png" />. In the theory of four-dimensional manifolds a certain simply-connected almost-parallelizable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d031010102.png" /> (usually called a Rokhlin manifold) plays an especially important role; its signature is 16, cf. [[#References|[6]]]. In the known examples of Rokhlin manifolds, the minimum value of the two-dimensional [[Betti number|Betti number]] is 22. The second manifold is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d031010103.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d031010104.png" /> is the graph indicated above, and the tubular neighbourhood of the diagonal in the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d031010105.png" /> is taken as the block. The boundary of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d031010106.png" /> thus obtained is a [[Dodecahedral space|dodecahedral space]] which is not simply connected.
+
The PL-manifolds $  {\widehat{M}  } {}  ^ {4k} $
 +
and $  {\widehat{K}  } {}  ^ {4k} $
 +
are often used in the topology of manifolds. These manifolds are obtained by adding a cone over the boundary of, respectively, the Milnor manifolds $  M  ^ {4k} $
 +
and the Kervaire manifolds $  K  ^ {4k+} 2 $.  
 +
In the theory of four-dimensional manifolds a certain simply-connected almost-parallelizable manifold $  W  ^ {4} $(
 +
usually called a Rokhlin manifold) plays an especially important role; its signature is 16, cf. [[#References|[6]]]. In the known examples of Rokhlin manifolds, the minimum value of the two-dimensional [[Betti number|Betti number]] is 22. The second manifold is $  W  ^ {4} ( \Gamma ) $,  
 +
where $  \Gamma $
 +
is the graph indicated above, and the tubular neighbourhood of the diagonal in the product $  S  ^ {2} \times S  ^ {2} $
 +
is taken as the block. The boundary of the manifold $  Q  ^ {3} = \partial  W  ^ {4} ( \Gamma ) $
 +
thus obtained is a [[Dodecahedral space|dodecahedral space]] which is not simply connected.
  
The three-dimensional dendritic manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031010/d031010107.png" /> belong to the class of so-called Seifert manifolds. Not all three-dimensional manifolds are dendritic manifolds; the [[Poincaré conjecture|Poincaré conjecture]] holds for dendritic manifolds. In particular, three-dimensional lens spaces (cf. [[Lens space|Lens space]]) are obtained by glueing two blocks only.
+
The three-dimensional dendritic manifolds $  M  ^ {3} = \partial  W  ^ {4} ( T) $
 +
belong to the class of so-called Seifert manifolds. Not all three-dimensional manifolds are dendritic manifolds; the [[Poincaré conjecture|Poincaré conjecture]] holds for dendritic manifolds. In particular, three-dimensional lens spaces (cf. [[Lens space|Lens space]]) are obtained by glueing two blocks only.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Kervaire, "A manifold which does not admit any differentiable structure" ''Comment. Math. Helv.'' , '''34''' (1960) pp. 257–270 {{MR|139172}} {{ZBL|0145.20304}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Kervaire, J. Milnor, "Groups of homotopy spheres. I" ''Ann. of Math.'' , '''77''' : 3 (1963) pp. 504–537 {{MR|0148075}} {{ZBL|0115.40505}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.W. Milnor, "Differential topology" , ''Lectures on modern mathematics'' , '''II''' , Wiley (1964) pp. 165–183 {{MR|0178474}} {{ZBL|0142.40803}} {{ZBL|0123.16201}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Hirzebruch, W.D. Neumann, S.S. Koh, "Differentiable manifolds and quadratic forms" , M. Dekker (1971) {{MR|0341499}} {{ZBL|0226.57001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> W.B. Browder, "Surgery on simply-connected manifolds" , Springer (1972) {{MR|0358813}} {{ZBL|0239.57016}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R. Mandelbaum, "Four-dimensional topology: an introduction" ''Bull. Amer. Math. Soc.'' , '''2''' : 1 (1980) pp. 1–159 {{MR|0551752}} {{ZBL|0476.57005}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Kervaire, "A manifold which does not admit any differentiable structure" ''Comment. Math. Helv.'' , '''34''' (1960) pp. 257–270 {{MR|139172}} {{ZBL|0145.20304}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Kervaire, J. Milnor, "Groups of homotopy spheres. I" ''Ann. of Math.'' , '''77''' : 3 (1963) pp. 504–537 {{MR|0148075}} {{ZBL|0115.40505}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.W. Milnor, "Differential topology" , ''Lectures on modern mathematics'' , '''II''' , Wiley (1964) pp. 165–183 {{MR|0178474}} {{ZBL|0142.40803}} {{ZBL|0123.16201}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Hirzebruch, W.D. Neumann, S.S. Koh, "Differentiable manifolds and quadratic forms" , M. Dekker (1971) {{MR|0341499}} {{ZBL|0226.57001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> W.B. Browder, "Surgery on simply-connected manifolds" , Springer (1972) {{MR|0358813}} {{ZBL|0239.57016}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R. Mandelbaum, "Four-dimensional topology: an introduction" ''Bull. Amer. Math. Soc.'' , '''2''' : 1 (1980) pp. 1–159 {{MR|0551752}} {{ZBL|0476.57005}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The technique described in the article above and leading to so-called "dendritic manifolds" (a phrase not often used in the West) is known as [[Surgery|surgery]], plumbing or as the technique of spherical modification.
 
The technique described in the article above and leading to so-called "dendritic manifolds" (a phrase not often used in the West) is known as [[Surgery|surgery]], plumbing or as the technique of spherical modification.

Revision as of 17:32, 5 June 2020


A smooth odd-dimensional manifold of a special type which is the boundary of an even-dimensional manifold constructed from fibrations over spheres by a glueing scheme specified by some graph (tree).

Let $ p _ {i} : E _ {i} ^ {2n} \rightarrow S _ {i} ^ {n} $, $ i = 1, 2 \dots $ be a fibration over $ n $- spheres with as fibre the $ n $- ball $ D ^ {n} $ and as structure group the group $ \mathop{\rm SO} _ {n} $, and let $ B _ {i} ^ {n} $ be the closed standard $ n $- ball in the $ n $- sphere $ S _ {i} ^ {n} $; then

$$ p _ {i} ^ {-} 1 ( B _ {i} ^ {n} ) \approx B _ {i} ^ {n} \times D _ {i} ^ {n} , $$

where $ D _ {i} ^ {n} $ is the fibre $ p _ {i} $. Let

$$ \gamma _ {ij} : B _ {i} ^ {n} \times D _ {i} ^ {n} \rightarrow B _ {j} ^ {n} \times D _ {j} ^ {n} ,\ i = 1, 2 , $$

be a homeomorphism realizing the glueing of two fibrations $ p _ {i} $, $ p _ {j} $ and mapping each $ n $- ball $ B ^ {n} \times x $ from $ B _ {i} ^ {n} \times D _ {i} ^ {n} $ into some ball $ y \times D ^ {n} $ from $ B _ {j} ^ {n} \times D _ {j} ^ {n} $( the glueing alters the factors of the direct product $ B ^ {n} \times D ^ {n} $). The result of glueing two fibrations $ p _ {i} $, $ p _ {j} $ is the $ 2n $- dimensional manifold $ E _ {i} ^ {2n} \cup _ {\gamma _ {ij} } E _ {j} ^ {2n} $ which, as a result of "angle smoothing" , is converted to a smooth manifold.

The fibrations $ E _ {i} ^ {2n} $ are considered as "structural blocks" from which it is possible to construct, by pairwise glueing, the resulting smooth manifold as follows. Let $ T $ be a one-dimensional finite complex (a graph). Each vertex of $ T $ is brought into correspondence with a block $ E _ {i} ^ {2n} $; next, $ k $, $ k = 1 , 2 \dots $ non-intersecting $ n $- balls $ B _ {i _ {k} } ^ {k} $ are selected in $ S _ {i} ^ {n} $, where $ k $ is equal to the branching index of the respective vertex, and the glueing is performed according to the scheme indicated by $ T $. The manifold with boundary thus obtained is denoted by $ W ^ {2n} ( T ) $( neglecting the dependence on the choice of $ E _ {i} ^ {2n} $). If $ T $ is a tree, and therefore the graph is without cycles, the boundary $ \partial W ^ {2n} ( T ) = M ^ {2n-} 1 $ is said to be a dendritic manifold.

If $ T $ is a tree, $ W ^ {2n} ( T ) $ has the homotopy type of a bouquet of $ k $ spheres, where $ k $ is the number of vertices of $ T $.

The dendritic manifold $ M ^ {2n-} 1 = \partial W ^ {2n} ( T ) $ is an integral homology $ ( 2n - 1 ) $- sphere if and only if the determinant of the matrix of the integral bilinear intersection $ ( - 1 ) ^ {n} $- form defined on the lattice of $ n $- dimensional homology groups $ H _ {n} ( W ^ {2n} , \mathbf Z ) $ equals $ \pm 1 $. If this condition is met, the manifold $ W ^ {2n} ( T ) $ is called a plumbing.

If $ T $ is a tree and $ n \geq 3 $, $ \partial W ^ {2n} ( T ) $ is simply connected; if $ W ^ {2n} $ is a plumbing, the boundary $ \partial W ^ {2n} $ is a homotopy sphere if $ n \geq 3 $.

If the plumbing $ W ^ {4k} $ is parallelizable, the diagonal of the intersection matrix of $ 2k $- dimensional cycles is occupied by even numbers; in such a case the signature of the intersection matrix is divisible by 8. The plumbing $ W ^ {4k} $ is parallelizable if and only if all the fibrations over $ S ^ {2k} $ used in constructing $ W ^ {4k} $ are stably trivial; e.g., if all fibrations used in constructing $ W ^ {4k} $ are tangent bundles on discs over $ 2k $- dimensional spheres, the plumbing $ W ^ {4k} $ is parallelizable. The plumbing $ W ^ {4k+} 2 $ will be parallelizable if and only if any fibration $ E _ {i} ^ {4k+} 2 $ used as a block in the construction of $ W ^ {4k+} 2 $ is either trivial or is a tubular neighbourhood of the diagonal in the product $ S ^ {2k+} 1 \times S ^ {2k+} 1 $, i.e. is a tangent bundle on discs over $ S ^ {2k+} 1 $. If the plumbing $ W ^ {4k+} 2 $ is parallelizable, its intersection matrix can be reduced to the symplectic form consisting of blocks

$$ \left \| \begin{array}{rr} 0 & 1 \\ - 1 & 0 \\ \end{array} \right \| $$

situated along the main diagonal.

Especially important plumbings are the Milnor manifolds of dimension $ 4k $, $ k > 1 $, and the Kervaire manifolds of dimension $ 4k + 2 $, $ k \geq 0 $. The Milnor manifolds are constructed as follows: A few copies of the tubular neighbourhood $ E ^ {4k} $ of the diagonal in the product $ S ^ {2k} \times S ^ {2k} $ are taken as blocks, while the graph $ T $ is of the form

Figure: d031010a

Under these conditions the manifold $ W ^ {4k} ( T ) $ realizes a quadratic form of order eight, in which every element on the main diagonal equals 2, while the signature equals 8.

In constructing the Kervaire manifolds $ K ^ {4k+} 2 $ one takes two copies of the block obtained as the tubular neighbourhood $ E ^ {4k+} 2 $ of the diagonal in the product $ S ^ {2k+} 1 \times S ^ {2k+} 1 $. They are glued together so that the intersection matrix has the form

$$ \left \| \begin{array}{rr} 0 & 1 \\ - 1 & 0 \\ \end{array} \right \| . $$

The boundary of a Milnor manifold $ \partial M ^ {4k} $( a Milnor sphere) is never diffeomorphic to the standard sphere $ S ^ {4k-} 1 $. As regards Kervaire manifolds, this problem has not yet (1987) been conclusively solved. If $ 2k + 1 \neq 2 ^ {i} - 1 $, then the boundary of a Kervaire manifold $ \partial K ^ {4k+} 2 $( a Kervaire sphere) is always non-standard; if $ 2k + 1 = 2 ^ {i} - 1 $, one obtains the standard sphere $ S ^ {4k+} 1 $ for $ 1 \leq i \leq 6 $, while for other $ i $ it remains unsolved (cf. Kervaire invariant).

The Kervaire manifolds $ K ^ {4k+} 2 $ of dimension 2, 6 or 14 are products of spheres $ S ^ {2k+} 1 \times S ^ {2k+} 1 $, $ k = 0 , 1 , 3 $ respectively, after an open cell has been discarded, while all other Kervaire manifolds are not homeomorphic to the products of spheres with a discarded cell.

The PL-manifolds $ {\widehat{M} } {} ^ {4k} $ and $ {\widehat{K} } {} ^ {4k} $ are often used in the topology of manifolds. These manifolds are obtained by adding a cone over the boundary of, respectively, the Milnor manifolds $ M ^ {4k} $ and the Kervaire manifolds $ K ^ {4k+} 2 $. In the theory of four-dimensional manifolds a certain simply-connected almost-parallelizable manifold $ W ^ {4} $( usually called a Rokhlin manifold) plays an especially important role; its signature is 16, cf. [6]. In the known examples of Rokhlin manifolds, the minimum value of the two-dimensional Betti number is 22. The second manifold is $ W ^ {4} ( \Gamma ) $, where $ \Gamma $ is the graph indicated above, and the tubular neighbourhood of the diagonal in the product $ S ^ {2} \times S ^ {2} $ is taken as the block. The boundary of the manifold $ Q ^ {3} = \partial W ^ {4} ( \Gamma ) $ thus obtained is a dodecahedral space which is not simply connected.

The three-dimensional dendritic manifolds $ M ^ {3} = \partial W ^ {4} ( T) $ belong to the class of so-called Seifert manifolds. Not all three-dimensional manifolds are dendritic manifolds; the Poincaré conjecture holds for dendritic manifolds. In particular, three-dimensional lens spaces (cf. Lens space) are obtained by glueing two blocks only.

References

[1] M. Kervaire, "A manifold which does not admit any differentiable structure" Comment. Math. Helv. , 34 (1960) pp. 257–270 MR139172 Zbl 0145.20304
[2] M. Kervaire, J. Milnor, "Groups of homotopy spheres. I" Ann. of Math. , 77 : 3 (1963) pp. 504–537 MR0148075 Zbl 0115.40505
[3] J.W. Milnor, "Differential topology" , Lectures on modern mathematics , II , Wiley (1964) pp. 165–183 MR0178474 Zbl 0142.40803 Zbl 0123.16201
[4] F. Hirzebruch, W.D. Neumann, S.S. Koh, "Differentiable manifolds and quadratic forms" , M. Dekker (1971) MR0341499 Zbl 0226.57001
[5] W.B. Browder, "Surgery on simply-connected manifolds" , Springer (1972) MR0358813 Zbl 0239.57016
[6] R. Mandelbaum, "Four-dimensional topology: an introduction" Bull. Amer. Math. Soc. , 2 : 1 (1980) pp. 1–159 MR0551752 Zbl 0476.57005

Comments

The technique described in the article above and leading to so-called "dendritic manifolds" (a phrase not often used in the West) is known as surgery, plumbing or as the technique of spherical modification.

How to Cite This Entry:
Dendritic manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dendritic_manifold&oldid=46624
This article was adapted from an original article by M.A. Shtan'ko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article