View source for Dendritic manifold

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

<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>

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

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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.

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.

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" />.

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.

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 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

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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

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031010a.gif" />

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.

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

<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>

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 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 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 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.

====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>



====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.