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 $p _ {i} : E _ {i} ^ {2n} \rightarrow S _ {i} ^ {n}$, $i = 1, 2 \dots$ 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