For a finite geometric left regular band B, we will use the following special case of Rota's cross-cut theorem [26, 6] to provide a simplicial complex homotopy equivalent to the order complex [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of B\{1}.

Then K is homotopy equivalent to the order complex [DELTA](P).

1 would therefore imply that, for such [sigma] and [tau], intervals [1, [tau]] and [[sigma], [tau]] are each either contractible or

homotopy equivalent to a single sphere (of dimension [absolute value of [tau]] - 3 and [absolute value of [tau]] - [absolute value of [sigma]] - 2, respectively).

However, it is still open whether MacP(k, n) is

homotopy equivalent to Gr(k, n).

0]) whose proper part is homotopy equivalent to [sub.

0]) of a rank d arrangement A with base chamber c0 is homotopy equivalent to [S.

The complex [DELTA] is called shellable if there exists a shelling order on its facets; it can be shown that any pure d-dimensional shellable simplicial complex is homotopy equivalent to a wedge of spheres, all of dimension d.

Moreover, the complex [DELTA] is homotopy equivalent to a wedge of [h.

Furthermore, the order complex of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is homotopy equivalent to a wedge of spheres of dimension k - 2.

We now will use Quillen's fiber lemma to show that the chain complex [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is homotopy equivalent to the simplicial complex [[DELTA].

For a d-dimensional simplicial complex we have the following implications: shellable [right arrow] constructible [right arrow] homotopy Cohen-Macaulay [right arrow]

homotopy equivalent to a wedge of d-dimensional spheres.

n]) \ {e} is

homotopy equivalent to a wedge of (n - 2)-dimensional spheres and Cohen-Macaulay over Z.