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}.

Keywords: Asymmetric reaction, nonhomogeneous differential opertator, C-condition, critical groups,

homotopy equivalent, mountain pass theorem.

Hence by Lemma 5.7 we obtain a pointed G-action on a pointed space [Y.sub.*]

homotopy equivalent to X.

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

homotopy equivalent to Gr(k, n).

A Stein manifold is

homotopy equivalent to a finite dimensional CW-complex and when this complex is finite then [E.sub.n](C(X)) = [bar.[E.sub.n](C(X))].

In this situation, we prove [bar.R](A, [r.sub.0]) is

homotopy equivalent to a (rk A - 3)-sphere.

If G is finite then, rationally, it is

homotopy equivalent to a product of Eilenberg-MacLane spaces as an H-space, implying that it is homotopy commutative.

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.

In fact, B C(E) is

homotopy equivalent to a functor in the variable E [member of] [A.sub.p](G).

Furthermore, the order complex of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is

homotopy equivalent to a wedge of spheres of dimension k - 2.

The naive converse of this result is not true; namely there are spaces X and Y that are not

homotopy equivalent but have isomorphic homotopy groups.