Isotopy classes of imbeddings

Abstract

Hu [3] has shown that X* and P(X*) are homotopically equivalent. In [4], the author has shown that if Y is a triod, then P( Y*) is a circle, and that up to homeomorphism the triod is the only tree (finite, contractible, 1-dimensional polyhedron) with this property. It is also shown in [4] that if X is a tree, then H1(X*, Z) where Z is the integers, is a free abelian group. T. R. Brahana suggested to the author that if X is a tree, then there might be a connection between the number of generators of H1(X*, Z) and the number of isotopy classes of imbeddings of the triod in X and that we might be able to extend this to higher dimensions. In ?2, we obtain a formula for computing the number of isotopy classes of imbeddings of the triod in a tree and show that there is a definite relation between this number and the 1-dimensional Betti number of the deleted product of the tree. We show that up to homeomorphism there are at least two finite, contractible, 2-dimensional polyhedra, C and 0, which have the property that P(C*) and P(O*) are homeomorphic to the 2-sphere. There is at least one more finite, contractible, 2-dimensional polyhedron A whose deleted product has the homotopy type of the 2-sphere. However C can be imbedded in both 0 and A, and in ?4, we prove a collection of theorems which give a combinatorial method for computing the number of isotopy classes of imbeddings of C in a finite, contractible, 2-dimensional polyhedron. The connection between the number of isotopy classes of imbeddings of C in a finite, contractible, 2-dimensional polyhedron X and the 2-dimensional Betti number of the deleted product of X is to be investigated in a forthcoming paper.