If X is a topological space, let Dx denote the subset of XXX consisting of the set of all points of the form (x, x), where xoCX. Then the deleted product space, X*, of X is the space X XX -Dx with the relative topology. It follows from a theorem of Eilenberg (see [1, p. 43]) that for a connected, finite, 1-dimensional polyhedron X, X* is arcwise connected if and only if X is not an arc. In this paper we prove the following theorem: If X is a connected, finite, 1-dimensional polyhedron which is not an arc, then Ik(X*) = O for all k > 1. DEFINITION 1. If X is a connected, finite, 1-dimensional polyhedron and A and B are subpolyhedra of X, let P(A XB -Dx) -U{rxslr is a simplex of A, s is a simplex of B, and rflss=0}. REMARK 1. Let X be a connected, finite, 1-dimensional polyhedron which is not an arc. If X does not have a vertex of order > 3, then X is a simple closed curve. If X does have a vertex of order > 3, let A' be a triod in X. Then it is clear that there is a subdivision X' of X such that: (1) each simplex of A' is a simplex of X', (2) X'consists of a finite number of 1-simplexes, ri, * * *, rk, and (3) X' can be realized by starting with A' and adding one 1-simplex rj at a time so that either rjrl(UJrk) is a single vertex or rjfr(UJ1 rk) consists of two vertices vi, v2, where each vi (i = 1, 2) is a vertex of order one in U`:l rk. In this paper, we shall assume that such a subdivision of X has been made. Definition 1 and Remark 1 may be found in [3]. It is shown in [4] that if X is a connected, finite, 1-dimensional polyhedron, then there is a deformation retraction of X* onto P(X*). DEFINITION 2. A space X is said to be aspheric if llk(X) = 0 for all k>1.