ABSTRACT. Let K be a proper rectilinear triangulation of a 2-simplex S in the plane and L(K) be the space of all homeomorphisms of S which are linear on each simplex of K and are fixed on Bd(S). The author shows in this paper that L(K) with the compact open topology is simply-connected. This is a generalization of a result of S. S. Cairns in 1944 that the space L(K) is pathwise connected. Both results will be used in Part II of this paper to show that nQ(L 2) = nfL2) = 0 where L is a space of p.l. homeomorphisms of an zz-simplex, a space introduced by R. Thorn in his study of the smoothings of combinatorial manifolds. In his study of the smoothings of Brouwer manifolds, S. S. Cairns considered a space of triangulations of a 2-simplex. He showed that two isomorphic (rectilinear) triangulations of a 2-simplex S, each having only three vertices on Bd(S), may be deformed continuously onto each other LlJ, [2j. To be more precise, we shall let 5 be a fixed 2-simplex in the Euclidean plane. For a simplicial subdivision K of S, let L(K) be the space of all homeomorphisms from S onto S which are linear on each simplex of K and are pointw,ise fixed on Bd(s). We shall give LÍK) the compact open topology. Furthermore, a simplicial subdivision K of S is called a proper subdivision if K has only three vertices on Bd ÍS). Cairns ' theorem may then be stated as follows: If K is a proper subdivision of S, the space L(K) is path-connected. In this part of the paper, we shall prove the following. Main theorem. For any proper subdivision K of S, the space LÍK) is simply connected. Using this theorem, we shall show in Part II  that 77n(L2) = 77j(L2) = 0 where L is a space of p.l. homeomorphisms on an zz-simplex, the space introduced by R. Thorn  and studied by N. H. Kuiper  in connection with th
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