Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX TOPOLOGY OF SOBOLEV MAPPINGS IV Fengbo Hang Department of Mathematics Michigan State University East Lansing, MI 48824 Fanghua Lin Courant Institute New York University 251 Mercer Street New York, NY 10012 (Communicated by Aim Sciences) Abstract. We will classify the path connected components of spaces of Sobolev maps between manifolds and study the strong and weak density of smooth maps in the spaces of Sobolev maps for the case the domain manifold has nonempty boundary and Dirichlet problems. 1. Introduction. This is a sequel to [6, 7, 8]. The main aim is to classify the path connected components of spaces of Sobolev maps between manifolds and study the density of smooth maps in the space of Sobolev maps similar to those in [7] for the case domain manifold has boundary and Dirichlet problems. We will emphasize those analytical techniques different from the no boundary cases. These techniques have their own interest in variational problems for maps between manifolds. 2. Some preparations. Let M be a smooth compact Riemannian manifold. At the beginning of section 2 of [7], we used Lipschitz triangulation to mean that one has a simplicial complex K and a bi-Lipschitz map h : |K|→ M . Sometime the name “Lipschitz triangulation” can be confused as “a triangulation which is Lipschitz”. To avoid this, we will use the name “bi-Lipschitz triangulation” instead. Similar conventions apply to bi-Lipschitz cubeulation and bi-Lipschitz rectilinear cell decomposition. For basics about rectilinear cell complex and simplicial complex, we refer readers to [17] (appendix II) and [10]. For basics of homotopy theory, we refer readers to [16]. Let (X, A) be a pair of topological spaces, that is, X be a topological space and A be a subspace of X, Y be a topological space and f ∈ C (A, Y ), we write C f (X, Y )= {u ∈ C (X, Y ): u| A = f } , [X, Y ; f ] rel.A = C f (X, Y ) / ∼ rel.A . 2000 Mathematics Subject Classification. Primary: 58D15, 46T10; Secondary: 41A29. Key words and phrases. Density of smooth maps, path connected components, Dirichlet bound- ary conditions, k-homotopy classes, obstruction theory. The research of the first author is supported by National Science Foundation Grant DMS– 0209504. The research of the second author is supported by National Science Foundation Grant DMS–0201443. 1