2002/11/06 by Tatsuhiko Yagasaki, Yagasaki, Tatsuhiko
Mathematics · #57N05 #57N20 #57N35 #Algebraic structures and combinatorial models #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GN #math.GT #msc:57N05 #msc:57N20 #msc:57N35
paper · pdf · doi:10.48550/arxiv.math/0211104
31 pages, 3 figures
arxiv created 2002/11/06 · openalex publication_date 2002/11/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose M is a connected PL 2-manifold and X is a compact connected subpolyhedron of M (X ≠ 1pt, a closed 2-manifold). Let E(X, M) denote the space of topological embeddings of X into M with the compact-open topology and let E(X, M)0 denote the connected component of the inclusion iX : X ⊂ M in E(X, M). In this paper we classify the homotopy type of E(X, M)0 in term of the subgroup G = Im[iX_∗ : π1(X) → π1(M)]. We show that if G is not a cyclic group and M ≠ T2, T2 then E(X, M)0 ≃ ∗, if G is a nontrivial cyclic group and M ≠ P2, T2, K2 then E(X, M)0 ≃ S1, and when G = 1, if X is an arc or M is orientable then E(X, M)0 ≃ ST(M) and if X is not an arc and M is nonorientable then E(X, M)0 ≃ ST(M). Here S1 is the circle, T2 is the torus, P2 is the projective plane and K2 is the Klein bottle. The symbol ST(M) denotes the tangent unit circle bundle of M with respect to any Riemannian metric of M and M denotes the orientation double cover of M.