Atlas: Classification of Homotopy Types of Embedding Spaces into 2-Manifolds by Tatsuhiko Yagasaki

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Geometric Topology II
September 29 - October 5, 2002
Inter-University Center, Dubrovnik; Department of Mathematics, University of Zagreb
Dubrovnik, Croatia

Organizers
Ivan Ivansic, University of Zagreb;, James E. Keesling, University of Florida;, Alexander N. Dranishnikov, University of Florida;, Sime Ungar, University of Zagreb

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Classification of Homotopy Types of Embedding Spaces into 2-Manifolds
by
Tatsuhiko Yagasaki
Kyoto Institute of Technology

This talk gives a complete description of the homotopy types of the connected components of spaces of embeddings of compact polyhedra into 2-manifolds.

Suppose M is a connected 2-manifold and X is a compact connected subpolyhedron of M (X is not 1 pt, a closed 2-manifold).

Let E(X, M) denote the space of topological embeddings of X into M with the compact-open topology and E(X, M)_0 denote the connected component of Inclusion i_X of X into M in E(X, M).

Homotopy type of E(X, M)_0 can be classified in term of Subgroup G = Im[(i_X)_* : pi_1(X) --> pi_1(M)].

Main statements are: (1) If G is not a cyclic group and M not = T, K, then E(X, M)_0 simeq 1 pt. (2) If G is a nontrivial cyclic group and M is not P, T, K, then E(X, M)_0 simeq S^1.

(3) In Case G = 1, (i) if X is an arc or M is orientable, then E(X, M)_0 simeq ST(M), (ii) if X is not an arc and M is nonorientable, then E(X, M)_0 simeq ST(M~). Here S^1 is Circle, T is Torus, P is Projective plane and K is Klein bottle. ST(M) denotes Tangent unit circle bundle of M with respect to any Riemannian metric of M and M~ denotes Orientation double cover of M.

Date received: August 15, 2002