surfacepreserving

Surfacepreserving is a term used in mathematics, particularly in topology and geometry, to describe a transformation, embedding, or automorphism that preserves a specified surface within a space. Concretely, let M be a manifold and S ⊂ M a surface. A map f: M → M is surface-preserving if f(S) = S. Depending on the setting, one might require f to fix S setwise, fix S pointwise, preserve orientation on S, or preserve the boundary ∂M if S = ∂M.

Variations of the concept arise from choosing the category (homeomorphisms, diffeomorphisms, or more general embeddings) and the level of preservation (setwise versus pointwise, orientation-preserving, etc.). The collection of surface-preserving maps forms a subgroup of the appropriate mapping class group or automorphism group, consisting of all maps that send the surface S to itself. In many contexts, one also considers maps that preserve not only S but its embedding class or its collision with other substructures.

Examples include the identity map, which is trivially surface-preserving, and any homeomorphism of M that sends S onto itself. In the study of 3-manifolds and Heegaard splittings, surface-preserving automorphisms play a role in understanding how a chosen Heegaard surface can be carried into itself by ambient symmetries.

See also: mapping class group, diffeomorphism, isotopy, Dehn twist, Heegaard splitting, boundary. Note that the phrase surfacepreserving is not universally standardized and may be used with varying emphasis on setwise versus pointwise preservation.