There are two relevant equivalence relations on the set of immersions from a manifold \(M^ m\) into Euclidean space \({\mathbb{R}}^ n:\) regular homotopy and a second coming from the action of diffeomorphisms of M on it. Imposing both relations defines the term ”immersed manifold”. This paper achieves a nice geometrical classification of immersions of a closed surface into \({\mathbb{R}}^ 3\), as well as of immersed surfaces.
rmmersions are distinguished by a \({\mathbb{Z}}/4\)-valued (linking number) quadratic form on \(H_ 1(M; {\mathbb{Z}}/2)\). The action of the diffeomorphism group of M is reflected by that of the automorphism group of the form. Quadratic forms are classified by dimension and a \({\mathbb{Z}}/8\)-valued Arf invariant. The outcome is a description of the monoid (by connected sum) of immersed surfaces by generators (torus, ”twisted torus”, Boy surface and its ”mirror image”) and relations. Furthermore it is shown that bordism of immersed surfaces is completely classified by the Arf invariant.