English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let Imm[X,Y ] be the set of regular homotopy classes of immersions of a manifold X in a manifold Y , and Emb[X,Y ] denote the subset of Imm[X,Y ] consisting of all regular homotopy classes containing an embedding. Smale [6] has given a 1-1 correspondence (the Smale invariant) s : Imm[Sn,RN ]→ πn(VN,n), where VN,n is the Stiefel manifold of all n-frames in RN . Hirsch [2] has generalized this to the case of immersions of an arbitrary manifold in an arbitrary manifold. These results solve the problem of the number of regular homotopy classes in terms of homotopy theory, but do not succeed in finding representatives for each class or determining which classes are represented by an embedding. According to Hughes [4], Imm[Sn,RN ] has a group structure under connected sum and the Smale invariant actually gives a group isomorphism. [4] gives explicit generators of Imm[S3,R4] and Imm[S3,R5]. Hughes-Melvin [5] determine which classes of Imm[Sn,Rn+2] are represented by an embedding, and prove that Emb[Sn,Rn+2] is isomorphic to Z if n ≡ 3 mod 4, and to 0 otherwise. Furthermore, [5] proves that the regular homotopy class of an embedding Sn ↪→ Rn+2(n ≡ 3 mod 4) can be completely determined by the signature of its oriented “Seifert” manifold. For example, in the case n = 3, we have the following diagram:

Embeddings of Z2-homology 3-spheres in R5up to regular homotopy

Embeddings of Z₂-homology 3-spheres in R⁵up to regular homotopy