Formulae for order one invariants of immersions of surfaces

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The universal order 1 invariant fU of immersions of a closed orientable surface into R3, whose existence has been established in [T. Nowik, Order one invariants of immersions of surfaces into 3-space, Math. Ann. 328 (2004) 261-283], is the direct sumfU = under(⊕, n ∈ Z) fnH ⊕ under(⊕, n ∈ Z) fnT ⊕ M ⊕ Q where each fnH, fnT is a Z valued invariant and M, Q are Z / 2 valued invariants. An explicit formula for the value of Q on any embedding has been given in [T. Nowik, Automorphisms and embeddings of surfaces and quadruple points of regular homotopies, J. Differential Geom. 58 (2001) 421-455]. In the present work we give explicit formulae for the value of each fnH, fnT on all immersions, and for the value of M on any embedding.

Original languageEnglish
Pages (from-to)358-372
Number of pages15
JournalAdvances in Mathematics
Issue number2
StatePublished - 10 Nov 2006

Bibliographical note

Funding Information:
E-mail address: URL: 1 Partially supported by the Minerva Foundation.


  • Finite order invariants
  • Immersions of surfaces


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