RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find

 Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 5, Pages 3–62 (Mi izv2681)

The construction of combinatorial manifolds with prescribed sets of links of vertices

A. A. Gaifullin

M. V. Lomonosov Moscow State University

Abstract: To every oriented closed combinatorial manifold we assign the set (with repetitions) of isomorphism classes of links of its vertices. The resulting transformation $\mathcal{L}$ is the main object of study in this paper. We pose an inversion problem for $\mathcal{L}$ and show that this problem is closely related to Steenrod's problem on the realization of cycles and to the Rokhlin–Schwartz–Thom construction of combinatorial Pontryagin classes. We obtain a necessary condition for a set of isomorphism classes of combinatorial spheres to belong to the image of $\mathcal{L}$. (Sets satisfying this condition are said to be balanced.) We give an explicit construction showing that every balanced set of isomorphism classes of combinatorial spheres falls into the image of $\mathcal{L}$ after passing to a multiple set and adding several pairs of the form $(Z,-Z)$, where $-Z$ is the sphere $Z$ with the orientation reversed. Given any singular simplicial cycle $\xi$ of a space $X$, this construction enables us to find explicitly a combinatorial manifold $M$ and a map $\varphi\colon M\to X$ such that $\varphi_*[M]=r[\xi]$ for some positive integer $r$. The construction is based on resolving singularities of $\xi$. We give applications of the main construction to cobordisms of manifolds with singularities and cobordisms of simple cells. In particular, we prove that every rational additive invariant of cobordisms of manifolds with singularities admits a local formula. Another application is the construction of explicit (though inefficient) local combinatorial formulae for polynomials in the rational Pontryagin classes of combinatorial manifolds.

DOI: https://doi.org/10.4213/im2681

Full text: PDF file (960 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2008, 72:5, 845–899

Bibliographic databases:

UDC: 515.164.3
MSC: 52B70, 57R95, 55R40

Citation: A. A. Gaifullin, “The construction of combinatorial manifolds with prescribed sets of links of vertices”, Izv. RAN. Ser. Mat., 72:5 (2008), 3–62; Izv. Math., 72:5 (2008), 845–899

Citation in format AMSBIB
\Bibitem{Gai08} \by A.~A.~Gaifullin \paper The construction of combinatorial manifolds with prescribed sets of links of vertices \jour Izv. RAN. Ser. Mat. \yr 2008 \vol 72 \issue 5 \pages 3--62 \mathnet{http://mi.mathnet.ru/izv2681} \crossref{https://doi.org/10.4213/im2681} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2473771} \zmath{https://zbmath.org/?q=an:1156.52009} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2008IzMat..72..845G} \elib{http://elibrary.ru/item.asp?id=20358650} \transl \jour Izv. Math. \yr 2008 \vol 72 \issue 5 \pages 845--899 \crossref{https://doi.org/10.1070/IM2008v072n05ABEH002422} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000261096200001} \elib{http://elibrary.ru/item.asp?id=13595512} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-56849122703} 

• http://mi.mathnet.ru/eng/izv2681
• https://doi.org/10.4213/im2681
• http://mi.mathnet.ru/eng/izv/v72/i5/p3

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Alexander A. Gaifullin, “Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class”, Proc. Steklov Inst. Math., 268 (2010), 70–86
2. A. A. Aizenberg, V. M. Buchstaber, “Nerve complexes and moment–angle spaces of convex polytopes”, Proc. Steklov Inst. Math., 275 (2011), 15–46
3. Alexander Gaifullin, “Universal realisators for homology classes”, Geom. Topol, 17:3 (2013), 1745
4. A. A. Gaifullin, “Small covers of graph-associahedra and realization of cycles”, Sb. Math., 207:11 (2016), 1537–1561
•  Number of views: This page: 687 Full text: 174 References: 43 First page: 18