Multiplicity of a space over another space. (English) Zbl 1276.18010

Summary: We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another. Based on this multiplicity, we define a pseudo distance on the class of objects. We define and study several multiplicities in the category of topological spaces and continuous maps, the category of groups and homomorphisms, the category of finitely generated \(R\)-modules and \(R\)-linear maps over a principal ideal domain \(R\), and the neighbourhood category of oriented knots in the 3-sphere.


18D99 Categorical structures
13C05 Structure, classification theorems for modules and ideals in commutative rings
20E99 Structure and classification of infinite or finite groups
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M99 General low-dimensional topology
Full Text: DOI arXiv Euclid


[1] W. Adkins and S. Weintraub, Algebra, An Approach via Module Theory, Grad. Texts in Math., 136 , Springer-Verlag, 1992. · Zbl 0768.00003
[2] S. Bogatyi, J. Fricke and E. Kudryavtseva, On multiplicity of mappings between surfaces, Geom. Topol. Monogr., 14 (2008), 49-62. · Zbl 1145.55001 · doi:10.2140/gtm.2008.14.49
[3] M. Gromov, Singularities, expanders and topology of maps, Part 2: From combinatorics to topology via algebraic isoperimetry, Geom. Funct. Anal., 20 (2010), 416-526. · Zbl 1251.05039 · doi:10.1007/s00039-010-0073-8
[4] K. Kanno and K. Taniyama, Braid presentation of spatial graphs, Tokyo J. Math., 33 (2010), 509-522. · Zbl 1210.57008 · doi:10.3836/tjm/1296483485
[5] R. N. Karasev, Multiplicity of continuous maps between manifolds, preprint, · Zbl 1153.52002
[6] R. Nikkuni, Private communication, 2010.
[7] M. Ozawa, Waist and trunk of knots, Geom. Dedicata, 149 (2010), 85-94. · Zbl 1220.57002 · doi:10.1007/s10711-010-9466-y
[8] D. Rolfsen, Knots and Links, Mathematics Lecture Series, 7 , Publish or Perish, Inc., Berkeley, 1976. · Zbl 0339.55004
[9] K. Taniyama, Multiplicity of a space over another space, Proceedings of Intelligence of Low Dimensional Topology, 2007, pp.,157-161.
[10] K. Taniyama, Multiplicity of a space over another space, (in Japanese), Gakujutsu Kenkyu, School of Education, Waseda University, Series of Mathematics, 56 (2008), 1-4. · Zbl 1276.18010
[11] K. Taniyama, Multiplicity distance of knots, In: Intelligence of Low-dimensional Topology, Sūrikaisekikenkyūsho Kôkyurôku, 1716 , Res. Inst. Math. Sci., Kyoto, 2010, pp.,37-42.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.