Journal of the Mathematical Society of Japan

Multiplicity of a space over another space


Full-text: Open access


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.

Article information

J. Math. Soc. Japan, Volume 64, Number 3 (2012), 823-849.

First available in Project Euclid: 24 July 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D99: None of the above, but in this section
Secondary: 13C05: Structure, classification theorems 20E99: None of the above, but in this section 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M99: None of the above, but in this section

multiplicity category topological space group module knot


TANIYAMA, Kouki. Multiplicity of a space over another space. J. Math. Soc. Japan 64 (2012), no. 3, 823--849. doi:10.2969/jmsj/06430823.

Export citation


  • W. Adkins and S. Weintraub, Algebra, An Approach via Module Theory, Grad. Texts in Math., 136, Springer-Verlag, 1992.
  • S. Bogatyi, J. Fricke and E. Kudryavtseva, On multiplicity of mappings between surfaces, Geom. Topol. Monogr., 14 (2008), 49–62.
  • M. Gromov, Singularities, expanders and topology of maps, Part 2: From combinatorics to topology via algebraic isoperimetry, Geom. Funct. Anal., 20 (2010), 416–526.
  • K. Kanno and K. Taniyama, Braid presentation of spatial graphs, Tokyo J. Math., 33 (2010), 509–522.
  • R. N. Karasev, Multiplicity of continuous maps between manifolds, preprint, arXiv: 1002.0660
  • R. Nikkuni, Private communication, 2010.
  • M. Ozawa, Waist and trunk of knots, Geom. Dedicata, 149 (2010), 85–94.
  • D. Rolfsen, Knots and Links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Berkeley, 1976.
  • K. Taniyama, Multiplicity of a space over another space, Proceedings of Intelligence of Low Dimensional Topology, 2007, pp.,157–161.
  • 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.
  • K. Taniyama, Multiplicity distance of knots, In: Intelligence of Low-dimensional Topology, Sūrikaisekikenkyūsho Kôkyurôku, 1716, Res. Inst. Math. Sci., Kyoto, 2010, pp.,37–42.