Singular and Simplicial Homology Groups


  • Hisham Ali Ahmed Ghanim
  • Mohammed Ali Bashir Hussein Department of Mathematics, Faculty of Mathematical Science and Statistics, Al Neelain University, Khartoum, Sudan
  • Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman Department of Mathematics, Faculty of Education, Omdurman Islamic University, Omdurman, Sudan



Boundary operator, Simplicial complex, Singular homology


     This paper identified the homology groups  of -simplex, -dimensional simplicial complex and -complex. We show that a -simplex  is a point or a vertex, -simplex  is a line or an edge, a -simplex  is defined to be a triangle with its interior included and a -simplex  is a solid tetrahedron. It is easy to continue to any -simplex . The aims of this paper are to construct and defined Singular Homology Groups and Simplicial Homology Groups in terms of -dimensional simplicial complex  and a topological space . We followed the historical analysis mathematical method. We found that simplexes are building blocks of polyhedron, both approaches yield the same results, and the homology groups  are quotient groups of -cycle groups  and -boundary groups .


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Allan J. Sieradski, An Introduction to Topology and Homotopy, PWS-Kent Publishing Company, Boston, 1992, 63-94.

Allen Hatcher, Algebraic Topology, Cambridge University Press, USA, 2002, 103-111.

Andrew H. Wallance, An Introduction to Algebraic Topology, Pergmon Press, New York, 2016, 175-183.

B. K. Lahiri, A First Course in Algebraic Topology, Alpha Science International Ltd, UK, 2000, 11-106.

D. Chatterjee, Topology. General and Algebraic, New Age International (p) Ltd, India, 2007, 146.

Glen E. Bredon, Topology and Geometry, Springer-Verlag, New York, Inc., 1993, 173.

James R. Munkres, Elements of Algebraic Topology, The Benjamin/ Cumming Publishing Company, Inc., California, 1984, 38-39.

James W. Vick, Homology Theory An Introduction to Algebraic Topology, 2end ed, Springer-Verlag, USA, 1994, 8-17.

M. S. Narasimhan - S. Ramanan - R. Sridharan - K. Varadarajan, Algebraic Topology, Tata Institute of Fundamental Research, Bombay, 2011, 29.

Mahima Ranjan Adhikari, Basic Algebraic Topology and Its Applications, Springer, India, 2016, 348.

Mikio Nakahara, Geometry. Topology and Physics, 2end ed, IOP Publishing Ltd, London, 2003, 118-125.

Steven H. Weintraub, Fundamentals of Algebraic Topology, Springer International Publishing, Switzerland, 2014, 2-57.

Tammo Tom Dieck, Algebraic Topology, European Mathematical Society Publishing House, Germany, 2008, 223-224.

V. V. Prasolov, Elements of Homology Theory, Volume 81, American Mathematical Society, USA, 2007, 11.

William Fulton, Algebraic Topology A First Course, Springer-Verlag, New York, 1995, 93.



How to Cite

Ghanim, H. A. A. ., Bashir Hussein, M. A. ., & Abdel Rahman, A. R. A. R. A. G. . . (2021). Singular and Simplicial Homology Groups. EPH - International Journal of Mathematics and Statistics, 7(8), 01–10.