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Homotopy Theory: Fundamental Group and Higher Homotopy Groups

In: Basic Topology 3

Author

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  • Mahima Ranjan Adhikari

    (Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC))

Abstract

This chapter officially inaugurates homotopy theory to begin a study of algebraic topology by conveying the basic concepts of homotopy and fundamental groups born through the work of H. Poincaré (1854–1912) in his land-marking ‘Analysis Situs,’ Paris, 1895, and also discusses higher homotopy groups constructed in 1935 by H. Hurewicz (1904–1956) in his paper [Hurewicz, 1935], which are natural generalizations of fundamental groups. Homotopy theory studies those properties of topological spaces and continuous maps which are invariants under homotopic maps, called homotopy invariants. Finally, this chapter presents some interesting applications of homotopy, fundamental and higher homotopy groups in analysis, geometry, algebra, matrix theory, atmospheric science, vector field and extension problems and some others.

Suggested Citation

  • Mahima Ranjan Adhikari, 2022. "Homotopy Theory: Fundamental Group and Higher Homotopy Groups," Springer Books, in: Basic Topology 3, chapter 0, pages 27-161, Springer.
    • Handle: RePEc:spr:sprchp:978-981-16-6550-9_2
    DOI: 10.1007/978-981-16-6550-9_2
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