Some Topics from the Topology of ℝn


As we have already stated, topology is the study of the properties shared by a geometric figure and all its bi-continuous transformations, i.e., the study of invariants by homeomorphisms. Its origin dates back to the problem of Königsberg bridges and Euler’s theorem about polyhedra, to Riemann’s work on the geometric representation of functions, to Betti’s work on the notion of multiconnectivity and, most of all, to the work of J. Henri Poincaré (1854–1912). Starting from his research on differential equations in mechanics, Poincaré introduced relevant topological notions and, in particular, the idea of associating to a geometric figure (using a rule that is common to all figures) an algebraic object, such as a group, that is a topological invariant for the figure and that one could compute. The fundamental group and homology groups are two important examples of algebraic objects introduced by Poincaré: this is the beginning of combinatorial or algebraic topology. With the development of what we call today general topology due to, among others, René-Louis Baire (1874–1932), Maurice Fréchet (1878–1973), Frigyes Riesz (1880–1956), Felix Hausdorff (1869– 1942), Kazimierz Kuratowski (1896–1980), and the interaction between general and algebraic topology due to L. E. Brouwer (1881–1966), James Alexander (1888–1971), Solomon Lefschetz (1884–1972), Pavel Alexandroff (1896–1982), Pavel Urysohn (1898–1924), Heinz Hopf (1894–1971), L. Agranovich Lyusternik (1899–1981), Lev G. Schnirelmann (1905–1938), Harald Marston Morse (1892–1977), Eduard Čech (1893–1960), the study of topology in a wide sense is consolidated and in fact receives new incentives thanks to the work of Jean Leray (1906–1998), Élie Cartan (1869– 1951), Georges de Rham (1903–1990). Clearly, even a short introduction to these topics would deviate us from our course; therefore we shall confine ourselves to illustrating some fundamental notions and basic results related to the topology of ℝn, to the notion of dimension and, most of all, to the existence of fixed points.


Base Point Fundamental Group Universal Covering Homotopy Class Continuous Extension 


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© Birkhäuser Boston 2007

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