Rudolf Fritsch

Bio: Rudolf Fritsch is an academic researcher from Ludwig Maximilian University of Munich. The author has contributed to research in topics: Brouwer fixed-point theorem & Projective geometry. The author has an hindex of 7, co-authored 37 publications receiving 246 citations. Previous affiliations of Rudolf Fritsch include Saarland University & University of Konstanz.

Homotopy inverses for nerve

Abstract: Whitehead [19] introduced the category of CW complexes as the appropriate category in which to do homotopy theory. Eilenberg, Mac Lane and Zilber ([1], [2]) defined the notion of simplicial set in the early 50's and Kan ([5], [6], [7]) introduced the necessary conditions to do homotopy theory in this category. The equivalence of these categories under adjoint functors (see [14], [6], [5], [4]) played an important role in the development of geometric topology. In the late 60's, Quillen [16] used the notion of classifying space for a small category [17], and showed the importance of doing homotopy theory in the category of small categories. Latch [8] recently showed that the category of small categories and the category of simplicial sets were equivalent \"up to homotopy,\" but not by using adjoint functors . In this paper, adjoint pairs are given and a general criteria for such adjoint functors to induce a \"homotopy equivalence\" are announced. The homotopic category of K, the category of (semi-) simplicial sets, is equivalent to the homotopy category of W, the category of spaces of homotopy type of a CW complex [4, VII, 1 ] , via a pair of adjoint functors. Moreover, in [8], the homotopic categories of K and Cat, the category of small categories, are shown to be equivalent via the pair N: Cat —• K and r: fC —̂ Cat, where N is the standard embedding nerve functor and T is the category of simplices functor. As in the case for K and W, one would like to replace T by the left adjoint of nerve, categorical realization c: K — Cat; however, c is \"wildly wrong\" with respect to homotopy since it maps certain spheres to contractible categories. In this announcement, we give conditions for other \"reasonable\" functors from K to Cat to be (weak) homotopy inverses for nerve. The functor T: K —* Cat above and the functor A: K —• Cat used in [11] are examples of such homotopy inverses. We only consider functors from K to Cat having right adjoints. Under very weak homotopy conditions, these right adjoints are homotopy equivalent to nerve

The Four-Color Theorem: History, Topological Foundations, and Idea of Proof

Abstract: It's History.- Topological maps.- Topological Version of The Four-Color Theorem.- From Topology to Combinatorics.- The Combinatorial Version of The Four-Color Theorem.- Reducibility.- The Quest for Unavoidable Sets.

The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle

Abstract: (1994). The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle. Mathematics Magazine: Vol. 67, No. 3, pp. 188-205.

An introduction to classical real analysis, by Karl R. Stromberg. Pp 575. $29·95. 1981. ISBN 0-534-98012-0 (Wadsworth)

Abstract: * Preliminaries* Numbers* Sequences and series* Limits and continuity* Differentiation* The elementary transcendental functions* Integration* Infinite series and infinite products* Trigonometric series* Bibliography* Other works by the author* Index