Algebraic Topology: Constructions, Retractions, and Fixed Point Theory

The aim of the textbook is two-fold: first to serve as an introductory graduate course in Algebraic Topology and then to provide an application-oriented presentation of some fundamental concepts in Algebraic Topology to the fixed point theory.

A simple approach based on point-set Topology is used throughout to introduce many standard constructions of fundamental and homological groups of surfaces and topological spaces. The approach does not rely on Homological Algebra. The constructions of some spaces using the quotient spaces such as the join, the suspension, and the adjunction spaces are developed in the setting of Topology only.

The computations of the fundamental and homological groups of many surfaces and topological spaces occupy large parts of the book (sphere, torus, projective space, Mobius band, Klein bottle, manifolds, adjunctions spaces). Borsuk's theory of retracts which is intimately related to the problem of the extendability of continuous functions is developed in details. This theory together with the homotopy theory, the lifting and covering maps may serve as additional course material for students involved in General Topology.

The book comprises 280 detailed worked examples, 320 exercises (with hints or references), 80 illustrative figures, and more than 80 commutative diagrams to make it more oriented towards applications (maps between spheres, Borsuk-Ulam Theory, Fixed Point Theorems, …) As applications, the book offers some existence results on the solvability of some nonlinear differential equations subject to initial or boundary conditions.

The book is suitable for students primarily enrolled in Algebraic Topology, General Topology, Homological Algebra, Differential Topology, Differential Geometry, and Topological Geometry. It is also useful for advanced undergraduate students who aspire to grasp easily some new concepts in Algebraic Topology and Applications. The textbook is practical both as a teaching and research document for Bachelor, Master students, and first-year PhD students since it is accessible to any reader with a modest understanding of topological spaces.

The book aspires to fill a gap in the existing literature by providing a research and teaching document which investigates both the theory and the applications of Algebraic Topology in an accessible way without missing the main results of the topics covered.

Dr S. Djebali received a Master degree (1983) and a PhD degree (1987) in Mathematics from Paris XI University (France,) then a Science Doctorate (Habilitation) from Algiers University (2001).

S. Djebali is specialized in Mathematical Analysis with a major in Fixed point Theory and Applications to Differential Equations (Index fixed point and topological degree, Topological structure of solution set for differential equations and inclusions). Yet, the author has skills in Nonlinear Analysis (Variational and Topological methods), Applied Analysis (mainly in Integro-Differential Equations and Inclusions), and Algebraic Topology.

Web localization and visibility:

Research gate: https://www.researchgate.net/profile/Smail\_Djebali/

https://members.imamu.edu.sa/staff/ibdjebali/en/Pages/default.aspx

MathSciNet ID: 643955

Zentrablatt: djebali.smail

dblp: 02/5426

Orcid ID: http://orcid.org/0000-0002-2318-1989

Publons ID: 1751998 (WoS: A-1503-2012)

Scopus Author ID: 14064587500

Google Scholar Profile (H-index=16 – i10-index=36 – 1045 citations):

http:scholar.google.com/citations?user=mwyKeMsAAAAJ&hl=en&oi=sra

Previous books by Authors (including year, title, publisher):

1. Existence and Structure of Solution Sets for Impulsive Differential Inclusions: a Survey" (joint work with L. Gorniewicz and A. Ouahab), Lecture Notes of the Schauder Center for Nonlinear Studies (Torun, Poland), Vol. 13, Feb. 2012 (pp. 148)
2. Solution Sets for Differential Equations and Inclusions", (joint book with L. Gorniewicz and A. Ouahab), "De Gruyter Series in Nonlinear Analysis and Applications, 18", Walter De Gruyter, Berlin, Germany, 2012, xix+453 pp. appeared 14 November 2012) ISBN: 978-3-11-029356-2. DOI: 0.1515/9783110293562. http://www.degruyter.com/view/product/184567.
3. Fixed Point Theorems with Applications. (joint book with Karima Mebarki, Svetlin G. Georgiev, and Khaled Zennir). Copyright Year 2023. ISBN 9781032464961. May 12, 2023 Forthcoming by Chapman & Hall. 430 Pages