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This monograph aims to give a self-contained introduction into the whole field of topological analysis: Requiring essentially only basic knowledge of elementary calculus and linear algebra, it provides all required background from topology, analysis, linear and nonlinear functional analysis, and multivalued maps, containing even basic topics like separation axioms, inverse and implicit function theorems, the Hahn-Banach theorem, Banach manifolds, or the most important concepts of continuity of multivalued maps. Thus, it can be used as additional material in basic courses on such topics. The main intention, however, is to provide also additional information on some fine points which are usually not discussed in such introductory courses. The selection of the topics is mainly motivated by the requirements for degree theory which is presented in various variants, starting from the elementary Brouwer degree (in Euclidean spaces and on manifolds) with several of its famous classical consequences, up to a general degree theory for function triples which applies for a large class of problems in a natural manner. Although it has been known to specialists that, in principle, such a general degree theory must exist, this is the first monograph in which the corresponding theory is developed in detail.

Topological degree. --- Topological spaces. --- Fredholm operators. --- Algebraic topology. --- Topology --- Operators, Fredholm --- Linear operators --- Spaces, Topological --- Degree, Topological --- Degree (Topology) --- Degree theory --- Algebraic topology --- Analysis. --- Banach Manifold. --- Fredholm. --- Hahn-Banach Theorem. --- Implicit Function Theorem. --- Inverse Function Theorem. --- Linear Functional Analysis. --- Multivalued Map. --- Nonlinear Functional Analysis. --- Nonlinear Inclusion. --- Separation Axiom. --- Topology. --- Triple Degree.

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When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied. The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors." This mathematical masterpiece will appeal to all mathematicians working in algebraic topology.

Homology theory. --- Abelian group. --- Additive group. --- Algebra homomorphism. --- Algebraic topology. --- Anticommutativity. --- Associative algebra. --- Associative property. --- Axiom. --- Betti number. --- C0. --- Category of modules. --- Change of rings. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Commutative property. --- Commutative ring. --- Cyclic group. --- Derived functor. --- Diagram (category theory). --- Differential operator. --- Direct limit. --- Direct product. --- Direct sum of modules. --- Direct sum. --- Duality (mathematics). --- Endomorphism. --- Epimorphism. --- Equivalence class. --- Exact category. --- Exact sequence. --- Existential quantification. --- Explicit formulae (L-function). --- Factorization. --- Field of fractions. --- Finite group. --- Finitely generated module. --- Free abelian group. --- Free monoid. --- Functor. --- Fundamental group. --- G-module. --- Galois theory. --- Global dimension. --- Graded ring. --- Group algebra. --- Hereditary ring. --- Hochschild homology. --- Homological algebra. --- Homology (mathematics). --- Homomorphism. --- Homotopy. --- Hyperhomology. --- I0. --- Ideal (ring theory). --- Inclusion map. --- Induced homomorphism. --- Injective function. --- Injective module. --- Integral domain. --- Inverse limit. --- Left inverse. --- Lie algebra. --- Linear differential equation. --- Mathematical induction. --- Maximal ideal. --- Module (mathematics). --- Monoidal category. --- Natural transformation. --- Noetherian ring. --- Noetherian. --- Permutation. --- Polynomial ring. --- Pontryagin duality. --- Product topology. --- Projective module. --- Quotient algebra. --- Quotient group. --- Quotient module. --- Right inverse. --- Ring (mathematics). --- Ring of integers. --- Separation axiom. --- Set (mathematics). --- Special case. --- Spectral sequence. --- Subalgebra. --- Subcategory. --- Subgroup. --- Subring. --- Summation. --- Tensor product. --- Theorem. --- Topological space. --- Topology. --- Trivial representation. --- Unification (computer science). --- Universal coefficient theorem. --- Variable (mathematics). --- Zero object (algebra).