A PRACTICAL PERCEPTION OF HOMOLOGY

Bulletin of the American Mathematical Society, 1979

Bulletin of the American Mathematical Society, 2006

Transactions of the American Mathematical Society, 1962

The mapping functor F plays a role in topology analogous to that played by Horn in group theory. We seek an analogue of the tensor product; this is provided by the reduced join /\; in fact, the reduced join and mapping functors are adjoint functors in the sense of Kan [13] because of the well-known relation F(XA Y, Z) m F(X, F(Y, Z)), valid for well-behaved spaces. Now if X is a space and E a spectrum, then E/\X is again a spectrum and we define the generalized homology groups Hn(X;E) = irn(E/\X). We prove that the generalized homology groups satisfy the Eilenberg-Steenrod axioms, except for the dimension axiom. With an appropriate notion of pairing of spectra, we can define cup-and cap-products. Using these we then prove an Alexander duality theorem. Moreover, we characterize the class of manifolds satisfying Poincaré duality for arbitrary spectra; it includes the n-manifolds of J. H. C. Whitehead [35] and of Milnor [19]. The results of this paper were announced in [32]. §2 is devoted to general preliminaries, and §3 to homology and homotopy properties of the reduced join. Most of the results of these sections are well-License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1962] GENERALIZED HOMOLOGY THEORIES 229 objects and maps of "Wo! the terms "free space" and "free map" will refer to objects and maps of "W. Let n be a positive integer. By "w-ad" we shall mean an «-ad (X; Xx, ■ • • , Xn-x\ #0) having the same homotopy type as some CW-wad

The work is centered on the entire, general, and the concepts of the structures and the fundamentals of the Homological Algebra. . . .

Suppose that m ≤ n. The Milnor hypersurface in CP m × CP n is the space Consider the space This is an example of an elliptic curve. It is homeomorphic to the torus S 1 × S 1 . is an open neighbourhood of A, and that we have a homeomorphism Hom(A, B) → U given by α → graph(α) = (1 + α)(A). A complete flag in V is a sequence of complex subspaces 0 = W0 < W1 < . . . < W d = V such that dim(W k ) = k for all k. The space of complete flags is written Flag(V ); it is again a compact manifold, of dimension d 2 -d. Suppose we have two spaces X and Y , and thus projections this is called the external product of a and b. This construction gives a map µ : The map µ is an isomorphism if each group H r (X ) is free and finitely generated (this is a special case of the Künneth theorem). , given by a → (i * (a), j * (a)). This is an isomorphism, by a degenerate case of the Mayer-Vietoris axiom. We say that X is contractible if it is homotopy equivalent to a point. If so, then H * (X ) = H 0 (X ) = Z. For example, R n is contractible.

1989

Two structures are called homologous if they represent corresponding parts of organisms which are built according to the same body plan (23, 33). The existence of corresponding structures in different species is explained by derivation from a common ancestor that had the same structure as the two species compared (25, 35). The eye of a cow is homologous to the eye of a fish but not to the eye of a squid. Homology is assessed regardless of shape or function.

Frontiers in Neuroscience, 2015

Faunes M, Francisco Botelho J, Ahumada Galleguillos P and Mpodozis J (2015) On the hodological criterion for homology. Front. Neurosci. 9:223.

Biology and Philosophy, 2007

Homology is one of the most important concepts in biology .

This book provides a comprehensive introduction to foundational mathematical concepts, structured across five chapters that seamlessly integrate set theory, category theory, and cohomology of groups. It is designed to serve as a resource for students, researchers, and professionals seeking a robust foundation in algebra, topology, and related mathematical structures. Chapter 1 explores the Fundamentals of Set Theory, delving into the nature and properties of sets, relationships between sets, and methods of representation. Set operations such as union, intersection, difference, and complement are discussed in detail, with a focus on their theoretical underpinnings through commutative, associative, and distributive laws, as well as De Morgan's laws. An exercise section consolidates understanding and facilitates practical application. Chapter 2 introduces the Theory of Categories, laying the groundwork for understanding objects, morphisms, and their relationships within categories and subcategories. Fundamental concepts such as monomorphisms, epimorphisms, and the universal properties of products, coproducts, pullbacks, and pushouts are explored. The role of kernels, cokernels, and zero morphisms in exact categories is also examined, emphasizing their importance in algebraic structures. Chapter 3 builds on these foundations with Advanced Concepts in Category Theory. It highlights functor categories, natural transformations, and quotient categories, providing a deeper insight into their construction and properties. Equalizers, coequalizers, and exact functors are characterized, and their applica