Algebraic Topology - Hatcher

homology In this book our attention will be focused on compact, simply connected, dif-ferentiable 4-manifolds. The restriction to the simply connected case certainly rules out many interesting examples :indeed it is well-known that any finitely presented group can occur as the fundamental group of a 4-manifold.Furthermore the techniques we will develop in the body of the book are in reality rather insensitive to the fundamental group and much of our discussion can easily be generalized. The main issues however can be reached more quickly in the simply connected case. We shall see that for many purposes 4-manifolds with trivial fundamental group are of beguiling simplicitly but nevertheless the most basic questions about the differential topology of these manifolds lead us into new uncharted waters where the results described in this book serve at present,as isolated markers. After the fundamental group we have the homology and cohomology groups of a 4-manifold. For a closed oriented 4-manifold Poincare duality gives an isomorphism between homology and cohomology in complementary dimensions i and 4-i. So, when X is simply connected the first and third homology groups vanish and all the homological information is contained in H 2. The universal coefficient theorem for cohomology implies that when H 1 is zero H 2 is a free abelian group.In turn by Poincare duality the homology group is free. There are three concrete ways in which we can realize 2-dimensional ho-mology or cohomology classes on a 4-manifold and it is useful to be able to translate easily between them. The first is complex line bundles complex vector bundles of rank 1. On any space X a line bundle L is determined up to the bundle isomorphism by its Chern class and this sets up a bijection between the isomorphism classes of line bundles. The second realization is by smoothly embedded two-dimensional oriented surfaces in the manifold X. Such a surface carries a fundamental homology class ,given a line bundle L we can choose a general smooth section of the bundle whose zero set is a surface representing the homology class dual to c 1 (L) Third we have the de Rham representation of real cohomology classes by differential forms. Let X be a compact oriented simply connected four-manifold. The Poincare duality isomorphism between homology and cohomology is equivalent to a bi-linear form Q. This is the intersection form of the manifold. It is a unimodular symmetric form.Geometrically two oriented surfaces in X placed in general position will meet in a finite set of points. To each point we associate a sign ±1 according to the matching of the orientations in the isomorphism of the tangent bundles at that point. The intersection number is given by the total number of points counted with the signs. The pairing passes to homology to yield the form Q.Going over to cohomology the form translates into the cup product. Thus the form is an invariant of the oriented homotopy type of X. In terms of de Rham cohomology if w 1 and w 2 are closed 2-forms representing classes dual to the surfaces the intersection number is given by the integral X w 1 ∧ w 2 .

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.

Jahresbericht der Deutschen Mathematiker-Vereinigung, 2012

Bulletin of the American Mathematical Society, 1990

The notion of classification of structure arises in many areas of mathematics, and a common classification is "up to homotopy," or in terms of "deformation." For this reason, techniques of homotopy theory, and in particular the fundamental group and higher homotopy groups, are important and have been applied across a range of mathematical disciplines. Algebraic Homotopy, which we refer to as AH, has in the Introduction the following quotation from J. H. C. Whitehead's address to the International Congress of Mathematicians at Harvard in 1950 [W 7]:

Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry, 2022

for all p ≥ 0. Furthermore, this assignment is functorial (more precisely, it is a functor from the category of cochain complexes and chain maps to the category of abelian groups and their homomorphisms). At first glance cohomology appears to be very abstract so it is natural to look for explicit examples. A way to obtain a cochain complex is to apply the operator (functor) Hom Z (-, G) to a chain complex C, where G is any abelian group. Given a fixed abelian group A, for any abelian group B we denote by Hom Z (B, A) the abelian group of all homomorphisms from B to A. Given any two abelian groups B and C, for any homomorphism f : The map Hom Z (f, A) is also denoted by Hom Z (f, id A ) or even Hom Z (f, id). Observe that the effect of Hom Z (f, id) on ϕ is to precompose ϕ with f . If f : B → C and g : C → D are homomorphisms of abelian groups, a simple computation shows that Hom R (g 1.1. EXACT SEQUENCES, CHAIN COMPLEXES, HOMOLOGY, COHOMOLOGY 21 Observe that Hom Z (f, id) and Hom Z (g, id) are composed in the reverse order of the composition of f and g. It is also immediately verified that Hom Z (id A , id) = id Hom Z (A,G) . We say that Hom Z (-, id) is a contravariant functor (from the category of abelian groups and group homomorphisms to itself). Then given a chain complex we can form the cochain complex obtained by applying Hom Z (-, G), and denoted Hom Z (C, G). The coboundary map d p is given by which means that for any f ∈ Hom Z (C p , G), we have Thus, for any (p + 1)-chain c ∈ C p+1 we have We obtain the cohomology groups H p (Hom Z (C, G)) associated with the cochain complex Hom Z (C, G). The cohomology groups H p (Hom Z (C, G)) are also denoted H p (C; G). This process was applied to the simplicial chain complex C * (K) associated with a simplicial complex K by Alexander and Kolmogoroff to obtain the simplicial cochain complex Hom Z (C * (K); G) denoted C * (K; G) and the simplicial cohomology groups H p (K; G) of the simplicial complex K; see Section 5.6. Soon after, this process was applied to the singular chain complex S * (X; Z) of a space X to obtain the singular cochain complex Hom Z (S * (X; Z); G) denoted S * (X; G) and the singular cohomology groups H p (X; G) of the space X; see Section 4.8. Given a continuous map f : X → Y , there is an induced chain map f : S * (Y ; G) → S * (X; G) between the singular cochain complexes S * (Y ; G) and S * (X; G), and thus homomorphisms of cohomology f * : H p (Y ; G) → H p (X; G). Observe the reversal: f is a map from X to Y , but f * maps H p (Y ; G) to H p (X; G). We say that the map X → (H p (X; G)) p≥0 is a contravariant functor from the category of topological spaces and continuous maps to the category of abelian groups and their homomorphisms. So far our homology groups have coefficients in Z, but the process of forming a cochain complex Hom Z (C, G) from a chain complex C allows the use of coefficients in any abelian obtained by applying Hom R (-, G), and denoted Hom R (C, G).

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 concerns with the introductions , the, foundations, as well as the fundamentals and foundational details and the entire general concepts of the Algebraic Topology as a course . . .

Cladistics, 2012

Homology in cladistics is reviewed. The definition of important terms is explicated in historical context. Homology is not synonymous with synapomorphy: it includes symplesiomorphy, and Hennig clearly included both plesiomorphy and synapomorphy as types of homology. Homoplasy is error, in coding, and is analogous to residual error in simple regression. If parallelism and convergence are to be distinguished, homoplasy would be evidence of the former and analogy evidence of the latter. We discuss whether there is a difference between molecular homology and morphological homology, character state homology, nested homology (additive characters), and serial homology. We conclude by proposing a global definition of homology.

2016

Springer eBooks, 1984

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

A Concise Course in Algebraic Topology [J. P. May]

The homotopy hypothesis was originally stated by Grothendieck [13]: topological spaces should be &quot;equivalent&quot; to (weak) ∞-groupoids, which give algebraic representatives of homotopy types. Much later, several authors developed geometrizations of computational models, e.g. for rewriting , distributed systems, (homotopy) type theory etc. But an essential feature in the work set up in concurrency theory, is that time should be considered irreversible, giving rise to the field of directed algebraic topology. Following the path proposed by Porter, we state here a directed homotopy hypothesis: Grandis&#39; directed topological spaces should be &quot;equivalent&quot; to a weak form of topologically enriched categories, still very close to (∞,1)-categories. We develop, as in ordinary algebraic topology, a directed homotopy equivalence and a weak equivalence, and show invariance of a form of directed homology.

In this article we will study a generalization of the homotopy theory we know from algebraic topology. We discuss the abstract tools needed for this generalization, namely model categories and their homotopy categories. We will apply our general setting to topological spaces to find the familiar homotopy theory.