Homology, Cohomology, and Sheaf Cohomology Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: jean@cis.upenn.edu c Jean Gallier Please, do not reproduce without permission of the authors September 24, 2020 X p has the structure of a graded ring for each space X. {\displaystyle \mathbf {p} } and A fundamental result by Brown, Whitehead, and Adams says that every generalized homology theory comes from a spectrum, and likewise every generalized cohomology theory comes from a spectrum. {\displaystyle I^{\mathbf {p} }H_{i}(X)} {\displaystyle i(V)} ∗ of X; this is justified in that the class In their 1952 book, Foundations of Algebraic Topology, they proved that the existing homology and cohomology theories did indeed satisfy their axioms. the inclusion map, the intersection complex For every sheaf of abelian groups E on a topological space X, one has cohomology groups Hi(X,E) for integers i. n By building context and familiarity with examples, the text offers an ideal starting point for those entering the field. ( . X C H {\displaystyle X\setminus X_{n-k}} f = ] {\displaystyle (n-i)} ; For ) X {\displaystyle V} is given by starting with the constant sheaf on the open set Δ i {\displaystyle (n-i)} − subscheme ∂ H . ≠ = i It is common to take A to be a commutative ring R; then the cohomology groups are R-modules. , R ⊂ ) This result can be stated more simply in terms of cohomology. That definition suggests various generalizations. = ∗ defined by a cubic homogeneous polynomial This space has the remarkable property that it is a classifying space for cohomology: there is a natural element u of Hj(K(A,j),A), and every cohomology class of degree j on every space X is the pullback of u by some continuous map X → K(A,j). Q Given a smooth elliptic curve They introduced an equivalence relation for allowable cycles (where only "allowable boundaries" are equivalent to zero), and called the group. n They furthermore showed that the intersection of an i- and an At a 1935 conference in Moscow, Andrey Kolmogorov and Alexander both introduced cohomology and tried to construct a cohomology product structure. The singular intersection homology groups (with perversity p). ( is defined by cap product with the fundamental class of X. ) ) Q X k ; U The groups Ci are zero for i negative. | ; Intersection Homology & Perverse Sheaves is suitable for graduate students with a basic background in topology and algebraic geometry. is contained in the X Starting in the 1950s, sheaf cohomology has become a central part of algebraic geometry and complex analysis, partly because of the importance of the sheaf of regular functions or the sheaf of holomorphic functions. In particular, in the case of the constant sheafon X associated to an abelian group A, the resulting groups H (X,A) coincide with singular cohomology for X a manifold or CW complex (though not for arbitrary spaces X).
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