S. Eilenberg and N. E. Steenrod, Foundations of Algebraic Topology," Princeton Univ. Press, Princeton, NJ, 1952.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... structures and produces the convergence of methods in generic domains such as the geometric modeling and computational geometry, and the computer imagery and discrete geometry [35, 27, 36, 18, 19, 28] The aim of the present study is to revisit the classic cellular structure of the CW complexes [11, 55, 52, 45]. We investigate to which extent the underlying topological structures, the semisimplicial complexes, and the corresponding semisimplicial constructions (product, quotient and adjunction) form a basis in the modeling space of interest. The motivation is to establish and illustrate the genuineness ....

....boundary reconstruction of discrete objects. Finally, a synthesis of the related results and future work are reported. 2 Basic Notions Introductory material on general topology could be found in [3, 30, 70, 22, 26, 33] A more advanced topic textbooks in algebraic (or combinatorial) topology are [11, 31]. The terminology from the topology of CW complexes mostly follows the textbook of Lundell [45] The exceptions would be explicitly referenced. A good reading on the subject could also be found in [52] Closed Euclidean n cell E . is a homeomorphic image of the Euclidean n cube I , the ....

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Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Princeton University Press, 1952.

....b . The set K is partially ordered by the relation , and for any pair of complexes K a , K b there exist a complex K g in K , which is a refinement of both [Whitney, 1957] remember that our domains are homeomorphic to polytopes) This property makes of K a directed set [Hocking and Young, 1988; Eilenberg and Steenrod, 1952]. For each complex K a in K let us consider the set C p (K a ) of all the physical p cochains on K a , with a given physical dimension (with our choice of the space where the cochains take their values, C p (K a ) is a vector space, but to keep thing simple, we will consider only the group ....

Eilenberg, S., and Steenrod, N. (1952). Foundations of Algebraic Topology, Princeton University Press, Princeton, NJ.

....dimensional Morse theory in analogy with Floer homology may be found in M. Schwarz [25] While many basic concepts of Morse homology can be found in the classical investigations of Milnor, Smale, and Thom, its presentation as an axiomatic homology theory in the sense of Eilenberg and Steeenrod [7] is given for the first time in [25] One consequence of this axiomatic approach is the uniqueness result for Morse homology and its natural equivalence with other axiomatic homol15 ogy theories defined on a suitable category of topological spaces. Witten s isomorphism is then a corollary of ....

S. Eilenberg and N. Steenrod. Foundations of Algebraic Topology. Princeton Uni. Press, Princeton, 1952.

....dR;Z (X n Y; C ) is in fact nothing but the relative cohomology group H i (XnY; Xn (Y [ Z) C ) Consider the space X n (Y Z) and its open covering by the two sets X n Y and X n Z. It follows from [4] Example 17.1, that this is an exact triad for homology with integer coefficients, and from [3], Theorem 11.4, that the same holds for cohomology with coefficients in C . This means that the natural inclusion of pairs (X n Y; X n (Y [ Z) X n (Y Z) X n Z) induces a natural isomorphism between the groups H i (X n Y; X n (Y [ Z) C ) and H i (X n (Y Z) X nZ; C ) This in turn ....

S. Eilenberg and N. Steenrod. Foundations of Algebraic Topology. Princeton University Press, 1952. 52 ULI WALTHER

....will ever be supplanted. With this language at hand, Eilenberg and Steenrod were able to formulate their axiomatization of ordinary homology and cohomology theory. The axioms were announced in 1945 [ES45] but their celebrated book The foundations of algebraic topology did not appear until 1952 [ES52], by which time its essential ideas were well known to workers in the eld. It should be recalled that Eilenberg had set the stage with his fundamentally important 1940 paper [Eil40] in which he de ned singular homology and cohomology as we know them today. I will say a little about the axioms ....

S. Eilenberg and N.E. Steenrod. The foundations of algebraic topology. Princeton University Press. 1952.

....H i dR;Z (XnY; C ) is in fact nothing but the relative cohomology group H i (XnY; Xn(Y [ Z) C ) Consider the space Xn(Y Z) and its open covering by the two sets XnY and XnZ. It follows from [14] Example 17.1, that this is an exact triad for homology with integer coefficients, and from [9], Theorem 11.4, that the same holds for cohomology with coefficients in C . This means that the natural inclusion of pairs (XnY; Xn(Y [ Z) Xn(Y Z) XnZ) induces an isomorphism H i (XnY; Xn(Y [ Z) C ) H i (Xn(Y Z) XnZ; C ) This implies that instead of H i dR;Z (XnY; C ) we may ....

S. Eilenberg and N. Steenrod. Foundations of Algebraic Topology. Princeton University Press, 1952.

....above problems in a consistent manner. Our framework will hinge on various mathematical and physical disciplines among which the theory of systems of partial differential and integral equations [57] the theory of dynamical systems [58] differential and integral geometry [59] algebraic topology [60, 61, 62, 63, 64], category theory [65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77] thermodynamics and statistical mechanics [78] renormalisation theory [39] and dynamic scale space theory [56] Of course, the actual development and deployment of a sustainable system is also depending heavily on the input ....

.... rich methods to systematically retrieve such a set of equivalences based on: ffl Differential and Integral Geometry [79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95] 3 ffl Lie Group Theory applied to PDES [96, 97, 57, 98, 99, 100] ffl Algebraic and Differential Topology [60, 101, 61, 62, 63, 64]. In the following we briefly summarise the differential and integral geometric method for obtaining such a set of equivalences. However, in Section 2.2 we will see that only a set of equivalences consistent with an encoding of a natural system can come about by taking advantage of the ....

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S. Eilenberg and S. Steenrod. Foundations of Algebraic Topology. Princeton University Press, Princeton, 1952.

....the first to introduce the general tensor product and torsion product Tor of abelian group into homological algebra. However, such a modern formulation of Cech s result (and the name Tor, due to Eilenberg around 1950) did not appear in print before 1951 (Expos e 10 of [C50] see also p. 161 of [ES]) We note a contemporary variant in passing: Steenrod proved a Universal Coefficient Theorem for cohomology with coefficients in a compact topological group; see [S36] in this context the Universal Coefficient group is the character group R=Z of Z. The fourth great advance in 1935 was the ....

.... A = F=R amounted to a free resolution of A, and his formulas were equivalent to the modern calculation of Ext(A; B) as the cokernel of Hom(F;B) Hom(R;B) But working with free resolutions was still a decade away ( Hf44, F46] and using them to calculate Ext(A; B) was even further in the future ([ES]) We now turn to 1941. That year, Saunders Mac Lane gave a series of lectures on group extensions at the University of Michigan. According to [M88] most of the lectures concerned applications to Galois groups and class field theory, but Mac Lane ended with a calculation of the abelian ....

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S. Eilenberg and N. Steenrod, Foundations of algebraic topology, Princeton U. Press, Princeton, 1952.

....of an exact sequence is the best one can hope for. 0 B A C 0 i j Figure 4: A short exact sequence : A i Gamma1 A i A i 1 : ff i Gamma1 ff i Figure 5: A long exact sequence The use of exact sequences was initiated in algebraic topology by Eilenberg and Steenrod (Eilenberg and Steenrod, 1952). It illustrates a basic notational device which, as we will see, has grown into a whole (informal) mechanism of proof in this particular field of mathematics, as well as in category theory, algebraic geometry and formal logics. More generally, many situations give rise to long exact sequences : ....

Eilenberg, S. & Steenrod, N., Foundations of algebraic topology, Princeton University Press, Princeton, NJ, 1952.

.... [0; 1] C defined by H(x; t) 1 Gamma t)oe(x) tx: It is known that for every finite union C of nonempty closed convex sets lying strictly inside D; and every continuous g : C C; the Lefschetz number (g) with respect to the singular homology over the rationals) is well defined (see [9,11] for details) Lemma 2. Let C be a complete family of nonempty closed convex sets lying strictly inside D; and let g : jCj jCj be continuous with (g) 6= 0: Then g(C) C 6= for at least one C 2 C: Proof. See the proof of the Lemma in [11] Lemma 2 also yields the following fact. Lemma 3. ....

S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton Univ. Press, Princeton, NJ, 1952.

....in 1964. But there are versions of Poincar e and Alexander duality for homology and 13 cohomology of sheaves, for local coefficients, for cohomology with compact supports; all of which play very important roles in various branches of mathematics besides topology. Finally we discuss the book of Eilenberg Steenrod, 1952), which does not mention Poincar e duality, and yet it sets in motion several ideas which improve the theorem. In Eilenberg Steenrod (1945) the axioms for homology and cohomology theory were published. The proofs were deferred to the book. With the axioms, we see that for CW complexes at least, ....

....Cech homology, and one does not see it used today in algebraic topology. It is ironic that Cech s name is given to a cohomology theory he did not define, yet the fact that Cech first realized that the Poincar e duality isomorphism could be expressed by the cap product has all but been forgotten. Eilenberg and Steenrod s book (1952) effected a revolution in mathematical notation. Perhaps not since Descartes La g eom etrie has a book influenced how we write Mathematics. One knew they were looking at mathematics before 1600 because of the geometric diagrams with vertices and sides labeled by alphabetic letters. La g eom etrie ....

Eilenberg, S. and Steenrod, N. (1952), Foundations of Algebraic Topology, Princeton Univ. Press.

....and debate on the development of ideas. The origins of category theory help to explain its utility. It arose from attempting to explain the meaning of the word natural in mathematics, and with a strong impetus from the axiomatic approach to homology theories, developed by Eilenberg and Steenrod [6]. The original paper on the subject by Eilenberg and Mac Lane [5] has an interesting discussion of the word natural in terms of the map V V of a vectorspace into its double dual. To define natural required a definition of functors, and to define functors required a definition of category. ....

S.Eilenberg and N.Steenrod, Foundations of algebraic topology, Princeton University Press, (1952).

....one of the applications of the interface library is to support the efficient interpolation of piecewise smooth functions, it would be beneficial to allow arbitrary positive integer values for the dimension dim. In this case, the interface is a generalization of the notion of a simplicial complex [10] of dimension dim Gamma 1 imbedded in R dim . Multiple INTERFACEs and their uses, even within a single computation, prevent dim from being a globally defined variable. Each of the elementary objects in an INTERFACE (including the INTERFACE itself) has support routines, for initializing, ....

Eilenberg, S., Steenrod, N.: Foundations of Algebraic Topology. Princeton University Press, Princeton, 1952

....one of the applications of the interface library is to support the efficient interpolation of piecewise smooth functions, it would be beneficial to allow arbitrary positive integer values for the dimension dim. In this case, the interface is a generalization of the notion of a simplicial complex [16] of dimension dim Gamma 1 embedded in R dim . Multiple INTERFACEs and their uses, even within a single computation, prevent dim from being a globally defined variable. Each of the elementary objects in an INTERFACE (including the INTERFACE itself) has support routines, for initializing, ....

S. Eilenberg and N. Steenrod. Foundations of Algebraic Topology. Princeton University Press, Princeton, 1952.

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S. Eilenberg and N. E. Steenrod, Foundations of Algebraic Topology," Princeton Univ. Press, Princeton, NJ, 1952.

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S. Eilenberg and N. Steenrod, Foundations of algebraic topology, Princeton Univ. Press, Princeton, N.J., 1952.

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S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton University Press, NJ, 1952.

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S. Eilenberg and N. Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, NJ, 1952.

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S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton University Press, Princeton, New Jersey, 1952.

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S. Eilenberg and N. E. Steenrod, Foundations of algebraic topology, Princeton University Press, 1952.

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S. Eilenberg and N. E. Steenrod, Foundations of algebraic topology, Princeton University Press, 1952.

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S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology. Princeton, New Jersey: Princeton University Press, 1952.

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Eilenberg and, S, Steenrod, N.: Foundation of Algebraic Topology, Princeton Press, 1952

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