On the Groups H(Π, n), I

Eilenberg, Samuel; Lane, Saunders Mac

doi:10.2307/1969820

Cited by 371 publications

(406 citation statements)

References 6 publications

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“…as follows, using shuffles [14]. Take elements This lemma, of course, is the usual one for shuffle-products (see [14]); it depends on the fact that A is central in r.…”

Section: C1mentioning

confidence: 99%

On the Non-Existence of Elements of Hopf Invariant One

Adams¹

1960

The Annals of Mathematics

704

592

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“…as follows, using shuffles [14]. Take elements This lemma, of course, is the usual one for shuffle-products (see [14]); it depends on the fact that A is central in r.…”

Section: C1mentioning

confidence: 99%

On the Non-Existence of Elements of Hopf Invariant One

Adams¹

1960

The Annals of Mathematics

704

592

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“…Now, we recover all the algebraic machinery underlying in Discrete Morse Theory, establishing a new framework for dealing with special chain complexes associated to finite cell complexes and we show that trees are a convenient combinatorial tool for solving the homological computation problem. This integral operator, can also be called chain homotopy operator (Eilenberg andMac Lane, 1953,1954). We will represent an integral operator by an arrow from the cell of lower dimension to the cell of higher dimension (see Fig.…”

Section: Algebraic Discrete Morse Theorymentioning

confidence: 99%

“…The integral chain equivalence relation can be seen as the natural extension of the classical chain homotopy equivalence between chain complexes to the integral case (see, for example, (Eilenberg andMac Lane, 1953,1954)). …”

Section: Algebraic Discrete Morse Theorymentioning

confidence: 99%

Homological optimality in Discrete Morse Theory through chain homotopies

Molina‐Abril

Real

2012

Pattern Recognition Letters

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a b s t r a c tMorse theory is a fundamental tool for analyzing the geometry and topology of smooth manifolds. This tool was translated by Forman to discrete structures such as cell complexes, by using discrete Morse functions or equivalently gradient vector fields. Once a discrete gradient vector field has been defined on a finite cell complex, information about its homology can be directly deduced from it. In this paper we introduce the foundations of a homology-based heuristic for finding optimal discrete gradient vector fields on a general finite cell complex K. The method is based on a computational homological algebra representation (called homological spanning forest or HSF, for short) that is an useful framework to design fast and efficient algorithms for computing advanced algebraic-topological information (classification of cycles, cohomology algebra, homology A(1)-coalgebra, cohomology operations, homotopy groups, . . .). Our approach is to consider the optimality problem as a homology computation process for a chain complex endowed with an extra chain homotopy operator.

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“…It is well known that, given an abelian group A, the Eilenberg-MacLane complexes K(A, n), n ≥ 0, has as q-simplices the normalized n-cocycles of the representable simplicial set [q] = Hom (−, [q]) with coefficients in A [17], where denotes, as usual, the category of ordered sets [n] = {0, 1, . .…”

Section: The N-nerve Of a Symmetric Categorical Groupmentioning

confidence: 99%

Simplicial Cohomology with Coefficients in Symmetric Categorical Groups

Carrasco

Martínez-Moreno

2004

Applied Categorical Structures

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Abstract. In this paper we introduce and study a cohomology theory {H n (−, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A, n)} n≥0 , which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n ≥ 3, the functor K(−, n) is right adjoint to the functor ℘ n , where ℘ n (X • ) is defined as the fundamental groupoid of the n-loop complex n (X • ). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with π i (Y ) = 0 for all i = n, n + 1 and n ≥ 3; and also we obtain a classification theorem for those spaces:Mathematics Subject Classifications (2000): 18D10, 18G50, 18G60.

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On the Groups H(Π, n), I

Cited by 371 publications

References 6 publications

On the Non-Existence of Elements of Hopf Invariant One

On the Non-Existence of Elements of Hopf Invariant One

Homological optimality in Discrete Morse Theory through chain homotopies

Simplicial Cohomology with Coefficients in Symmetric Categorical Groups

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