Survey Article: An elementary illustrated introduction to simplicial sets

Friedman, Greg

doi:10.1216/rmj-2012-42-2-353

Cited by 100 publications

(91 citation statements)

References 17 publications

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“…Here we introduce them briefly, with emphasis on the ideas and intuition, referring to Friedman [18] for a very friendly thorough introduction, to [10,34] for older compact sources, and to [20] for a more modern and comprehensive treatment.…”

Section: Simplicial Setsmentioning

confidence: 99%

“…Here we sketch some basic examples of simplicial sets; again, we won't provide all details, referring to [18]. Let ∆ n denote the standard n-dimensional simplex regarded as a simplicial set.…”

Section: Simplicial Setsmentioning

confidence: 99%

“…Intuitively, one takes disjoint geometric simplices corresponding to the nondegenerate simplices of X, and glues them together according to the identifications implied by the face and degeneracy operators (we again refer to the literature, especially to [18], for a formal definition).…”

Section: Geometric Realizationmentioning

confidence: 99%

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Computing All Maps into a Sphere

Čadek

Krčál

Matoušek

et al. 2014

J. ACM

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Given topological spaces X, Y , a fundamental problem of algebraic topology is understanding the structure of all continuous maps X → Y . We consider a computational version, where X, Y are given as finite simplicial complexes, and the goal is to compute [X, Y ], i.e., all homotopy classes of such maps.We solve this problem in the stable range, where for some d ≥ 2, we have dimThe algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology).In contrast, [X, Y ] is known to be uncomputable for general X, Y , since for X = S 1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y . In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A → Y and ask whether it extends to a map X → Y , or computing the Z 2 -index -everything in the stable range. Outside the stable range, the extension problem is undecidable.

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Section: Simplicial Setsmentioning

confidence: 99%

Section: Simplicial Setsmentioning

confidence: 99%

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Computing All Maps into a Sphere

Čadek

Krčál

Matoušek

et al. 2014

J. ACM

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show abstract

“…We refer to [Fri11] for a very friendly thorough introduction to simplicial sets. Intuitively, a simplicial set can be thought of as a kind of hybrid or compromise between a simplicial complex (more special) on the one hand and a cell complex (more general) on the other hand.…”

Section: Simplicial Setsmentioning

confidence: 99%

“…Like a simplicial complex, every simplicial set X defines a topological space |X|, the geometric realization of X, which is unique up to homeomorphism. More specifically, |X| is a cell complex with one n-cell for every nondegenerate n-simplex of X, and these cells are glued together according to the identifications implied by the face and degeneracy operators (we omit the precise definition of the attachments, since we will not really use it and refer to the literature, e.g., to [Fri11] or [FP90,Sec. 4…”

Section: 2]mentioning

confidence: 99%

Extendability of Continuous Maps Is Undecidable

Čadek

Krčál

Matoušek

et al. 2013

Discrete Comput Geom

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We consider two basic problems of algebraic topology, the extension problem and the computation of higher homotopy groups, from the point of view of computability and computational complexity.The extension problem is the following: Given topological spaces X and Y , a subspace A ⊆ X, and a (continuous) map f : A → Y , decide whether f can be extended to a continuous mapf : X → Y . All spaces are given as finite simplicial complexes and the map f is simplicial.Recent positive algorithmic results, proved in a series of companion papers, show that for (k − 1)-connected Y , k ≥ 2, the extension problem is algorithmically solvable if the dimension of X is at most 2k − 1, and even in polynomial time when k is fixed.Here we show that the condition dim X ≤ 2k − 1 cannot be relaxed: for dim X = 2k, the extension problem with (k − 1)-connected Y becomes undecidable. Moreover, either the target space Y or the pair (X, A) can be fixed in such a way that the problem remains undecidable.Our second result, a strengthening of a result of Anick, says that the computation of π k (Y ) of a 1-connected simplicial complex Y is #P-hard when k is considered as a part of the input.

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A Topological Approach to Cheeger‐Gromov Universal Bounds for von Neumann ρ‐Invariants

Choon

2015

Comm Pure Appl Math

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Using deep analytic methods, Cheeger and Gromov showed that for any smooth .4k 1/-manifold there is a universal bound for the von Neumann L 2 -invariants associated to arbitrary regular covers. We present a proof of the existence of a universal bound for topological .4k 1/-manifolds, using L 2 -signatures of bounding 4k-manifolds. We give explicit linear universal bounds for 3-manifolds in terms of triangulations, Heegaard splittings, and surgery descriptions. We show that our explicit bounds are asymptotically optimal. As an application, we give new lower bounds of the complexity of 3-manifolds that can be arbitrarily larger than previously known lower bounds. As ingredients of the proofs that seem interesting on their own, we develop a geometric construction of efficient 4-dimensional bordisms of 3-manifolds over a group and develop an algebraic topological notion of uniformly controlled chain homotopies.

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Survey Article: An elementary illustrated introduction to simplicial sets

Cited by 100 publications

References 17 publications

Computing All Maps into a Sphere

Computing All Maps into a Sphere

Extendability of Continuous Maps Is Undecidable

A Topological Approach to Cheeger‐Gromov Universal Bounds for von Neumann ρ‐Invariants

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