A User’s Guide to Algebraic Topology

Dodson, C. T. J.; Parker, Phillip E.

doi:10.1007/978-1-4615-6309-9

Cited by 77 publications

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“…See Dodson and Parker (1997), page 25. Instead of R p , we might take any contractible topological space Θ, due to the pathwise connectedness of S p−1 ; see, for instance, Exercise 19(ii), page 26 of Rotman (1988).…”

Section: And Such That the Restriction Of ξ To S Is Not Homotopic To mentioning

confidence: 99%

On depth and deep points: a calculus

Mizera¹

2002

Ann. Statist.

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Abstract. For a general definition of depth in data analysis a differential-like calculus is constructed in which the location case (the framework of Tukey's median) plays a fundamental role similar to that of linear functions in the mathematical analysis. As an application, a lower bound for maximal regression depth is proved in the general multidimensional case-as conjectured by Rousseeuw and Hubert, and others. This lower bound is demonstrated to have an impact on the breakdown point of the maximum depth estimator.

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Section: And Such That the Restriction Of ξ To S Is Not Homotopic To mentioning

confidence: 99%

On depth and deep points: a calculus

Mizera¹

2002

Ann. Statist.

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“…Clearly we have a homomorphism π 1 (|K|, e G ) −→ π 1 (G) which sends an edge loop in K into the τ -loop it determines in G. This is well defined since if two edge loops are homotopic in |K| then they are obviously τ -homotopic in G. The argument in the previous paragraph shows that the homomorphism π 1 (|K|, e G ) −→ π 1 (G) is surjective. Now as explained in [2] Chapter 3, Subsection 3.5.3, the fundamental group of a (finite) simplicial complex is finitely generated. Hence π 1 (G) is also finitely generated.…”

Section: The Main Resultsmentioning

confidence: 99%

The universal covering homomorphism in o‐minimal expansions of groups

Edmundo

Eleftheriou

2007

Mathematical Logic Qtrly

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Suppose that G is a definably connected, definable group in an ominimal expansion of an ordered group. We show that the o-minimal universal covering homomorphism p : G −→ G is a locally definable covering homomorphism and π 1 (G) is isomorphic to the o-minimal fundamental group π(G) of G defined using locally definable covering homomorphisms. * With partial support from the FCT (Fundação para a Ciência e Tecnologia), program POCTI (Portugal/FEDER-EU).

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“…where f * 0 = (1), f * i = (0) for i = 1, ..., n − 1 and f * n = (D) where D is the degree of the map f , see for more details [2]. From (2) we have that…”

Section: In This Setting Is Defined the Lefschetz Numbermentioning

confidence: 99%

“…, where D is the degree of the map f and f * i = (0) for i ∈ {0, ..., 2n}, i = 0, n, 2n (see for more details [2]). From (2) the Lefschetz zeta function of f is…”

Section: In This Setting Is Defined the Lefschetz Numbermentioning

confidence: 99%

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Periods of continuous maps on some compact spaces

Guirao

Llibre

2017

Journal of Difference Equations and Applications

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Abstract. The objective of this paper is to provide information on the set of periodic points of a continuous self-map defined in the following compact spaces: S n (the n-dimensional sphere), S n × S m (the product space of the n-dimensional with the m-dimensional spheres), CP n (the n-dimensional complex projective space) and HP n (the n-dimensional quaternion projective space). We use as main tool the action of the map on the homology groups of these compact spaces.

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A User’s Guide to Algebraic Topology

Cited by 77 publications

References 0 publications

On depth and deep points: a calculus

On depth and deep points: a calculus

The universal covering homomorphism in o‐minimal expansions of groups

Periods of continuous maps on some compact spaces

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