Cotorsion and wild homology

Herfort, Wolfgang; Hojka, Wolfram

doi:10.1007/s11856-017-1566-z

Cited by 15 publications

(26 citation statements)

References 14 publications

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“…In [HH13] it is shown that the abelianizations of ( ), ( 2 ), and ( ) are in fact all equal to each other; perhaps offering some support to Questions 1 and 2.…”

Section: Further Resultsmentioning

confidence: 91%

Archipelago groups

Conner

Hojka²,

Meilstrup

2015

Proc. Amer. Math. Soc.

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The classical archipelago is a non-contractible subset of 3 which is homeomorphic to a disk except at one non-manifold point. Its fundamental group, , is the quotient of the topologist's product of , the fundamental group of the shrinking wedge of countably many copies of the circle (the Hawaiian earring), modulo the corresponding free product. We show is locally free, not indicable, and has the rationals both as a subgroup and a quotient group. Replacing with arbitrary groups yields the notion of archipelago groups.Surprisingly, every archipelago of countable groups is isomorphic to either ( ) or ( 2 ), the cases where the archipelago is built from circles or projective planes respectively. We conjecture that these two groups are isomorphic and prove that for large enough cardinalities of G i , (G i ) is not isomorphic to either.

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“…In [HH13] it is shown that the abelianizations of ( ), ( 2 ), and ( ) are in fact all equal to each other; perhaps offering some support to Questions 1 and 2.…”

Section: Further Resultsmentioning

confidence: 91%

Archipelago groups

Conner

Hojka²,

Meilstrup

2015

Proc. Amer. Math. Soc.

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“…Herfort and Hojka proved that the class of cotorsion groups is the same as the class of abelian Higman-complete groups [23,Theorem 3]. Theorem 5.5.…”

Section: Homology Of Barratt-milnor Examplesmentioning

confidence: 99%

“…Proof. By [23,Section 2] it is sufficient to show that for every sequence of elements pb i q in h `πl pH n q ˘and every sequence of natural numbers pn i q, the infinite system of equations…”

Section: Homology Of Barratt-milnor Examplesmentioning

confidence: 99%

Inverse limit slender groups

Conner¹,

Herfort²,

Kent³

et al. 2021

Preprint

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Classically, an abelian group G is said to be slender if every homomorphism from the countable product Z N to G factors through the projection to some finite product Z n . Various authors have proposed generalizations to non-commutative groups, resulting in a plethora of similar but not completely equivalent concepts. In the first part of this work we present a unified treatment of these concepts and examine how are they related. In the second part of the paper we study slender groups in the context of co-small objects in certain categories, and give several new applications including the proof that certain homology groups of Barratt-Milnor spaces are cotorsion groups and a universal coefficients theorem for Čech cohomology with coefficients in a slender group.

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“…Thus G is the fundamental group of a space X in which each based loop has arbitrarily small representatives. Then by [15,Theorem 4], we know G satisfies the property of being Higman-complete. Moreover, the first singular homology H 1 (X) is isomorphic to the abelianization of G. Since we are assuming that G is isomorphic to the weak direct product W…”

mentioning

confidence: 99%

“…In particular, the abelianization of G is torsion-free. Then by [15,Corollary 5], since Ab(G) ∼ = H 1 (X) is torsion-free it must be algebraically compact. Now ℵ 0 p∈P A p is algebraically compact as a direct summand of the algebraically compact abelian group Ab(G).…”

mentioning

confidence: 99%

On topological homotopy groups and relation to Hawaiian groups

Mashayekhy

Babaee

Mirebrahimi

et al. 2020

Hacettepe Journal of Mathematics and Statistics

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By generalizing the whisker topology on the nth homotopy group of pointed space (X, x 0), denoted by π wh n (X, x 0), we show that π wh n (X, x 0) is a topological group if n ≥ 2. Also, we present some necessary and sufficient conditions for π wh n (X, x 0) to be discrete, Hausdorff and indiscrete. Then we prove that L n (X, x 0) the natural epimorphic image of the Hawaiian group H n (X, x 0) is equal to the set of all classes of convergent sequences to the identity in π wh n (X, x 0). As a consequence, we show that L n (X, x 0) ∼ = L n (Y, y 0) if π wh n (X, x 0) ∼ = π wh n (Y, y 0), but the converse does not hold in general, except for some conditions. Also, we show that on some classes of spaces such as semilocally n-simply connected spaces and n-Hawaiian like spaces, the whisker topology and the topology induced by the compact-open topology of n-loop space coincide. Finally, we show that n-SLT paths can transfer π wh n and hence L n isomorphically along its points.

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Cotorsion and wild homology

Cited by 15 publications

References 14 publications

Archipelago groups

Archipelago groups

Inverse limit slender groups

On topological homotopy groups and relation to Hawaiian groups

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