Geodesic
Relationship among various Vietoris-type and microsimplicial homology theories
Archivum mathematicum, Tome 57 (2021) no. 3, pp. 131-150 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, we clarify the relationship among the Vietoris-type homology theories and the microsimplicial homology theories, where the latter are nonstandard homology theories defined by M.C. McCord (for topological spaces), T. Korppi (for completely regular topological spaces) and the author (for uniform spaces). We show that McCord’s and our homology are isomorphic for all compact uniform spaces and that Korppi’s and our homology are isomorphic for all fine uniform spaces. Our homology shares many good properties with Korppi’s homology. As an example, we outline a proof of the continuity of our homology with respect to uniform resolutions. S. Garavaglia proved that McCord’s homology is isomorphic to Vietoris homology for all compact topological spaces. Inspired by this result, we prove that our homology is isomorphic to uniform Vietoris homology for all precompact uniform spaces and that Korppi’s homology is isomorphic to normal Vietoris homology for all pseudocompact completely regular topological spaces.
In this paper, we clarify the relationship among the Vietoris-type homology theories and the microsimplicial homology theories, where the latter are nonstandard homology theories defined by M.C. McCord (for topological spaces), T. Korppi (for completely regular topological spaces) and the author (for uniform spaces). We show that McCord’s and our homology are isomorphic for all compact uniform spaces and that Korppi’s and our homology are isomorphic for all fine uniform spaces. Our homology shares many good properties with Korppi’s homology. As an example, we outline a proof of the continuity of our homology with respect to uniform resolutions. S. Garavaglia proved that McCord’s homology is isomorphic to Vietoris homology for all compact topological spaces. Inspired by this result, we prove that our homology is isomorphic to uniform Vietoris homology for all precompact uniform spaces and that Korppi’s homology is isomorphic to normal Vietoris homology for all pseudocompact completely regular topological spaces.
DOI : 10.5817/AM2021-3-131
Classification : 54J05, 55N05, 55N35
Keywords: McCord homology; Korppi homology; $\mu $-homology; Vietoris homology; nonstandard analysis 
@article{10_5817_AM2021_3_131,
     author = {Imamura, Takuma},
     title = {Relationship among various {Vietoris-type} and microsimplicial homology theories},
     journal = {Archivum mathematicum},
     pages = {131--150},
     year = {2021},
     volume = {57},
     number = {3},
     doi = {10.5817/AM2021-3-131},
     mrnumber = {4306173},
     zbl = {07396179},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2021-3-131/}
}
TY  - JOUR
AU  - Imamura, Takuma
TI  - Relationship among various Vietoris-type and microsimplicial homology theories
JO  - Archivum mathematicum
PY  - 2021
SP  - 131
EP  - 150
VL  - 57
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2021-3-131/
DO  - 10.5817/AM2021-3-131
LA  - en
ID  - 10_5817_AM2021_3_131
ER  - 
%0 Journal Article
%A Imamura, Takuma
%T Relationship among various Vietoris-type and microsimplicial homology theories
%J Archivum mathematicum
%D 2021
%P 131-150
%V 57
%N 3
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2021-3-131/
%R 10.5817/AM2021-3-131
%G en
%F 10_5817_AM2021_3_131
Imamura, Takuma. Relationship among various Vietoris-type and microsimplicial homology theories. Archivum mathematicum, Tome 57 (2021) no. 3, pp. 131-150. doi: 10.5817/AM2021-3-131

[1] Dowker, C.H.: Homology groups of relations. Ann. of Math. (2) 56 (1952), no. 1, 84–95. | DOI | MR

[2] Garavaglia, S.: Homology with equationally compact coefficients. Fund. Math. 100 (1978), no. 1, 89–95. | DOI | MR

[3] Imamura, T.: Nonstandard homology theory for uniform spaces. Topology Appl. 209 (2016), 22–29, Corrigendum in DOI:10.13140/RG.2.2.36585.75368. | DOI | MR

[4] Isbell, J.R.: Uniform Spaces. Mathematical Surveys and Monographs, vol. 12, American Mathematical Society, Providence, 1964. | MR | Zbl

[5] Korppi, T.: A non-standard homology theory with some nice properties. Dubrovnik VI - Geometric Topology, September–October 2007.

[6] Korppi, T.: On the homology of compact spaces by using non-standard methods. Topology Appl. 157 (2010), 2704–2714. | DOI | MR

[7] Korppi, T.: A new microsimplicial homology theory. viXra:1205.0081, 2012.

[8] Mardešić, S., Segal, J.: Shape Theory. North-Holland Mathematical Library, vol. 26, North-Holland, Amsterdam-New York-Oxford, 1982. | MR

[9] McCord, M.C.: Non-standard analysis and homology. Fund. Math. 74 (1972), no. 1, 21–28. | DOI | MR

[10] Robinson, A.: Non-standard Analysis. Studies in Logic and the Foundations of Mathematics, vol. 42, North-Holland, Amsterdam, 1966. | MR

[11] Stroyan, K.D., Luxemburg, W.A.J.: Introduction to The Theory of Infinitesimals. Pure and Applied Mathematics, vol. 72, Academic Press, New York-San Francisco-London, 1976. | MR

[12] Wattenberg, F.: Nonstandard analysis and the theory of shape. Fund. Math. 98 (1978), no. 1, 41–60. | DOI | MR

Cité par Sources :