Pub Date : 2022-01-01DOI: 10.4310/hha.2022.v24.n1.a3
Sajjad Mohammadi
{"title":"The homotopy types of $SU(n)$-gauge groups over $S^{2m}$","authors":"Sajjad Mohammadi","doi":"10.4310/hha.2022.v24.n1.a3","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n1.a3","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70435013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4310/hha.2022.v24.n2.a4
D. Jojic, G. Panina, R. Živaljević
{"title":"Optimal colored Tverberg theorems for prime powers","authors":"D. Jojic, G. Panina, R. Živaljević","doi":"10.4310/hha.2022.v24.n2.a4","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a4","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70435065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-16DOI: 10.4310/HHA.2023.v25.n1.a8
Niles Johnson, Donald Yau
There is a free construction from multicategories to permutative categories, left adjoint to the endomorphism multicategory construction. The main result shows that these functors induce an equivalence of homotopy theories. This result extends a similar result of Thomason, that permutative categories model all connective spectra.
{"title":"Multicategories model all connective spectra","authors":"Niles Johnson, Donald Yau","doi":"10.4310/HHA.2023.v25.n1.a8","DOIUrl":"https://doi.org/10.4310/HHA.2023.v25.n1.a8","url":null,"abstract":"There is a free construction from multicategories to permutative categories, left adjoint to the endomorphism multicategory construction. The main result shows that these functors induce an equivalence of homotopy theories. This result extends a similar result of Thomason, that permutative categories model all connective spectra.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42232059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-05DOI: 10.4310/hha.2022.v24.n2.a20
César Galindo, Pablo Peláez
We show that the orthogonal spectral sequence introduced by the second author is strongly convergent in Voevodsky's triangulated category of motives DM over a field k. In the context of the Morel-Voevodsky motivic stable homotopy category we provide concrete examples where the spectral sequence is not strongly convergent, and give a criterion under which the strong convergence still holds. This criterion holds for Voevodsky's slices, and as a consequence we obtain a spectral sequence which converges strongly to the E1-term of Voevodsky's slice spectral sequence.
{"title":"On the convergence of the orthogonal spectral sequence","authors":"César Galindo, Pablo Peláez","doi":"10.4310/hha.2022.v24.n2.a20","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a20","url":null,"abstract":"We show that the orthogonal spectral sequence introduced by the second author is strongly convergent in Voevodsky's triangulated category of motives DM over a field k. In the context of the Morel-Voevodsky motivic stable homotopy category we provide concrete examples where the spectral sequence is not strongly convergent, and give a criterion under which the strong convergence still holds. This criterion holds for Voevodsky's slices, and as a consequence we obtain a spectral sequence which converges strongly to the E1-term of Voevodsky's slice spectral sequence.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46301850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-06DOI: 10.4310/hha.2023.v25.n1.a17
Yusuke Tajima, M. Yoshinaga
Recently, Asao and Izumihara introduced CW-complexes whose homology groups are isomorphic to direct summands of the graph magnitude homology group. In this paper, we study the homotopy type of the CW-complexes in connection with the diagonality of magnitude homology groups. We prove that the Asao-Izumihara complex is homotopy equivalent to a wedge of spheres for pawful graphs introduced by Y. Gu. The result can be considered as a homotopy type version of Gu's result. We also formulate a slight generalization of the notion of pawful graphs and find new non-pawful diagonal graphs of diameter $2$.
{"title":"Magnitude homology of graphs and discrete Morse theory on Asao–Izumihara complexes","authors":"Yusuke Tajima, M. Yoshinaga","doi":"10.4310/hha.2023.v25.n1.a17","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a17","url":null,"abstract":"Recently, Asao and Izumihara introduced CW-complexes whose homology groups are isomorphic to direct summands of the graph magnitude homology group. In this paper, we study the homotopy type of the CW-complexes in connection with the diagonality of magnitude homology groups. We prove that the Asao-Izumihara complex is homotopy equivalent to a wedge of spheres for pawful graphs introduced by Y. Gu. The result can be considered as a homotopy type version of Gu's result. We also formulate a slight generalization of the notion of pawful graphs and find new non-pawful diagonal graphs of diameter $2$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48686829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-21DOI: 10.4310/hha.2022.v24.n2.a17
Maximilian Stegemeyer
The free loop space of a Lie group is homeomorphic to the product of the Lie group itself and its based loop space. We show that the coproduct on the homology of the free loop space that was introduced by Goresky and Hingston splits into the diagonal map on the group and a so-called based coproduct on the homology of the based loop space. This result implies that the coproduct is trivial for even-dimensional Lie groups. Using results by Bott and Samelson, we show that the coproduct is trivial as well for a large family of simply connected Lie groups. CONTENTS
{"title":"On the string topology coproduct for Lie groups","authors":"Maximilian Stegemeyer","doi":"10.4310/hha.2022.v24.n2.a17","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a17","url":null,"abstract":"The free loop space of a Lie group is homeomorphic to the product of the Lie group itself and its based loop space. We show that the coproduct on the homology of the free loop space that was introduced by Goresky and Hingston splits into the diagonal map on the group and a so-called based coproduct on the homology of the based loop space. This result implies that the coproduct is trivial for even-dimensional Lie groups. Using results by Bott and Samelson, we show that the coproduct is trivial as well for a large family of simply connected Lie groups. CONTENTS","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45209466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-20DOI: 10.4310/hha.2022.v24.n2.a18
Y. Muranov, A. Szczepkowska, V. Vershinin
We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy invariance of the introduced path homology groups. We provide examples of computation of these homology groups.
{"title":"Path homology of directed hypergraphs","authors":"Y. Muranov, A. Szczepkowska, V. Vershinin","doi":"10.4310/hha.2022.v24.n2.a18","DOIUrl":"https://doi.org/10.4310/hha.2022.v24.n2.a18","url":null,"abstract":"We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy invariance of the introduced path homology groups. We provide examples of computation of these homology groups.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49395011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-03DOI: 10.4310/hha.2023.v25.n1.a10
F. Muro
We define an obstruction to the formality of a differential graded algebra over a graded operad defined over a commutative ground ring. This obstruction lives in the derived operadic cohomology of the algebra. Moreover, it determines all operadic Massey products induced on the homology algebra, hence the name of derived universal Massey product.
{"title":"Derived universal Massey products","authors":"F. Muro","doi":"10.4310/hha.2023.v25.n1.a10","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a10","url":null,"abstract":"We define an obstruction to the formality of a differential graded algebra over a graded operad defined over a commutative ground ring. This obstruction lives in the derived operadic cohomology of the algebra. Moreover, it determines all operadic Massey products induced on the homology algebra, hence the name of derived universal Massey product.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41935531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-25DOI: 10.4310/hha.2023.v25.n1.a4
Jonas Frey, Nima Rasekh
We prove that every locally Cartesian closed $infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.
{"title":"Constructing coproducts in locally Cartesian closed $infty$-categories","authors":"Jonas Frey, Nima Rasekh","doi":"10.4310/hha.2023.v25.n1.a4","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a4","url":null,"abstract":"We prove that every locally Cartesian closed $infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47014450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-19DOI: 10.4310/hha.2023.v25.n1.a12
R. Huang
We show that in a fibration the coformality of the base space implies the coformality of the total space under reasonable conditions, and these conditions can not be weakened. The result is partially dual to the classical work of Lupton cite{Lup} on the formality within a fibration. Our result has two applications. First, we show that for certain cofibrations, the coformality of the cofiber implies the coformality of the base. Secondly, we show that the total spaces of certain spherical fibrations are Koszul in the sense of Berglund cite{Ber}.
{"title":"Coformality around fibrations and cofibrations","authors":"R. Huang","doi":"10.4310/hha.2023.v25.n1.a12","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a12","url":null,"abstract":"We show that in a fibration the coformality of the base space implies the coformality of the total space under reasonable conditions, and these conditions can not be weakened. The result is partially dual to the classical work of Lupton cite{Lup} on the formality within a fibration. Our result has two applications. First, we show that for certain cofibrations, the coformality of the cofiber implies the coformality of the base. Secondly, we show that the total spaces of certain spherical fibrations are Koszul in the sense of Berglund cite{Ber}.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47230291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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