The theories in our zoo are all bordism groups, which generalize the case of smooth manifolds by allowing singularities. There are many concepts of manifolds with singularities one could use here. For our pupose the objects the author introduced some years ago, which are called stratifolds, work particularly well. The zoo comes from forcing certain strata indexed by the subset $A$ to be empty. Special cases are ordinary singular homology and singular bordism. Despite their simple construction computations of these groups seem to be very complicated. We give a few simple examples. Thus there are no interesting applications so far and the zoo looks a bit like a curiosity. But one never knows for what these theories might be good in the future. We mention a concrete question which might be useful in connection with the Griffith group consisting of algebraic cycles in a smooth algebraic variety over the complex numbers which vanish in singular homology. I dedicate these notes to my friend Egbert Brieskorn. Egbert is (in a very different way like our common teacher Hirzebruch) a person which had a great influence on me. When I had to make a complicated decision I often had him in front of my eyes and asked myself: What would Egbert suggest? Conversations with him were always intense and fruitful. I miss him very much.
{"title":"A zoo of geometric homology theories","authors":"M. Kreck","doi":"10.5427/jsing.2018.18n","DOIUrl":"https://doi.org/10.5427/jsing.2018.18n","url":null,"abstract":"The theories in our zoo are all bordism groups, which generalize the case of smooth manifolds by allowing singularities. There are many concepts of manifolds with singularities one could use here. For our pupose the objects the author introduced some years ago, which are called stratifolds, work particularly well. The zoo comes from forcing certain strata indexed by the subset $A$ to be empty. Special cases are ordinary singular homology and singular bordism. \u0000Despite their simple construction computations of these groups seem to be very complicated. We give a few simple examples. Thus there are no interesting applications so far and the zoo looks a bit like a curiosity. But one never knows for what these theories might be good in the future. We mention a concrete question which might be useful in connection with the Griffith group consisting of algebraic cycles in a smooth algebraic variety over the complex numbers which vanish in singular homology. \u0000I dedicate these notes to my friend Egbert Brieskorn. Egbert is (in a very different way like our common teacher Hirzebruch) a person which had a great influence on me. When I had to make a complicated decision I often had him in front of my eyes and asked myself: What would Egbert suggest? Conversations with him were always intense and fruitful. I miss him very much.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87095242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study homological stability for spaces ${rm Hom}(mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $ngeqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, ${rm Comm}(G)$ and ${rm B_{com}} G$, introduced by Cohen-Stafa and Adem-Cohen-Torres-Giese respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability - in particular, the theory of ${rm FI}_W$-modules developed by Church-Ellenberg-Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.
{"title":"Homological Stability for Spaces of Commuting Elements in Lie Groups","authors":"D. Ramras, Mentor Stafa","doi":"10.1093/IMRN/RNAA094","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA094","url":null,"abstract":"In this paper we study homological stability for spaces ${rm Hom}(mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $ngeqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, ${rm Comm}(G)$ and ${rm B_{com}} G$, introduced by Cohen-Stafa and Adem-Cohen-Torres-Giese respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability - in particular, the theory of ${rm FI}_W$-modules developed by Church-Ellenberg-Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89827610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-04-16DOI: 10.2140/pjm.2020.305.757
Xuan Zhao, Hongzhu Gao
In this paper we determine the rational homotopy type of the classifying space of a generic Kac-Moody group by computing its rational cohomology ring. As an application we determine the rational homology Hopf algebra of the generic Kac-Moody group.
{"title":"The rational cohomology Hopf algebra of a\u0000generic Kac–Moody group","authors":"Xuan Zhao, Hongzhu Gao","doi":"10.2140/pjm.2020.305.757","DOIUrl":"https://doi.org/10.2140/pjm.2020.305.757","url":null,"abstract":"In this paper we determine the rational homotopy type of the classifying space of a generic Kac-Moody group by computing its rational cohomology ring. As an application we determine the rational homology Hopf algebra of the generic Kac-Moody group.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79426364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-04-10DOI: 10.2140/agt.2020.20.1691
Clover May
Let $C_2$ be the cyclic group of order two. We present a structure theorem for the $RO(C_2)$-graded Bredon cohomology of $C_2$-spaces using coefficients in the constant Mackey functor $underline{mathbb{F}_2}.$ We show that, as a module over the cohomology of the point, the $RO(C_2)$-graded cohomology of a finite $C_2$-CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. The shifts are by elements of $RO(C_2)$ corresponding to actual (i.e. non-virtual) $C_2$-representations. This decomposition lifts to a splitting of genuine $C_2$-spectra.
{"title":"A structure theorem for RO(C2)–graded Bredon\u0000cohomology","authors":"Clover May","doi":"10.2140/agt.2020.20.1691","DOIUrl":"https://doi.org/10.2140/agt.2020.20.1691","url":null,"abstract":"Let $C_2$ be the cyclic group of order two. We present a structure theorem for the $RO(C_2)$-graded Bredon cohomology of $C_2$-spaces using coefficients in the constant Mackey functor $underline{mathbb{F}_2}.$ We show that, as a module over the cohomology of the point, the $RO(C_2)$-graded cohomology of a finite $C_2$-CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. The shifts are by elements of $RO(C_2)$ corresponding to actual (i.e. non-virtual) $C_2$-representations. This decomposition lifts to a splitting of genuine $C_2$-spectra.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"156 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73278096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Representation stability in the sense of Church-Farb is concerned with stable properties of representations of sequences of algebraic structures, in particular of groups. We study this notion on objects arising in toric topology. With a simplicial $G$-complex $K$ and a topological pair $(X, A)$, a $G$-polyhedral product $(X, A)^K$ is associated. We show that the homotopy decomposition [1] of $Sigma (X, A)^K$ is then $G$-equivariant. In the case of $Sigma_m$-polyhedral products, we give criteria on simplicial $Sigma_m$-complexes which imply representation stability of $Sigma_m$-representations ${H_i((X, A)^{K_m})}$.
{"title":"Simplicial G–complexes and representation\u0000stability of polyhedral products","authors":"X. Fu, Jelena Grbi'c","doi":"10.2140/agt.2020.20.215","DOIUrl":"https://doi.org/10.2140/agt.2020.20.215","url":null,"abstract":"Representation stability in the sense of Church-Farb is concerned with stable properties of representations of sequences of algebraic structures, in particular of groups. We study this notion on objects arising in toric topology. With a simplicial $G$-complex $K$ and a topological pair $(X, A)$, a $G$-polyhedral product $(X, A)^K$ is associated. We show that the homotopy decomposition [1] of $Sigma (X, A)^K$ is then $G$-equivariant. In the case of $Sigma_m$-polyhedral products, we give criteria on simplicial $Sigma_m$-complexes which imply representation stability of $Sigma_m$-representations ${H_i((X, A)^{K_m})}$.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"136 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88927991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-02-25DOI: 10.2140/agt.2018.18.2419
P. Hu, I. Kríz, P. Somberg
We investigate certain adjunctions in derived categories of equivariant spectra, including a right adjoint to fixed points, a right adjoint to pullback by an isometry of universes, and a chain of two right adjoints to geometric fixed points. This leads to a variety of interesting other adjunctions, including a chain of 6 (sometimes 7) adjoints involving the restriction functor to a subgroup of a finite group on equivariant spectra indexed over the trivial universe.
{"title":"On some adjunctions in equivariant stable homotopy theory","authors":"P. Hu, I. Kríz, P. Somberg","doi":"10.2140/agt.2018.18.2419","DOIUrl":"https://doi.org/10.2140/agt.2018.18.2419","url":null,"abstract":"We investigate certain adjunctions in derived categories of equivariant spectra, including a right adjoint to fixed points, a right adjoint to pullback by an isometry of universes, and a chain of two right adjoints to geometric fixed points. This leads to a variety of interesting other adjunctions, including a chain of 6 (sometimes 7) adjoints involving the restriction functor to a subgroup of a finite group on equivariant spectra indexed over the trivial universe.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"29 2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77386395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper contains an overview of background from stable homotopy theory used by Freed-Hopkins in their work on invertible extended topological field theories. We provide a working guide to the stable homotopy category, to the Steenrod algebra and to computations using the Adams spectral sequence. Many examples are worked out in detail to illustrate the techniques.
{"title":"A Guide for Computing Stable Homotopy Groups","authors":"A. Beaudry, Jonathan A. Campbell","doi":"10.1090/CONM/718/14476","DOIUrl":"https://doi.org/10.1090/CONM/718/14476","url":null,"abstract":"This paper contains an overview of background from stable homotopy theory used by Freed-Hopkins in their work on invertible extended topological field theories. We provide a working guide to the stable homotopy category, to the Steenrod algebra and to computations using the Adams spectral sequence. Many examples are worked out in detail to illustrate the techniques.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72822277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is shown that the Mayer-Vietoris sequence holds for the cohomology of complexes of Lie algebroids which are defined on simplicial complexes and satisfy the compatibility condition concerning restrictions to the faces of each simplex. The Mayer-Vietoris sequence will be obtained as a consequence of the extension lemma for piecewise smooth forms defined on complexes of Lie algebroids.
{"title":"Mayer-Vietoris sequence in cohomology of Lie algebroids on simplicial complexes","authors":"J. Oliveira","doi":"10.4134/CKMS.C170463","DOIUrl":"https://doi.org/10.4134/CKMS.C170463","url":null,"abstract":"It is shown that the Mayer-Vietoris sequence holds for the cohomology of complexes of Lie algebroids which are defined on simplicial complexes and satisfy the compatibility condition concerning restrictions to the faces of each simplex. The Mayer-Vietoris sequence will be obtained as a consequence of the extension lemma for piecewise smooth forms defined on complexes of Lie algebroids.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81011770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give an elementary proof of the fact that a pure-dimensional closed subvariety of a complex abelian variety has a signed intersection homology Euler characteristic. We also show that such subvarieties which, moreover, are local complete intersections, have a signed Euler-Poincar'e characteristic. Both results are special cases of the fact, proved by Franecki-Kapranov, that the Euler characteristic of a perverse sheaves on a semi-abelian variety is non-negative. Our arguments use in an essential way the stratified Morse theory of Goresky-MacPherson.
{"title":"On the signed Euler characteristic property for subvarieties of abelian varieties","authors":"E. Elduque, C. Geske, L. Maxim","doi":"10.5427/jsing.2018.17p","DOIUrl":"https://doi.org/10.5427/jsing.2018.17p","url":null,"abstract":"We give an elementary proof of the fact that a pure-dimensional closed subvariety of a complex abelian variety has a signed intersection homology Euler characteristic. We also show that such subvarieties which, moreover, are local complete intersections, have a signed Euler-Poincar'e characteristic. Both results are special cases of the fact, proved by Franecki-Kapranov, that the Euler characteristic of a perverse sheaves on a semi-abelian variety is non-negative. Our arguments use in an essential way the stratified Morse theory of Goresky-MacPherson.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83054303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christian Ausoni, K. Hess, Brenda Johnson, I. Moerdijk, J. Scherer
From a bimodule $M$ over an exact category $C$, we define an exact category $Cltimes M$ with a projection down to $C$. This construction classifies certain split square zero extensions of exact categories. We show that the trace map induces an equivalence between the relative $K$-theory of $Cltimes M$ and its relative topological cyclic homology. When applied to the bimodule $hom(-,-otimes_AM)$ on the category of finitely generated projective modules over a ring $A$ one recovers the classical Dundas-McCarthy theorem for split square zero extensions of rings.
从一个精确范畴C$上的双模M$，我们定义了一个精确范畴C$乘以M$，其投影到C$。这个构造对精确范畴的某些分裂的平方零扩展进行了分类。我们证明了迹映射在Cl乘以M$的相对K$-理论和它的相对拓扑循环同调之间推导出一个等价。当应用于环上有限生成的射影模范畴上的双模$ home (-，-otimes_AM)$时，恢复了环的分裂平方零扩展的经典Dundas-McCarthy定理。
{"title":"An Alpine Bouquet of Algebraic Topology","authors":"Christian Ausoni, K. Hess, Brenda Johnson, I. Moerdijk, J. Scherer","doi":"10.1090/CONM/708","DOIUrl":"https://doi.org/10.1090/CONM/708","url":null,"abstract":"From a bimodule $M$ over an exact category $C$, we define an exact category $Cltimes M$ with a projection down to $C$. This construction classifies certain split square zero extensions of exact categories. We show that the trace map induces an equivalence between the relative $K$-theory of $Cltimes M$ and its relative topological cyclic homology. When applied to the bimodule $hom(-,-otimes_AM)$ on the category of finitely generated projective modules over a ring $A$ one recovers the classical Dundas-McCarthy theorem for split square zero extensions of rings.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85631164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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