Pub Date : 2020-12-02DOI: 10.1016/j.topol.2020.107472
Soumen Sarkar, Jongbaek Song
{"title":"GKM theory for orbifold stratified spaces and application to singular toric varieties","authors":"Soumen Sarkar, Jongbaek Song","doi":"10.1016/j.topol.2020.107472","DOIUrl":"https://doi.org/10.1016/j.topol.2020.107472","url":null,"abstract":"","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90441906","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}
Let $G$ be a finite group and $mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen conjectured that $O_p(G)$ is nontrivial if $mathcal{A}_p(G)$ is contractible. We prove that $O_p(G)neq 1$ for any group $G$ admitting a $G$-invariant acyclic $p$-subgroup complex of dimension $2$. In particular, it follows that Quillen's conjecture holds for groups of $p$-rank $3$. We also apply this result to establish Quillen's conjecture for some particular groups not considered in the seminal work of Aschbacher--Smith.
{"title":"Acyclic 2-dimensional complexes and Quillen’s conjecture","authors":"K. I. Piterman, Iván Sadofschi Costa, A. Viruel","doi":"10.5565/publmat6512104","DOIUrl":"https://doi.org/10.5565/publmat6512104","url":null,"abstract":"Let $G$ be a finite group and $mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen conjectured that $O_p(G)$ is nontrivial if $mathcal{A}_p(G)$ is contractible. We prove that $O_p(G)neq 1$ for any group $G$ admitting a $G$-invariant acyclic $p$-subgroup complex of dimension $2$. In particular, it follows that Quillen's conjecture holds for groups of $p$-rank $3$. We also apply this result to establish Quillen's conjecture for some particular groups not considered in the seminal work of Aschbacher--Smith.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88649609","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 compute the compactly supported Euler characteristic of the space of degree $d$ irreducible polynomials in $n$ variables with real coefficients and show that the values are given by the digits in the so-called balanced binary expansion of the number of variables $n$.
{"title":"Euler characteristic of the space of real multivariate irreducible polynomials","authors":"Trevor Hyde","doi":"10.1090/proc/15849","DOIUrl":"https://doi.org/10.1090/proc/15849","url":null,"abstract":"We compute the compactly supported Euler characteristic of the space of degree $d$ irreducible polynomials in $n$ variables with real coefficients and show that the values are given by the digits in the so-called balanced binary expansion of the number of variables $n$.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85332387","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 prove a filtered version of the Homotopy Transfer Theorem which gives an A-infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the geometry and topology of complex manifolds, using the Hodge filtration, as well as to complex algebraic varieties, using mixed Hodge theory
{"title":"Filtered A-infinity structures in complex geometry","authors":"J. Cirici, A. Sopena","doi":"10.1090/proc/16009","DOIUrl":"https://doi.org/10.1090/proc/16009","url":null,"abstract":"We prove a filtered version of the Homotopy Transfer Theorem which gives an A-infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the geometry and topology of complex manifolds, using the Hodge filtration, as well as to complex algebraic varieties, using mixed Hodge theory","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73688266","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 : 2020-10-14DOI: 10.4310/hha.2021.v23.n2.a12
Kouyemon Iriye, D. Kishimoto
We characterize two-dimensional Golod complexes combinatorially by vertex-breakability and topologically by the fat-wedge filtration of a polyhedral product. Applying the characterization, we consider a difference between Golodness over fields and rings, which enables us to give a two-dimensional simple Golod complex over any field such that the corresponding moment-angle complex is not a suspension.
{"title":"Two-dimensional Golod complexes","authors":"Kouyemon Iriye, D. Kishimoto","doi":"10.4310/hha.2021.v23.n2.a12","DOIUrl":"https://doi.org/10.4310/hha.2021.v23.n2.a12","url":null,"abstract":"We characterize two-dimensional Golod complexes combinatorially by vertex-breakability and topologically by the fat-wedge filtration of a polyhedral product. Applying the characterization, we consider a difference between Golodness over fields and rings, which enables us to give a two-dimensional simple Golod complex over any field such that the corresponding moment-angle complex is not a suspension.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83602165","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}
For $G$ a finite group, we show that functions on fields for the 2-dimensional supersymmetric sigma model with background $G$-symmetry determine cocycles for complex analytic $G$-equivariant elliptic cohomology. Similar structures in supersymmetric mechanics determine cocycles for equivariant K-theory with complex coefficients. The path integral for gauge theory with a finite group constructs wrong-way maps associated to group homomorphisms. When applied to an inclusion of groups, we obtain the induced character formula of Hopkins, Kuhn, and Ravenel. For the homomorphism $Gto *$ we obtain Vafa's formula for gauging with discrete torsion. The image of equivariant Euler classes under gauging constructs modular form-valued invariants of representations that depend on a choice of string structure. We illustrate nontrivial dependence on the string structure for a 16-dimensional representation of the Klein 4-group.
{"title":"Equivariant elliptic cohomology, gauged sigma models, and discrete torsion","authors":"Daniel Berwick-Evans","doi":"10.1090/tran/8527","DOIUrl":"https://doi.org/10.1090/tran/8527","url":null,"abstract":"For $G$ a finite group, we show that functions on fields for the 2-dimensional supersymmetric sigma model with background $G$-symmetry determine cocycles for complex analytic $G$-equivariant elliptic cohomology. Similar structures in supersymmetric mechanics determine cocycles for equivariant K-theory with complex coefficients. The path integral for gauge theory with a finite group constructs wrong-way maps associated to group homomorphisms. When applied to an inclusion of groups, we obtain the induced character formula of Hopkins, Kuhn, and Ravenel. For the homomorphism $Gto *$ we obtain Vafa's formula for gauging with discrete torsion. The image of equivariant Euler classes under gauging constructs modular form-valued invariants of representations that depend on a choice of string structure. We illustrate nontrivial dependence on the string structure for a 16-dimensional representation of the Klein 4-group.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74436656","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 : 2020-10-12DOI: 10.4310/HHA.2021.V23.N2.A8
D. Kishimoto, Nobuyuki Oda
A topological spherical space form is the quotient of a sphere by a free action of a finite group. In general, their homotopy types depend on specific actions of a group. We show that the monoid of homotopy classes of self-maps of a topological spherical space form is determined by the acting group and the dimension of the sphere, not depending on a specific action.
{"title":"Monoids of self-maps of topological spherical space forms","authors":"D. Kishimoto, Nobuyuki Oda","doi":"10.4310/HHA.2021.V23.N2.A8","DOIUrl":"https://doi.org/10.4310/HHA.2021.V23.N2.A8","url":null,"abstract":"A topological spherical space form is the quotient of a sphere by a free action of a finite group. In general, their homotopy types depend on specific actions of a group. We show that the monoid of homotopy classes of self-maps of a topological spherical space form is determined by the acting group and the dimension of the sphere, not depending on a specific action.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81628520","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 : 2020-08-27DOI: 10.15673/tmgc.v13i2.1781
Олександра Олександрівна Хохлюк, S. Maksymenko
Let $M, N$ the be smooth manifolds, $mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with weak $C^{r}$ Whitney topology, and $mathcal{B} subset mathcal{C}^{r}(M,N)$ an open subset. It is proved that for $0leq r
设$M, N$为光滑流形，$mathcal{C}^{r}(M,N)$为具有弱$C^{r}$ Whitney拓扑的${C}^{r}$映射空间，$mathcal{B} subset mathcal{C}^{r}(M,N)$为开放子集。证明了对于$0leq r
{"title":"Smooth approximations and their applications to homotopy types","authors":"Олександра Олександрівна Хохлюк, S. Maksymenko","doi":"10.15673/tmgc.v13i2.1781","DOIUrl":"https://doi.org/10.15673/tmgc.v13i2.1781","url":null,"abstract":"Let $M, N$ the be smooth manifolds, $mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with weak $C^{r}$ Whitney topology, and $mathcal{B} subset mathcal{C}^{r}(M,N)$ an open subset. It is proved that for $0leq r<sleqinfty$ the inclusion $mathcal{B} cap mathcal{C}^{s}(M,N) subset mathcal{B}$ is a weak homotopy equivalence. It is also established a parametrized variant of such a result. In particular, it is shown that for a compact manifold $M$, the inclusion of the space of $mathcal{C}^{s}$ isotopies $[0,1]times M to M$ fixed near ${0,1}times M$ into the space of loops $Omega(mathcal{D}^{r}(M), mathrm{id}_{M})$ of the group of $mathcal{C}^{r}$ diffeomorphisms of $M$ at $mathrm{id}_{M}$ is a weak homotopy equivalence.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"112 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74414588","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 show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan correspondance of coalgebras in $infty$-categories. Moreover it shows that Shipley's zig-zag of Quillen equivalences provides an explicit symmetric monoidal equivalence of $infty$-categories for the stable Dold-Kan correspondance. We study homotopy coherent coalgebras associated to a monoidal model category and we show that these coalgebras cannot be rigidified. That is, their $infty$-categories are not equivalent to the Dwyer-Kan localizations of strict coalgebras in the usual monoidal model categories of spectra and of connective discrete $R$-modules.
{"title":"Coalgebras in the Dwyer-Kan localization of a model category","authors":"Maximilien P'eroux","doi":"10.1090/proc/15949","DOIUrl":"https://doi.org/10.1090/proc/15949","url":null,"abstract":"We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan correspondance of coalgebras in $infty$-categories. Moreover it shows that Shipley's zig-zag of Quillen equivalences provides an explicit symmetric monoidal equivalence of $infty$-categories for the stable Dold-Kan correspondance. We study homotopy coherent coalgebras associated to a monoidal model category and we show that these coalgebras cannot be rigidified. That is, their $infty$-categories are not equivalent to the Dwyer-Kan localizations of strict coalgebras in the usual monoidal model categories of spectra and of connective discrete $R$-modules.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"476 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84254178","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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