Ergun Yalcin
@w3.bilkent.edu.tr
Department of Mathematics
Bilkent University
RESEARCH, TEACHING, or OTHER INTERESTS
Geometry and Topology, Algebra and Number Theory
Scopus Publications
Scopus Publications
Ergün Yalçın
Springer Science and Business Media LLC
Ergün Yalçın
Elsevier BV
Matthew Gelvin and Ergün Yalçın
Elsevier BV
Muhammed Said Gündoğan and Ergün Yalçın
Springer Science and Business Media LLC
Olcay Coşkun and Ergün Yalçın
Elsevier BV
Cihan Okay and Ergün Yalçin
Mathematical Sciences Publishers
Given a spherical fibration $\\xi$ over the classifying space $BG$ of a finite group we define a dimension function for the $m-$fold fiber join of $\\xi$ where $m$ is some large positive integer. We show that the dimension functions satisfy the Borel-Smith conditions when $m$ is large enough. As an application we prove that there exists no spherical fibration over the classifying space of $\\text{Qd}(p)= (\\mathbb{Z}/p)^2\\rtimes\\text{SL}_2(\\mathbb{Z}/p)$ with $p-$effective Euler class, generalizing the result of \\"Ozg\\"un \\"Unl\\"u about group actions on finite complexes homotopy equivalent to a sphere. We have been informed that this result will also appear in a future paper as a corollary of a previously announced program on homotopy group actions due to Jesper Grodal.
Sune Precht Reeh and Ergün Yalçın
Springer Science and Business Media LLC
Ergün Yalçın
Elsevier BV
Ian Hambleton and Ergün Yalçin
International Press of Boston
Ian Hambleton and Ergün Yalçın
American Mathematical Society (AMS)
Let G G be a rank two finite group, and let H \\mathcal {H} denote the family of all rank one p p -subgroups of G G for which rank p ( G ) = 2 \\operatorname {rank}_p(G)=2 . We show that a rank two finite group G G which satisfies certain explicit group-theoretic conditions admits a finite G G -CW-complex X ≃ S n X\\simeq S^n with isotropy in H \\mathcal {H} , whose fixed sets are homotopy spheres. Our construction provides an infinite family of new non-linear G G -CW-complex examples.
Matthew Gelvin, Sune Precht Reeh, and Ergün Yalçın
Elsevier BV
Ian Hambleton and Ergün Yalçin
International Press of Boston
Let G be a finite group. The unit sphere in a finitedimensional orthogonal G-representation motivates the definition of homotopy representations, due to tom Dieck. We introduce an algebraic analogue and establish its basic properties, including the Borel–Smith conditions and realization by finite G-CW-complexes.
Özgün Ünlü and Ergün Yalçin
Cambridge University Press (CUP)
AbstractWe prove that if a finite group G acts smoothly on a manifold M such that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly on M × $\\mathbb{S}^{n_1}\\$ × … × $\\mathbb{S}^{n_k}$ for some positive integers n1, …, nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres, with trivial action on homology.
Ian Hambleton, Semra Pamuk, and Ergün Yalçin
European Mathematical Society - EMS - Publishing House GmbH
We give a general framework for studying G-CW complexes via the orbit category. As an application we show that the symmetric group G = S5 admits a nite G-CW complex X homotopy equivalent to a sphere, with cyclic isotropy subgroups.
Osman Berat Okutan and Ergün Yalçın
Mathematical Sciences Publishers
Ankara : The Department of Mathematics and the Graduate School of Engineering and Science of Bilkent University, 2012.
Ozgun Unlu and Ergun Yalcin
Indiana University Mathematics Journal
We prove that if a finite group $G$ has a representation with fixity $f$, then it acts freely and homologically trivially on a finite CW-complex homotopy equivalent to a product of $f+1$ spheres. This shows, in particular, that every finite group acts freely and homologically trivially on some finite CW-complex homotopy equivalent to a product of spheres.
Jonathan Pakianathan and Ergün Yalçın
Elsevier BV
Özgün Ünlü and Ergün Yalçın
Springer Science and Business Media LLC
A. Guclukan and E. Yalcin
Oxford University Press (OUP)
The subset complex (cid:2)(G) of a finite group G is defined as the simplicial complex whose simplices are non-empty subsets of G . The oriented chain complex of (cid:2)(G) gives a Z G -module extension of Z by ˜ Z , where ˜ Z is a copy of integers on which G acts via the sign representation of the regular representation. The extension class ζ G ∈ Ext | G |− 1 Z G ( Z , ˜ Z ) of this extension is called the Ext class or the Euler class of the subset complex (cid:2)(G) . This class was first introduced by Reiner and Webb [The combinatorics of the bar resolution in group cohomology, J. Pure Appl. Algebra 190 (2004), 291–327] who also raised the following question: What are the finite groups for which ζ G is non-zero? Inthispaper, we answer this question completely. We show that ζ G is non-zero if and only if G is an elementary abelian p -group or G is isomorphic to Z / 9, Z / 4 × Z / 4 or ( Z / 2 ) n × Z / 4 for some integer n ≥ 0. We obtain this result by first showing that ζ G is zero when G is a non-abelian group, then by calculating ζ G for specific abelian groups. The key ingredient in the proof is an observation by Mandell which says that the Ext class of the subset complex (cid:2)(G) is equal to the (twisted) Euler class of the augmentation module of the regular representation of G . We also give some applications of our results to group cohomology, to filtrations of modules and to the existence of Borsuk–Ulam type theorems.
Olcay Coşkun and Ergün Yalçın
Elsevier BV
Ergün Yalçın
American Mathematical Society (AMS)
In 1987, Serre proved that if G is a p-group which is not elementary abelian, then a product of Bocksteins of one dimensional classes is zero in the mod p cohomology algebra of G, provided that the product includes at least one nontrivial class from each line in H 1 (G, Fp). For p = 2, this gives that (σ G ) 2 = 0, where σ G is the product of all nontrivial one dimensional classes in H 1 (G, F 2 ). In this note, we prove that if G is a nonabelian 2-group, then σ G is also zero.
Jonathan Pakianathan and Ergün Yalçın
American Mathematical Society (AMS)
Let E E be a central extension of the form 0 → V → G → W → 0 0 \\to V \\to G \\to W \\to 0 where V V and W W are elementary abelian 2 2 -groups. Associated to E E there is a quadratic map Q : W → V Q: W \\to V , given by the 2 2 -power map, which uniquely determines the extension. This quadratic map also determines the extension class q q of the extension in H 2 ( W , V ) H^2(W,V) and an ideal I ( q ) I(q) in H 2 ( G , Z / 2 ) H^2(G, \\mathbb {Z} /2) which is generated by the components of q q . We say that E E is Bockstein closed if I ( q ) I(q) is an ideal closed under the Bockstein operator. We find a direct condition on the quadratic map Q Q that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map Q g l n : g l n ( F 2 ) → g l n ( F 2 ) Q_{\\mathfrak {gl}_n}: \\mathfrak {gl}_n (\\mathbb {F}_2)\\to \\mathfrak {gl}_n (\\mathbb {F}_2) given by Q ( A ) = A + A 2 Q(\\mathbb {A})= \\mathbb {A} +\\mathbb {A} ^2 yield Bockstein closed extensions. On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension 0 → M → G ~ → W → 0 0 \\to M \\to \\widetilde {G} \\to W \\to 0 for some Z / 4 [ W ] \\mathbb {Z} /4[W] -lattice M M . In this situation, one may write β ( q ) = L q \\beta (q)=Lq for a “binding matrix” L L with entries in H 1 ( W , Z / 2 ) H^1(W, \\mathbb {Z}/2) . We find a direct way to calculate the module structure of M M in terms of L L . Using this, we study extensions where the lattice M M is diagonalizable/triangulable and find interesting equivalent conditions to these properties.
Yusuf Civan and Ergün Yalçın
Elsevier BV
Serge Bouc and Ergün Yalçın
Elsevier BV
Robert Hartmann and Ergün Yalçın
Elsevier BV