Browsing by Author "Erdal, Mehmet Akif"
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Item Open Access Cobordism calculations with Adams and James spectral sequences(2010) Erdal, Mehmet AkifLet ξn : Z/p → U(n) be an n-dimensional faithful complex representation of Z/p and in : U(n)→O(2n) be inclusion for n ≥ 1. Then the compositions in ◦ ξn and jn ◦ in ◦ ξn induce fibrations on BZ/p where jn : O(2n) → O(2n + 1) is the usual inclusion. Let (BZ/p, f) be a sequence of fibrations where f2n : BZ/p→BO(2n) is the composition Bin ◦ Bξn and f2n+1 : BZ/p→BO(2n + 1) is the composition Bjn ◦Bin ◦Bξn. By Pontrjagin-Thom theorem the cobordism group Ωm(BZ/p, f) of m-dimensional (BZ/p, f) manifolds is isomorphic to π s m(MZ/p, ∗) where MZ/p denotes the Thom space of the bundle over BZ/p that pullbacks to the normal bundle of manifolds representing elements in Ωm(BZ/p, f). We will use the Adams and James Spectral Sequences to get information about Ωm(BZ/p, f), when p = 3.Item Open Access A model structure via orbit spaces for equivariant homotopy(Springer, 2019-06-26) Erdal, Mehmet Akif; Güçlükan İlhan, AslıLet G be discrete group and FF be a collection of subgroups of G. We show that there exists a left induced model structure on the category of right G-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on H-orbits for all H in FF. This gives a model categorical criterion for maps that induce weak equivalences on H-orbits to be weak equivalences in the FF-model structure.Item Open Access Monoid actions, their categorification and applications(2016-12) Erdal, Mehmet AkifWe study actions of monoids and monoidal categories, and their relations with (co)homology theories. We start by discussing actions of monoids via bi-actions. We show that there is a well-defined functorial reverse action when a monoid action is given, which corresponds to acting by the inverses for group actions. We use this reverse actions to construct a homotopical structure on the category of monoid actions, which allow us to build the Burnside ring of a monoid. Then, we study categorifications of the previously introduced notions. In particular, we study actions of monoidal categories on categories and show that the ideas of action reversing of monoid actions extends to actions of monoidal categories. We use the reverse action for actions of monoidal categories, along with homotopy theory, to define homology, cohomology, homotopy and cohomotopy theories graded over monoidal categories. We show that most of the existing theories fits into our setting; and thus, we unify the existing definitions of these theories. Finally, we construct the spectral sequences for the theories graded over monoidal categories, which are the strongest tools for computation of cohomology and homotopy theories in existence.Item Open Access On smooth manifolds with the homotopy type of a homology sphere(Elsevier, 2018) Erdal, Mehmet AkifIn this paper we study M(X), the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of X, via a map Ψ from M(X) into the quotient of K(X)=[X,BSO] by the action of the group of homotopy classes of simple self equivalences of X. The map Ψ describes which bundles over X can occur as normal bundles of manifolds in M(X). We determine the image of Ψ when X belongs to a certain class of homology spheres. In particular, we find conditions on elements of K(X) that guarantee they are pullbacks of normal bundles of manifolds in M(X).Item Open Access Semigroup actions on sets and the burnside ring(Springer Science, 2018) Erdal, Mehmet Akif; Ünlü, ÖzgünIn this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gr¨othendieck group.