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In this paper, we present relative retracts and we can say that these are multilevel retracts which either retain given properties depending on the level or not. Some properties are constant and are present on every level. These properties are especially important in regard to the theory of coincidence. The class of relative retracts consists of retracts in the sense of Borsuk, multiretracts and many fundamental retracts.

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. InHigher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

Since its introduction by Friedhelm Waldhausen in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing Waldhausen's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a \"desingularization,\" improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

We characterize the topological spaces of minimum cardinality which are weakly contractible but not contractible. This is equivalent to finding the non-dismantlable posets of minimum cardinality such that the geometric realization of their order complexes are contractible. Specifically, we prove that all weakly contractible topological spaces with fewer than nine points are contractible. We also prove that there exist (up to homeomorphism) exactly two topological spaces of nine points which are weakly contractible but not contractible.

This paper discusses two basic issues of functional integration: domains of integration and volume elements adapted to a given domain of integration. Two examples of domain of integration are given explicitly in Sections 2 and 3 respectively: the domain of integration is a space of contractible paths and the domain of integration is a space of Poisson paths. A property of volume element, presented in Section 3, namely the Koszul formula, valid on totally different geometries (riemannian, symplectic, grassman) can be used for some infinite dimensional geometries.

In this article the concept of classifying space of a group is generalized to a classifying space of an arbitrary permutation representation. An example of this classifying space is given by a generalization of the infinite join construction that defines the standard example of a classifying space of a group. In a previous paper of the author, the join of two permutation representations was defined, and it was shown that the cohomology ring of the join was trivial. In this paper the classifying space of the join of two permutation representations is shown to be the topological join of the two classifying spaces and from this the triviality of the cup-product is derived topologically.

The classical concept of a twisted product is examined within the framework of partial actions. Specifically, we demonstrate that the globalization of a partial action results in a twisted product. Additionally, we outline conditions for the metrizability of twisted products and prove certain homotopy and categorical properties. Moreover, sufficient conditions are established for the enveloping space to be an equivariant absolute neighborhood extensor.

Let M = S n /Γ and h be a nontrivial element of finite order p in π 1 ( Μ ), where the integers n , p ≥ 2, Γ is a finite abelian group which acts freely and isometrically on the n -sphere and therefore M is diffeomorphic to a compact space form. In this paper, we prove that there are infinitely many non-contractible closed geodesics of class [ h ] on the compact space form with C r -generic Finsler metrics, where 4 ≤ r ≤ ∞. The conclusion also holds for C r -generic Riemannian metrics for 2 ≤ r ≤ ∞. The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms.

Let and be Banach algebras, let be a Banach algebra epimorphism from  to  , and let  be a nonzero character on  . As is known if is -contractible (amenable) then is -contractible (amenable). We prove that the converse is true under some conditions. As an important application, we study the -contractibility and amenability of the convolution algebra of trace class operators , where is a locally compact quantum group, and is a nonzero character on  .

Let A and B be Banach algebras and B be an algebraic Banach A-bimodule. Then we examine pseudo-amenability, pseudo-contractibility and property F of generalized module extension Banach algebra A (ProQuest: ... denotes formula omited.) B. Indeed, we obtain essential and sufficient condition for pseudo-amenability, pseudo-constructibility and property F of A (ProQuest: ... denotes formula omited.) B in term of pseudo-amenability, pseudo-contractibility and property F of A and B, respectively.