A note on homotopy equivalences

Let M be a compact connected Riemannian manifold whose sectional curvature values are all nonpositive. Let T denote the fundamental group of M. We prove that any homotopy equivalence : N -> M from a compact closed manifold TV is homotopic to a homeomorphism, provided that m > 5 where m = dim M . We show that the surgery L-group L k+m (T, w { ) is isomorphic to the set of homotopy classes of k k

In this study, we introduce the concepts of fuzzy closure Hopf space and fuzzy closure Hopf group within the framework of fuzzy closure spaces, using homotopy theory. We investigate the relationships between the fuzzy closure Hopf group and its homotopy equivalence. Furthermore, we demonstrate the existence of a contravariant functor from the category of fuzzy closure Hopf spaces and the continuous functions, to the category of groups and homomorphisms. This is demonstrated by illustrating that the set of homotopy function classes among fuzzy closure Hopf groups constitutes a group.

Since we have removed 0 + jO from the complex plane, the Nyquist map D × Ω (ℂ \ {0 + j0}, if it exists, could have a nontrivial homotopy class. The set of all possible homotopy classes of all such maps depends essentially on D. This chapter addresses the question as to what homotopy classes can be expected for a given uncertainty space. In case of such uncontractible uncertainty spaces as toris—multichannel phase margin problems—the set of all homotopy classes is quite large. The next question is to determine which of these homotopy classes yield closed-loop stability. Finally, given an uncertainty structure D, it is shown how the loop function L(s) induces a homotopy class of the map.

This book introduces a new context for global homotopy theory, i.e., equivariant homotopy theory with universal symmetries. Many important equivariant theories naturally exist not just for a particular group, but in a uniform way for all groups in a specific class. Prominent examples are equivariant stable homotopy, equivariant $K$-theory or equivariant bordism. Global equivariant homotopy theory studies such uniform phenomena, i.e., the adjective `global' refers to simultaneous and compatible actions of all compact Lie groups. We give a self-contained treatment of unstable and stable global homotopy theory, modeled by orthogonal spaces respectively orthogonal spectra under global equivalences. Specific topics include the global stable homotopy category, operations on equivariant homotopy groups, global model structures, and ultra-commutative multiplications. The book includes many explicit examples and detailed calculations.

Homotopy theory and simplicial groupoids

Morita Equivalence of Twisted Crossed Products by Coactions

Detecting homotopy equivalences in base-point-free homotopy

The foundations of equivariant homotopy and cellular theory are examined; an equivariant Whitehead theorem is proved, and the classical results by Milnor about spaces with the homotopy-type of a CW complex are generalized to the equivariant case. The ambient group G is assumed compact Lie. Further results include equivariant cellular approximation and the procedure for replacement of an arbitrary G-space by a G-CW complex.

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’ alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an R O ( G ) R O(G) -grading. Our model for genuine proper G G -equivariant stable homotopy theory is the category of orthogonal G G -spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of G G . This class of π ∗ \pi _* -isomorphisms is part of a symmetric monoidal stable model structure, and the associated tensor triangulated homotopy category is compactly generated. Consequently, every orthogonal G G -spectrum represents an equivariant cohomology theory on the category of G G -spaces. These represented cohomology theories are designed to only depend on the ‘proper G G -homotopy type’, tested by fixed points under all compact subgroups. An important special case of our theory are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group theory; for example, the G G -sphere spectrum is a compact object in our triangulated equivariant homotopy category if the universal space for proper G G -actions has a finite G G -CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper G G -CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by G G -vector bundles. Via this description, we can identify the previously defined G G -cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal G G -spectra.

Every principal G-bundle over X is classified up to equivalence by a homotopy class X -> BG, where BG is the classifying space of G. On the other hand, for every nice topological space X Milnor constructed a strict model of its loop space (Omega) over tildeX, that is a group. Moreover, the morphisms of topological groups (Omega) over tildeX -> G generate all the G-bundles over X up to equivalence. In this paper, we show that the relation between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles over a fixed space, that is dual to the classification of bundles with a fixed group. Such a result clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space, which are very important in topological K-theory, group cohomology, and homotopy theory.

We study frameworks for the equivalence of abstract state-transition systems represented as posets. A basic notion of equivalence is proposed. A least fix-point operator transforms basic equivalences into strong equivalences (=Lagois Connections) which makes Lagois Connections into a category. In the absence of divergence, the two notions of equivalence coincide. We generalize these notions by adding a logical level to express divergence more precisely. Then both generalized notions of equivalence coincide even in the presence of divergence.KeywordsClosure OperatorOperational SemanticLogical LevelSemantic FunctionDenotational SemanticThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this last chapter we want to give some idea of the “anabelian” program of A. Grothendieck. The term anabelian should be read as “far from being abelian” and as we understand the matter, a group is far enough away from being abelian if all of its subgroups of finite index have a trivial center. The principal idea is the following: in topology, a space X of type K(π,1) is determined by its fundamental group π up to weak homotopy equivalence. If we require that X is a CW-complex, then X is already determined up to strong homotopy equivalence. The “anabelian” idea is that something similar should also be true for schemes, i.e. a scheme X which is an étale K(π,1) should be essentially reconstructible from its étale fundamental group. This is obviously not correct in general, but it should be true under certain conditions; for example, X should be absolutely finitely generated and $\pi^{\mathit{et}}_{1}(X)$ is supposed to be “anabelian”. The smallest constituents of this anabelian world are points, i.e. spectra of fields which are finitely generated over their prime fields. Here the étale fundamental group is just the absolute Galois group. Finite fields have an abelian absolute Galois group (in fact all these fields have the same Galois group $\hat{\mathbb{Z}}$ ), and so the first objects of interest are global fields. In §1,2 we will present some results on “anabelian properties” of global fields, which already existed before Grothendieck formulated his program. We will explain the general conjectures in §3.

We study the etale homotopy theory of Brauer-Severi varieties over fields of characteristic 0. We prove that the induced Galois representations on geometric homotopy invariants (e.g., l-adic cohomology or higher homotopy groups) are all isomorphic for Brauer-Severi varieties of the same dimension. If the base field has cohomological dimension smaller or equal 2 then we can show more in the case of Brauer-Severi curves: There is even an isomorphism between the Hochschild-Serre spectral sequences computing cohomology with local coefficients. Further, we study homotopy rational and homotopy fixed points on Brauer-Severi varieties and their connections to genuine rational points. In particular, we show that under a suitable assumption on the first profinite Chern class map an analogue of the weak section conjecture for Brauer-Severi varieties turns out to be true. We can give a counter example to this analogue without the extra assumption over p-adic local fields.

S. Ferry [7] constructed a compactum shape equivalent but not CE equivalent to the circle.We show that suspensions of shape equivalent spaces are CE equivalent A map p: X -F is called CE if p is surjective and each point-inverse has trivial shape.If A and F are ANR's, then p is a simple homotopy equivalence (T. A. Chapman [2, 3], J. West [8], cf.Ferry [6]).In contrast (Ferry [6]), homotopy equivalent compacta are CE equivalent, that is, connected by a sequence of CE maps A -A, -X2 -A3-F.

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