Algebraic Topology

Homotopy Theory is a fundamental branch of algebraic topology that studies topological spaces through their continuous deformations. Developed over the 20th century, it has evolved into a powerful tool with deep connections to algebra, geometry, and mathematical physics. This research paper explores the foundational concepts, key results, and diverse applications of Homotopy Theory in various mathematical disciplines.

The associations between single nucleotide polymorphisms (SNP) in the µ-calpain (CAPN1-316 and CANP1-4751) and calpastatin (CAST) genes and growth traits were evaluated in a population of Senepol, Charoláis and Senepol x Charolais bulls (n=99). In another study, associations were evaluated between these SNP and carcass traits of a subgroup of 42 animals submitted to three dietary treatments. The SNP CAPN1-316 was associated with daily weight gain at 205 and 240 d, and with weaning age, wherein animals with CG genotype were heavier, gained weight faster, and were younger at weaning than animals with GG genotype. The latter presented heavier Biceps femoris, Semitendinosus, and Gluteus spp. muscles, but CG animals had a superior Longissimus dorsiarea. The CAPN1-4751 gene was related to birth weight, Longissimus dorsi weight, and with muscle to bone ratio, and for these traits the TT genotype was superior to the CT. The CAPN1-316 and CAPN1-4751 C alleles were found to be correlated with...

Let M be a Mackey functor for a nite group G and let X be a pointed G-space. We dene a topological group F G (X;M), whose ho- motopy groups are isomorphic to the Bredon-Illman equivariant homology of X with coecients in a coecient system M associated to M. When M is a homological Mackey functor, we dene another topological group FG(X;M), whose homotopy groups are isomorphic to the Bredon-Illman equivariant homology of X with coecients in the covariant part of M. These topological groups are dened using simplicial groups F G (S(X);M) and FG(S(X);M), which have the same underlying groups, namely the groups of G-xed points F (Sn(X);M)G , where S(X) is the singular sim- plicial set of X. Furthermore, we study the transfer for nite covering G-maps and give its pullback property. We also analyze the composite of the transfer with the homomorphism induced by the projection map, in particular, in the case of (G; )-bundles. AMS Classication Primary 55N91,55R91; secondary 14F43,57M10 FG(X;M) ...

Dans la premiere partie de ce travail, on a etudie les grassmanniennes d'un espace de Hilbert separable, de dimension infinie, plus exactement le lien de la grassmannienne reguliere (hilbertienne) a ses composantes connexes, au groupe general lineaire restreint, et aux ouverts de l'atlas associes a sa structure hilbertienne. On a etudie aussi les composantes connexes d'une grassmannienne dense dans la grassmannienne reguliere, le lien de ses composantes connexes a sa decomposition en cellules de Schubert. A la fin de cette partie, on demontre le lien topologique qui existe entre les grassmanniennes de dimension infinie et les grassmanniennes de dimension finie. Dans la deuxieme partie, on a etudie le lien des groupes de lacets aux grassmanniennes, et l'equivalent de l'action de l'operateur vertex sur les elements de la grassmannienne associes a la fonction tau

Some methods have been used to express a finitely generated module over a principal ideal domain as a finite direct sum of its cyclic submodules. In this paper, we give an alternative technique to decompose a free module with finite rank over a principal ideal domain using eigen spaces of its endomorphism ring.

In this paper we discuss specific follow−up results from the upcoming paper about the 'theory of p−nomial triangles'. As they are strongly connected to 'p−nomial combinatorial number theory' and to 'probability calculus' we want to present some key developments. These include 'multivariate (inverse) continuous probabilities established via p−nomial combinatorial sums', topological probability 'Yvon spaces', which are not only constructed through different kind of −, but also with p − q−adic power series and product topologies, specific recurring 'Lechenault probability structures', the 'Vasnier elliptic curves in/on p−nomial manifolds' and the 'Pintenet surgery theory of m−dimensional p−nomial triangles'.

The operator χ of weak commutativity between isomorphic groups H and H ψ {H^{\psi}} , defined by χ ⁢ ( H ) = 〈 H , H ψ ∣ [ h , h ψ ] = 1 ⁢  for all  ⁢ h ∈ H 〉 , \chi(H)=\langle H,H^{\psi}\mid[h,h^{\psi}]=1\text{ for all }h\in H\rangle, is known to preserve group properties such as finiteness, solvability and polycyclicity. We introduce here the group construction ℰ ⁢ ( H ) = 〈 H , H ψ ∣ [ [ h 1 , h 2 ψ ] , h 3 - 1 ⁢ h 3 ψ ] = 1 ⁢  for all  ⁢ h 1 , h 2 , h 3 ∈ H 〉 ⁢ . {\mathcal{E}}(H)=\langle H,H^{\psi}\mid[[h_{1},h_{2}^{\psi}],h_{3}^{-1}h_{3}^{% \psi}]=1\text{ for all }h_{1},h_{2},h_{3}\in H\rangle\text{.} The group ℰ ⁢ ( H ) {{\mathcal{E}}(H)} maps onto χ ⁢ ( H ) {\chi(H)} and onto H ⊗ H {H\otimes H} , the non-abelian tensor square. The operator ℰ {{\mathcal{E}}} preserves solvability and preserves polycyclicity provided the abelianization H H ′ {\frac{H}{H^{\prime}}} is finite. Moreover, if H is perfect, then ℰ ⁢ ( H ) {{\mathcal{E}}(H)} is perfect and ℰ ⁢ ( H ) ≅ χ ⁢ ( H ) {{\mat...

It is known that the spectrum of the Gelfand pair $(U(n), H_{n})$ is homeomorphic to the Heisenberg fan. In this paper after defining a suitable notion of spectrum, we prove an analogous result for the generalized Gelfand pair $(U(p,q), H_{n})$, $p+q=n$.

The aim of this paper is to show that rigid syntomic cohomology – defined by Besser – is representable by a rational ring spectrum in the motivic homotopical sense. In fact, extending previous constructions, we exhibit a simple representability criterion and we apply it to several cohomologies in order to get our central result. This theorem gives new results for rigid syntomic cohomology such as h-descent and the compatibility of cycle classes with Gysin morphisms. Along the way, we prove that motivic ring spectra induce a complete Bloch–Ogus cohomological formalism and even more. Finally, following a general motivic homotopical philosophy, we exhibit a natural notion of rigid syntomic coefficients.

We find a necessary and sufficient condition for a compact 7-manifold to admit a $\tilde G_2$-structure. As a result we find a sufficient condition for an open 7-manifold to admit a closed 3-form of $\tilde G_2$-type. Comment: 4 pages, a new shorter exposition is given

We show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

In this note we provide a direct approach to the most basic operator in this theory namely the exterior derivative. The crucial ingredient is a calculus lemma based on determinants. We maintain the view that in a first course at least this direct approach is preferable to the more abstract one based on characterization of the exterior derivative in terms of its properties.

A emigração portuguesa cresceu nas vésperas do 25 de Abril, regrediu para níveis mínimos depois daquela data, voltou a crescer a partir de meados dos anos 80, acentuou-se no início do novo século, reduziu-se na crise financeira de 2008, atingiu de novo máximos históricos em 2013 e voltou a diminuir, lentamente, desde então. Em termos comparados, Portugal é hoje um país de repulsão migratória, característica que partilha com a maioria dos estados do Leste europeu. Neste artigo, apresentamos a história recente da emigração portuguesa, identificamos os mecanismos e episódios que a explicam e discutimos os problemas metodológicos envolvidos no seu estudo.