Journal of Homotopy and Related Structures最新文献

Let A be a finite abelian p-group of rank at least 2. We show that (E^0(BA)/I_{tr}), the quotient of the Morava E-cohomology of A by the ideal generated by the image of the transfers along all proper subgroups, contains p-torsion. The proof makes use of transchromatic character theory.

Let X be a simply connected space with finite-dimensional rational homotopy groups. Let (p_infty :UE rightarrow Bmathrm {aut}_1(X)) be the universal fibration of simply connected spaces with fibre X. We give a DG Lie algebra model for the evaluation map ( omega :mathrm {aut}_1(Bmathrm {aut}_1(X_mathbb {Q})) rightarrow Bmathrm {aut}_1(X_mathbb {Q})) expressed in terms of derivations of the relative Sullivan model of (p_infty ). We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space (Bmathrm {aut}_1(X_mathbb {Q})) as a consequence. We also prove that (mathbb {C} P^n_mathbb {Q}) cannot be realized as (Bmathrm {aut}_1(X_mathbb {Q})) for (n le 4) and X with finite-dimensional rational homotopy groups.

We consider the décalage construction ({{,mathrm{Dec},}}) and its right adjoint (T). These functors are induced on the category of simplicial objects valued in any bicomplete category ({mathcal {C}}) by the ordinal sum. We identify (T{{,mathrm{Dec},}}X) with the path object (X^{Delta [1]}) for any simplicial object X. We then use this formula to produce an explicit retracting homotopy for the unit (Xrightarrow T{{,mathrm{Dec},}}X) of the adjunction (({{,mathrm{Dec},}},T)). When ({mathcal {C}}) is a category of objects of an algebraic nature, we then show that the unit is a weak equivalence of simplicial objects in ({mathcal {C}}).

Manifold calculus is a form of functor calculus that analyzes contravariant functors from some categories of manifolds to topological spaces by providing analytic approximations to them. In this paper, using the technique of the h-principle, we show that for a symplectic manifold N, the analytic approximation to the Lagrangian embeddings functor (mathrm {Emb}_{mathrm {Lag}}(-,N)) is the totally real embeddings functor (mathrm {Emb}_{mathrm {TR}}(-,N)). More generally, for subsets ({mathcal {A}}) of the m-plane Grassmannian bundle ({{,mathrm{{Gr}},}}(m,TN)) for which the h-principle holds for ({mathcal {A}})-directed embeddings, we prove the analyticity of the ({mathcal {A}})-directed embeddings functor ({{,mathrm{Emb},}}_{{mathcal {A}}}(-,N)).

K. Borsuk in 1979, at the Topological Conference in Moscow, introduced the concept of capacity and depth of a compactum. In this paper we compute the capacity and depth of compact surfaces. We show that the capacity and depth of every compact orientable surface of genus (gge 0) is equal to (g+2). Also, we prove that the capacity and depth of a compact non-orientable surface of genus (g>0) is ([frac{g}{2}]+2).

The functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the self-enrichment of the category of abelian groups. If one considers enrichments into symmetric sequences or even bisymmetric sequences, one can produce an endomorphism operad or an endomorphism properad. In this note, we show that more generally, given a category enriched in a monoidal category , the functor that associates to a monoid in its category of representations in is adjoint to the functor that computes the endomorphism monoid of any functor with domain . After describing the first results of the theory we give several examples of applications.

We prove that the cyclic homology of a saturated (A_infty ) category admits the structure of a ‘polarized variation of Hodge structures’, building heavily on the work of many authors: the main point of the paper is to present complete proofs, and also explicit formulae for all of the relevant structures. This forms part of a project of Ganatra, Perutz and the author, to prove that homological mirror symmetry implies enumerative mirror symmetry.

We consider two families of spaces, X: the closed orientable Riemann surfaces of genus (g>0) and the classifying spaces of right-angled Artin groups. In both cases we compare the depth of the fundamental group with the depth of an associated Lie algebra, L, that can be determined by the minimal Sullivan algebra. For these spaces we prove that

and give precise formulas for the depth.

Lie algebras and groups equipped with a multiplication (mu ) satisfying some compatibility properties are studied. These structures are called symmetric Lie (mu )-algebras and symmetric (mu )-groups respectively. An equivalence of categories between symmetric Lie (mu )-algebras and symmetric Leibniz algebras is established when 2 is invertible in the base ring. The second main result of the paper is an equivalence of categories between simply connected symmetric Lie (mu )-groups and finite dimensional symmetric Leibniz algebras.