Journal of Homotopy and Related Structures最新文献

By careful analysis of the embedding of a simplicial set into its image under Kan’s (mathop {mathop {mathsf {Ex}}^infty }) functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a strong anodyne extension. From this description we can quickly deduce some basic facts about (mathop {mathop {mathsf {Ex}}^infty }) and hence provide a new construction of the Kan–Quillen model structure on simplicial sets, one which avoids the use of topological spaces or minimal fibrations.

We prove that the parallel transport of a flat (n-1)-gerbe on any given target space gives rise to an n-dimensional extended homotopy quantum field theory. In case the target space is the classifying space of a finite group, we provide explicit formulae for this homotopy quantum field theory in terms of transgression. Moreover, we use the geometric theory of orbifolds to give a dimension-independent version of twisted and equivariant Dijkgraaf–Witten models. Finally, we introduce twisted equivariant Dijkgraaf–Witten theories giving us in the 3-2-1-dimensional case a new class of equivariant modular tensor categories which can be understood as twisted versions of the equivariant modular categories constructed by Maier, Nikolaus and Schweigert.

An (A_n)-algebra (A= (A,m_1, m_2, ldots , m_n)) is a special kind of (A_infty )-algebra satisfying the (A_infty )-relations involving just the (m_i) listed. We consider obstructions to extending an (A_{n-1}) algebra to an (A_n)-algebra. We enhance the known techniques by extending the Bousfield–Kan spectral sequence to apply to the homotopy groups of the space of minimal (i.e.?(m_1=0))(A_infty )-algebra structures on a given graded projective module. We also consider the Bousfield–Kan spectral sequence for the moduli space of (A_infty )-algebras. We compute up to the (E_2) terms and differentials (d_2) of these spectral sequences in terms of Hochschild cohomology.

In this paper we study the mod 2 cohomology ring of the Grasmannian (widetilde{G}_{n,3}) of oriented 3-planes in ({mathbb {R}}^n). We determine the degrees of the indecomposable elements in the cohomology ring. We also obtain an almost complete description of the cohomology ring. This description allows us to provide lower and upper bounds on the cup length of (widetilde{G}_{n,3}). As another application, we show that the upper characteristic rank of (widetilde{G}_{n,3}) equals the characteristic rank of (widetilde{gamma }_{n,3}), the oriented tautological bundle over (widetilde{G}_{n,3}) if n is at least 8.

Motivated by the theory of representability classes by submanifolds, we study the rational homotopy theory of Thom spaces of vector bundles. We first give a Thom isomorphism at the level of rational homotopy, extending work of Félix-Oprea-Tanré by removing hypothesis of nilpotency of the base and orientability of the bundle. Then, we use the theory of weight decompositions in rational homotopy to give a criterion of representability of classes by submanifolds, generalising results of Papadima. Along the way, we study issues of formality and give formulas for Massey products of Thom spaces. Lastly, we link the theory of weight decompositions with mixed Hodge theory and apply our results to motivic Thom spaces.

In the first part of this paper, we propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber of the bundle. Afterwards, we define a family of invariants of principal bundles that detect the number of group reductions that a principal bundle admits. We prove that they fit into a long exact sequence of abelian groups, together with the cohomology of the base space and the cohomology of the classifying space of the structure group.

Let M be an orientable, simply-connected, closed, non-spin?4-manifold and let ({mathcal {G}}_k(M)) be the gauge group of the principal G-bundle over M with second Chern class (kin {mathbb {Z}}). It is known that the homotopy type of ({mathcal {G}}_k(M)) is determined by the homotopy type of ({mathcal {G}}_k({mathbb {C}}{mathbb {P}}^2)). In this paper we investigate properties of ({mathcal {G}}_k({mathbb {C}}{mathbb {P}}^2)) when (G=SU(n)) that partly classify the homotopy types of the gauge groups.

The main goal of this paper is to define an invariant (mc_{infty }(f)) of homotopy classes of maps (f:X rightarrow Y_{mathbb {Q}}), from a finite CW-complex X to a rational space (Y_{mathbb {Q}}). We prove that this invariant is complete, i.e. (mc_{infty }(f)=mc_{infty }(g)) if and only if f and g are homotopic. To construct this invariant we also construct a homotopy Lie algebra structure on certain convolution algebras. More precisely, given an operadic twisting morphism from a cooperad (mathcal {C}) to an operad (mathcal {P}), a (mathcal {C})-coalgebra C and a (mathcal {P})-algebra A, then there exists a natural homotopy Lie algebra structure on (Hom_mathbb {K}(C,A)), the set of linear maps from C to A. We prove some of the basic properties of this convolution homotopy Lie algebra and use it to construct the algebraic Hopf invariants. This convolution homotopy Lie algebra also has the property that it can be used to model mapping spaces. More precisely, suppose that C is a (C_infty )-coalgebra model for a simply-connected finite CW-complex X and A an (L_infty )-algebra model for a simply-connected rational space (Y_{mathbb {Q}}) of finite (mathbb {Q})-type, then (Hom_mathbb {K}(C,A)), the space of linear maps from C to A, can be equipped with an (L_infty )-structure such that it becomes a rational model for the based mapping space (Map_*(X,Y_mathbb {Q})).

We take a direct approach to computing the orbits for the action of the automorphism group (mathbb {G}_2) of the Honda formal group law of height 2 on the associated Lubin–Tate rings (R_2). We prove that ((R_2/p)_{mathbb {G}_2} cong mathbb {F}_p). The result is new for (p=2) and (p=3). For primes (pge 5), the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods.

We introduce and study a homology theory of crossed modules with coefficients in an abelian crossed module. We discuss the basic properties of these new homology groups and give some applications. We then restrict our attention to the case of integral coefficients. In this case we regain the homology of crossed modules originally defined by Baues and further developed by Ellis. We show that it is an instance of Barr–Beck comonadic homology, so that we may use a result of Everaert and Gran to obtain Hopf formulae in all dimensions.