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Abstract: 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. PubDate: 2019-03-13 DOI: 10.1007/s40062-019-00232-5

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Abstract: 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 \(k\in {\mathbb {Z}}\) . 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. PubDate: 2019-03-12 DOI: 10.1007/s40062-019-00233-4

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Abstract: We discuss two categorical characterizations of the class of acyclic maps between spaces. The first one is in terms of the higher categorical notion of an epimorphism. The second one employs the notion of a balanced map, that is, a map whose homotopy pullbacks along \(\pi _0\) -surjective maps define also homotopy pushouts. We also identify the modality in the homotopy theory of spaces that is defined by the class of acyclic maps, and discuss the content of the generalized Blakers–Massey theorem for this modality. PubDate: 2019-03-04 DOI: 10.1007/s40062-019-00231-6

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Abstract: Considering the potential equivariant formality of the left action of a connected Lie group K on the homogeneous space G / K, we arrive through a sequence of reductions at the case G is compact and simply-connected and K is a torus. We then classify all pairs (G, S) such that G is compact connected Lie and the embedded circular subgroup S acts equivariantly formally on G / S. In the process we provide what seems to be the first published proof of the structure (known to Leray and Koszul) of the cohomology rings PubDate: 2019-03-01 DOI: 10.1007/s40062-018-0207-5

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Abstract: In this paper we use topological tools to investigate the structure of the algebraic K-groups \(K_4(R)\) for \(R=Z[i]\) and \(R=Z[\rho ]\) where \(i := \sqrt{-1}\) and \(\rho := (1+\sqrt{-3})/2\) . We exploit the close connection between homology groups of \(\mathrm {GL}_n(R)\) for \(n\le 5\) and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which \(\mathrm {GL}_n(R)\) acts. Our main result is that \(K_{4} ({\mathbb {Z}}[i])\) and \(K_{4} ({\mathbb {Z}}[\rho ])\) have no p-torsion for \(p\ge 5\) . PubDate: 2019-03-01 DOI: 10.1007/s40062-018-0212-8

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Abstract: We revisit Auslander–Buchweitz approximation theory and find some relations with cotorsion pairs and model category structures. From the notion of relative generators, we introduce the concept of left Frobenius pairs \(({\mathcal {X}},\omega )\) in an abelian category \({\mathcal {C}}\) . We show how to construct from \(({\mathcal {X}},\omega )\) a projective exact model structure on \({\mathcal {X}}^\wedge \) , the subcategory of objects in \({\mathcal {C}}\) with finite \({\mathcal {X}}\) -resolution dimension, via cotorsion pairs relative to a thick subcategory of \({\mathcal {C}}\) . We also establish correspondences between these model structures, relative cotorsion pairs, Frobenius pairs, and Auslander–Buchweitz contexts. Some applications of this theory are given in the context of Gorenstein homological algebra, and connections with perfect cotorsion pairs, covering subcategories and cotilting modules are also presented and described. PubDate: 2019-03-01 DOI: 10.1007/s40062-018-0208-4

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Abstract: We use a simplicial product version of Quillen’s Theorem A to prove classical Waldhausen Additivity of \(wS_\bullet \) , which says that the “subobject” and “quotient” functors of cofiber sequences induce a weak equivalence \(wS_\bullet {\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}) \rightarrow wS_\bullet {\mathcal {A}}\times wS_\bullet {\mathcal {B}}\) . A consequence is Additivity for the Waldhausen K-theory spectrum of the associated split exact sequence, namely a stable equivalence of spectra \({\mathbf {K}}({\mathcal {A}}) \vee {\mathbf {K}}({\mathcal {B}}) \rightarrow {\mathbf {K}}({\mathcal {E}}({\mathcal {A}},{\mathcal {C}},{\mathcal {B}}))\) . This paper is dedicated to transferring these proofs to the quasicategorical setting and developing Waldhausen quasicategories and their sequences. We also give sufficient conditions for a split exact sequence to be equivalent to a standard one. These conditions are always satisfied by stable quasicategories, so Waldhausen K-theory sends any split exact sequence of pointed stable quasicategories to a split cofiber sequence. Presentability is not needed. In an effort to make the article self-contained, we recall all the necessary results from the theory of quasicategories, and prove a few quasicategorical results that are not in the literature. PubDate: 2019-03-01 DOI: 10.1007/s40062-018-0206-6

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Abstract: Building on our previous work, we show that the category of non-negatively graded chain complexes of \(\mathcal {D}_X\) -modules – where X is a smooth affine algebraic variety over an algebraically closed field of characteristic zero – fits into a homotopical algebraic context in the sense of Toën and Vezzosi. PubDate: 2019-03-01 DOI: 10.1007/s40062-018-0213-7

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Abstract: The existence of a model structure on the category \({\mathcal {D}}\) of diffeological spaces is crucial to developing smooth homotopy theory. We construct a compactly generated model structure on the category \({\mathcal {D}}\) whose weak equivalences are just smooth maps inducing isomorphisms on smooth homotopy groups. The essential part of our construction of the model structure on \({\mathcal {D}}\) is to introduce diffeologies on the sets \(\varDelta ^{p}\) \((p \ge 0)\) such that \(\varDelta ^{p}\) contains the \(k\mathrm{th}\) horn \(\varLambda ^{p}_{k}\) as a smooth deformation retract. PubDate: 2019-03-01 DOI: 10.1007/s40062-018-0209-3

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Abstract: This paper introduces an inductive tree notation for all the faces of polytopes arising from a simplex by truncations, which allows viewing face inclusion as the process of contracting tree edges. These polytopes, known as hypergraph polytopes or nestohedra, fit in the interval from simplices to permutohedra (in any finite dimension). This interval was further stretched by Petrić to allow truncations of faces that are themselves obtained by truncations. Our notation applies to all these polytopes. As an illustration, we detail the case of Petrić’s permutohedron-based associahedra. As an application, we present a criterion for determining whether edges of polytopes associated with the coherences of categorified operads correspond to sequential, or to parallel associativity. PubDate: 2019-03-01 DOI: 10.1007/s40062-018-0211-9

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Abstract: We prove that the normalization functor of the Dold-Kan correspondence does not induce a Quillen equivalence between Goerss’ model category of simplicial coalgebras and Getzler–Goerss’ model category of differential graded coalgebras. PubDate: 2019-03-01 DOI: 10.1007/s40062-018-0210-x

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Abstract: Tate cohomology (as well as Borel homology and cohomology) of connective K-theory for \(G=({\mathbb {Z}}/2)^n\) was completely calculated by Bruner and Greenlees (The connective K-theory of finite groups, 2003). In this note, we essentially redo the calculation by a different, more elementary method, and we extend it to \(p>2\) prime. We also identify the resulting spectra, which are products of Eilenberg–Mac Lane spectra, and finitely many finite Postnikov towers. For \(p=2\) , we also reconcile our answer completely with the result of [2], which is in a different form, and hence the comparison involves some non-trivial combinatorics. PubDate: 2019-01-10 DOI: 10.1007/s40062-018-00229-6

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Abstract: 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 \(Map_*(X,Y_\mathbb {Q})\) . PubDate: 2019-01-03 DOI: 10.1007/s40062-018-00230-z

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Abstract: 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 \(p\ge 5\) , the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods. PubDate: 2018-12-18 DOI: 10.1007/s40062-018-00228-7

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Abstract: We discuss a new strategy for the computation of the Hopf-cyclic cohomology of the Connes–Moscovici Hopf algebra \(\mathcal{H}_n\) . More precisely, we introduce a multiplicative structure on the Hopf-cyclic complex of \(\mathcal{H}_n\) , and we show that the van Est type characteristic homomorphism from the Hopf-cyclic complex of \(\mathcal{H}_n\) to the Gelfand–Fuks cohomology of the Lie algebra \(W_n\) of formal vector fields on \({\mathbb {R}}^n\) respects this multiplicative structure. We then illustrate the machinery for \(n=1\) . PubDate: 2018-12-01 DOI: 10.1007/s40062-018-0205-7

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Authors:Bogdan Gheorghe; Daniel C. Isaksen; Nicolas Ricka Abstract: We show that the Picard group \({{\mathrm{Pic}}}(\mathcal {A}_\mathbb {C}(1))\) of the stable category of modules over \(\mathbb {C}\) -motivic \(\mathcal {A}_\mathbb {C}(1)\) is isomorphic to \(\mathbb {Z}^4\) . By comparison, the Picard group of classical \(\mathcal {A}(1)\) is \(\mathbb {Z}^2 \oplus \mathbb {Z}/2\) . One extra copy of \(\mathbb {Z}\) arises from the motivic bigrading. The joker is a well-known exotic element of order 2 in the Picard group of classical \(\mathcal {A}(1)\) . The \(\mathbb {C}\) -motivic joker has infinite order. PubDate: 2018-04-20 DOI: 10.1007/s40062-018-0200-z

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Authors:Gennaro di Brino; Damjan Pištalo; Norbert Poncin Abstract: Homotopical geometry over differential operators is a convenient setting for a coordinate-free investigation of nonlinear partial differential equations modulo symmetries. One of the first issues one meets in the functor of points approach to homotopical \({{{\mathcal {D}}}}\) -geometry, is the question of a model structure on the category \({\mathtt{DGAlg({{{\mathcal {D}}}})}}\) of differential non-negatively graded \({{{\mathcal {O}}}}\) -quasi-coherent sheaves of commutative algebras over the sheaf \({{{\mathcal {D}}}}\) of differential operators of an appropriate underlying variety \((X,{{{\mathcal {O}}}})\) . We define a cofibrantly generated model structure on \({\mathtt{DGAlg({{{\mathcal {D}}}})}}\) via the definition of its weak equivalences and its fibrations, characterize the class of cofibrations, and build an explicit functorial ‘cofibration–trivial fibration’ factorization. We then use the latter to get a functorial model categorical Koszul–Tate resolution for \({{{\mathcal {D}}}}\) -algebraic ‘on-shell function’ algebras (which contains the classical Koszul–Tate resolution). The paper is also the starting point for a homotopical \({{{\mathcal {D}}}}\) -geometric Batalin–Vilkovisky formalism. PubDate: 2018-03-26 DOI: 10.1007/s40062-018-0202-x

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Authors:James Gillespie Abstract: Let R be any ring with identity and \( Ch (R)\) the category of chain complexes of (left) R-modules. We show that the Gorenstein AC-projective chain complexes of [1] are the cofibrant objects of an abelian model structure on \( Ch (R)\) . The model structure is cofibrantly generated and is projective in the sense that the trivially cofibrant objects are the categorically projective chain complexes. We show that when R is a Ding-Chen ring, that is, a two-sided coherent ring with finite self FP-injective dimension, then the model structure is finitely generated, and so its homotopy category is compactly generated. Constructing this model structure also shows that every chain complex over any ring has a Gorenstein AC-projective precover. These are precisely Gorenstein projective (in the usual sense) precovers whenever R is either a Ding-Chen ring, or, a ring for which all level (left) R-modules have finite projective dimension. For a general (right) coherent ring R, the Gorenstein AC-projective complexes coincide with the Ding projective complexes of [31] and so provide such precovers in this case. PubDate: 2018-03-20 DOI: 10.1007/s40062-018-0203-9

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Authors:Zhen Huan Abstract: Quasi-elliptic cohomology is a variant of Tate K-theory. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. In this paper we show how this theory is equipped with power operations. We also prove that the Tate K-theory of symmetric groups modulo a certain transfer ideal classify the finite subgroups of the Tate curve. PubDate: 2018-03-07 DOI: 10.1007/s40062-018-0201-y

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Authors:Benoit Fresse; Victor Turchin; Thomas Willwacher Abstract: We study the subcategory of topological operads \({\mathsf {P}}\) such that \({\mathsf {P}}(0) = *\) (the category of unitary operads in our terminology). We use that this category inherits a model structure, like the category of all operads in topological spaces, and that the embedding functor of this subcategory of unitary operads into the category of all operads admits a left Quillen adjoint. We prove that the derived functor of this left Quillen adjoint functor induces a left inverse of the derived functor of our category embedding at the homotopy category level. We deduce from this result that the derived mapping spaces associated to our model category of unitary operads are homotopy equivalent to the standard derived operad mapping spaces, which we form in the model category of all operads in topological spaces. We prove that analogous statements hold for the subcategory of k-truncated unitary operads within the model category of all k-truncated operads, for any fixed arity bound \(k\ge 1\) , where a k-truncated operad denotes an operad that is defined up to arity k. PubDate: 2018-02-13 DOI: 10.1007/s40062-018-0198-2