Let ΩX be the space of Moore loops on a finite, q-connected, n-dimensional CW complex X, and let R be a subring of Q containing 1/2. Let p(R) be the least non-invertible prime in R. For a graded R-module M of finite type, let FM = M / Torsion M. We show that the inclusion of the sub-Lie algebra P of primitive elements of FH_*(ΩX;R) induces an isomorphism of Hopf algebras UP = FH_*(ΩX;R), provided p(R) > n/q - 1. Furthermore, the Hurewicz homomorphism induces an embedding of F(π_*(ΩX)\otimes R) in P, with torsion cokernel. As a corollary, if X is elliptic, then FH_*(ΩX;R) is a finitely-generated R-algebra.
We give a construction of the universal enveloping $A_\infty$ algebra of a given $L_\infty$ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem, proposing a new $A_\infty$ model for simply connected rational homotopy types, and uncovering a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.
In his study of Halperin's toral-rank conjecture, M. R. Hilali conjectured that for any simply connected rationally elliptic space $X$, one must have $dimπ_*(X)\otimes \mathbb{Q} \leq dimH^*(X,\mathbb{Q})$. Let $(ΛV, d)$ denote a Sullivan minimal model of $X$ and $d_k$ the first non-zero homogeneous part of the differential $d$. In this paper, we use spectral sequence arguments to prove that if $(ΛV, d_k)$ is elliptic, then, there is no gaps in the $E_{\infty}$ term of the Milnor-Moore spectral sequence of $X$. Consequently, we confirm the Hilali conjecture when $V = V^{odd}$ or else when $k\g…
We introduce a linearly ordered lattice $μ(Grp)$ of torsion theories in simplicial groups. The torsion theories are defined where the torsion/torsion-free subcategories are given by the simplicial groups with bounded above/below Moore complex, respectively. These torsion theories extend naturally the torsion theories in internal groupoids in groups. Connections of this lattice with the homotopy groups are established since the homotopy groups of a simplicial group can be calculated as the quotients of torsion subojects.
We use covering space theory and the fundamental group of complements of complexified-real line arrangements to explore the associated Milnor fiber. This work yields a combinatorially determined upper bound on the rank of the first homology group of the Milnor fiber. Under certain combinatorial conditions, we then show that one may determine the exact rank of the group and show that it is torsion free.
We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion.
Let $X$ be a $4$-dimensional toric orbifold. If $H^3(X)$ has a non-trivial odd primary torsion, then we show that $X$ is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.
Let $X$ be a toric variety. Rationally Borel-Moore homology of $X$ is isomorphic to the homology of the Koszul complex $A^T_*(X)\otimes Λ^\x M$, where $A^T_*(X)$ is the equivariant Chow group and $M$ is the character group of $T$. Moreover, the same holds for coefficients which are the integers with certain primes inverted.
Let $M$ be any simply-connected Gorenstein space over any field. Félix and Thomas have extended to simply-connected Gorenstein spaces, the loop (co)products of Chas and Sullivan on the homology of the free loop space $H_*(LM)$. We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. As a consequence, we show that the Eilenberg-Moore spectral sequence converging to the loop homology of a Gorenstein space admits a multiplication and a comultiplication with shifted degree which are compatible with the loop…
We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as exceptional orbit varieties of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements and links. We do so by using the action of linear algebraic groups beginning with the complement, ins…
We prove an analogue of the Poincare'-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a B-infini-algebra. We construct a universal enveloping functor U2 from B-infini-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form U2(Prim H). We take advantage of the simple description of the free 2as-algebra in terms of planar trees to unravel the structure of the operad B-infini.
We prove that the relative K-groups associated with a nilpotent extension of Z/p^N Z-algebras and the bi-relative K-groups associated with a Milnor square of Z/p^N Z-algebras are p-primary torsion groups of bounded exponent. We also show that, in general, the cyclotomic trace map extends from Quillen K-theory to Bass completed non-connective algebraic K-theory.
Using the work of Dwyer, Weiss, and Williams we associate an invariant to any topologically trivial family of smooth h-cobordisms. This invariant is called the smooth structure class, and is closely related to the higher Franz--Reidemeister torsion of Igusa and Klein. We compute the smooth structure class in terms of a fiberwise generalized Morse function using fiberwise Poincaré--Hopf theory. This computation gives rise to a duality theorem for the smooth structure class that generalizes Milnor's duality theorem for the Whitehead torsion. From this result we deduce a vanishing theorem that im…
Let $X$ be a finite type simply connected rationally elliptic CW-complex with Sullivan minimal model $(ΛV, d)$ and let $k\geq 2$ the biggest integer such that $d=\sum_{i\geq k}d_i$ with $d_i(V)\subseteq Λ^iV$. We show that: $cat(X_{\mathbb{Q}}) = depht(ΛV, d_k)$ if and only if $(ΛV,d_{k})$ is elliptic. This result is obtained by introducing tow new spectral sequences that generalize the Milnor-Moore spectral sequence and its $\mathcal{E}xt$-version \cite{Mur94}. As a corollary, we recover a known result proved - with different methods - by L. Lechuga and A. Murillo in \cite{LM02} and G. Lupton…
Given any connected topological space $X$, assume that there exists an epimorphism $ϕ: π_1(X) \to \mathbb{Z}$. The deck transformation group $\mathbb{Z}$ acts on the associated infinite cyclic cover $X^ϕ$ of $X$, hence on the homology group $H_i(X^ϕ, \mathbb{C})$. This action induces a linear automorphism on the torsion part of the homology group as a module over the Laurent ring $\mathbb{C}[t,t^{-1}]$, which is a finite dimensional $\mathbb{C}$-vector space. We study the sizes of the Jordan blocks of this linear automorphism. When $X$ is a compact Kähler manifold, we show that all the Jordan …
For $G$ a topological group, existence theorems by Milnor (1956), Gelfand-Fuks (1968), and Segal (1975) of classifying spaces for principal $G$-bundles are generalized to $G$-spaces with torsion. Namely, any $G$-space approximately covered by tubes (a generalization of local trivialization) is the pullback of a universal space indexed by the orbit types of tubes and cardinality of the cover. For $G$ a Lie group, via a metric model we generalize the corresponding uniqueness theorem by Palais (1960) and Bredon (1972) for compact $G$. Namely, the $G$-homeomorphism types of proper $G$-spaces over …
