Weak Homotopy Research Articles

Oka manifolds

We give a positive answer to Gromov's question [Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989) 851–897, 3.4.(D), p. 881]: If every holomorphic map from a compact convex set in a Euclidean space C n to a certain complex manifold Y is a uniform limit of entire maps C n → Y , then Y enjoys the parametric Oka property. In particular, for any reduced Stein space X the inclusion O ( X , Y ) ↪ C ( X , Y ) of the space of holomorphic maps into the space of continuous maps is a weak homotopy equivalence. To cite this article: F. Forstnerič, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Relative Homotopy in Relational Structures

AbstractThe homotopy groups of a finite partially ordered set (poset) can be described entirely in the context of posets, as shown in a paper by B. Larose and C. Tardif. In this paper we describe the relative version of such a homotopy theory, for pairs (X, A) where X is a poset and A is a subposet of X. We also prove some theorems on the relevant version of the notion of weak homotopy equivalences for maps of pairs of such objects. We work in the category of reflexive binary relational structures which contains the posets as in the work of Larose and Tardif.

Loop structures in Taylor towers

We study spaces of natural transformations between homogeneous functors in Goodwillie’s calculus of homotopy functors and in Weiss’s orthogonal calculus. We give a description of such spaces of natural transformations in terms of the homotopy fixed point construction. Our main application uses this description in combination with the Segal Conjecture to obtain a delooping theorem for connecting maps in the Goodwillie tower of the identity and in the Weiss tower of BU.V/. The interest in such deloopings stems from conjectures made by the first and the third author [4] that these towers provide a source of contracting homotopies for certain projective chain complexes of spectra. 55P65; 55P47, 18G55 1 Introduction and notation In this paper, we study homogeneous functors in the sense of Weiss’s orthogonal calculus [13]. More precisely, we calculate the space of natural transformations between such functors, and we give a few examples and applications. Our main application is a delooping theorem for connecting maps in the Goodwillie tower of the identity functor for pointed spaces and in the Weiss tower of the functor V7!BU.V/. The motivation for this delooping theorem will be discussed later in this introduction. To state our results, we summarize the usual notation for this context. Let F be a functor from pointed spaces or from finite-dimensional real or complex vector spaces to pointed spaces or spectra. Goodwillie and Weiss calculus assign to such a functor F a “Taylor” tower of functors ! Pn F! Pn 1 F! ; together with a natural map from F to the homotopy inverse limit of the tower that is often a weak homotopy equivalence. The homotopy fiber of the map Pn F! Pn 1 F is customarily denoted Dn F and is referred to as “the nth homogeneous layer of F ,” while Pn F is referred to as “the nth Taylor polynomial of F .” Depending on whether

Complexification of real cycles and the Lawson suspension theorem

The space of totally real r-cycles of a totally real projective variety is embedded into the space of complex r-cycles by complexification. The holomorphic taffy argument in the proof of Lawson's suspension theorem is proved by using Chow forms, and this proof gives an analogous result for totally real cycle spaces. The Sturm theorem is used to derive a criterion for a real polynomials of degree d to have d distinct real roots, and this criterion is used to prove the openness of some subsets of real divisors. This enables us to prove that the suspension map induces a weak homotopy equivalence between two enlarged spaces of totally real cycle spaces.

On the homotopy type and the fundamental crossed complex of the skeletal filtration of a CW-complex

We prove that if $M$ is a CW-complex, then the homotopy type of the skeletal filtration of $M$ does not depend on the cell decomposition of $M$ up to wedge products with $n$-disks Dn, when the latter are given their natural CW-decomposition with unique cells of order $0, (n - 1$) and $n$, a result resembling J.H.C. Whitehead’s work on simple homotopy types. From the higher homotopy van Kampen Theorem (due to R. Brown and P.J. Higgins) follows an algebraic analogue for the fundamental crossed complex II$(M)$ of the skeletal filtration of $M$, which thus depends only on the homotopy type of $M$ (as a space) up to free product with crossed complexes of the type $\mathcal{D}^n \doteq \Pi (D^n), n \in \mathbb{N}$.This expands an old result (due to J.H.C. Whitehead) asserting that the homotopy type of $\Pi(M)$ depends only on the homotopy type of M. We use these results to define a homotopy invariant $I_{\mathcal{A}}$ of CW-complexes for each finite crossed complex $\mathcal{A}$. We interpret it in terms of the weak homotopy type of the function space $TOP ((M, *), (|\mathcal{A}|, *))$ where $|\mathcal{A}|$ is the classifying space of the crossed complex $\mathcal{A}$.

ON A NOTION OF MAPS BETWEEN ORBIFOLDS II: HOMOTOPY AND CW-COMPLEX

This is the second of a series of papers which is devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants — the homotopy groups, and (2) an analog of CW-complex theory. As a corollary of this machinery, the classical Whitehead theorem (which asserts that a weak homotopy equivalence is a homotopy equivalence) is extended to the orbifold category.

Homotopy algebras and the inverse of the normalization functor

In this paper, we investigate multiplicative properties of the classical Dold–Kan correspondence. The inverse of the normalization functor maps commutative differential graded algebras to E ∞ -algebras. We prove that it in fact sends algebras over arbitrary differential graded E ∞ -operads to E ∞ -algebras in simplicial modules and is part of a Quillen adjunction. More generally, this inverse maps homotopy algebras to weak homotopy algebras. We prove the corresponding dual results for algebras under the conormalization, and for coalgebra structures under the normalization resp. the inverse of the conormalization.

Auslander-Reiten theory over topological spaces

Auslander-Reiten triangles and quivers are introduced into algebraic topology. It is proved that the existence of Auslander-Reiten triangles characterizes Poincaré duality spaces, and that the Auslander-Reiten quiver is a weak homotopy invariant. The theory is applied to spheres whose Auslander-Reiten triangles and quivers are computed. The Auslander-Reiten quiver over the $d$-dimensional sphere turns out to consist of $d-1$ copies of ${\mathbb Z} A\_{\infty}$. Hence the quiver is a sufficiently sensitive invariant to tell spheres of different dimension apart.

Calculus III: Taylor Series

We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal n-excisive approximation, which may be thought of as its n-excisive part. Homogeneous functors, meaning n-excisive functors with trivial (n-1)-excisive part, can be classified: they correspond to symmetric functors of n variables that are reduced and 1-excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen's algebraic K-theory.

A Functional-Analytic Theory of Vertex (Operator) Algebras, II

For a finitely-generated vertex operator algebra V of central charge cℂ, a locally convex topological completion HV is constructed. We construct on HV a structure of an algebra over the operad of the \({{\frac{{c}}{{2}}^{{\rm{ th}}}}}\) power Detc/2 of the determinant line bundle Det over the moduli space of genus-zero Riemann surfaces with ordered analytically parametrized boundary components. In particular, HV is a representation of the semi-group of the \({{\frac{{c}}{{2}}^{{\rm{ th}}}}}\) power Detc/2(1) of the determinant line bundle over the moduli space of conformal equivalence classes of annuli with analytically parametrized boundary components. The results in Part I for ℤ-graded vertex algebras are also reformulated in terms of the framed little disk operad. Using May’s recognition principle for double loop spaces, one immediate consequence of such operadic formulations is that the compactly generated spaces corresponding to (or the k-ifications of) the locally convex completions constructed in Part I and in the present paper have the weak homotopy types of double loop spaces. We also generalize the results above to locally-grading-restricted conformal vertex algebras and to modules.

Framings of knots satisfying differential relations

This paper introduces the notion of a differential framing relation for knots in a three-dimensional manifold. There is a canonical map from the space of knots that satisfy a framing relation into the space of framed knots. Under reasonable assumptions this canonical map is a weak homotopy equivalence.

Automate parallèle à homotopie près (II)

One proves that the category of globular CW-complexes up to dihomotopy is equivalent to the category of flows up to weak dihomotopy. This theorem generalizes the classical theorem which states that the category of CW-complexes up to homotopy is equivalent to the category of topological spaces up to weak homotopy. To cite this article: P. Gaucher, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

A non-Hausdorff quaternion multiplication

We denote by (S3)′ the barycentric subdivision of the minimal model S3 of the three-dimensional sphere in the category of finite posets and order-preserving functions, op(X) is the poset obtained by reversing the order relations in a poset X. We describe a finite model of a quaternion multiplication in the form of a morphism op(S3)′×(S3)′→S3 that restricts to weak homotopy equivalences on the axes. For such multiplications a version of Hopf's construction can be defined that yields finite models of non-trivial homotopy classes.

A chain rule in the calculus of homotopy functors

We formulate and prove a chain rule for the derivative, in the sense of Goodwillie, of compositions of weak homotopy functors from simplicial sets to simplicial sets. The derivative spectrum dF(X) of such a functor F at a simplicial set X can be equipped with a right action by the loop group of its domain X, and a free left action by the loop group of its codomain Y = F(X). The derivative spectrum d(E o F)(X)$ of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives dE(Y) and dF(X), with respect to the two actions of the loop group of Y. As an application we provide a non-manifold computation of the derivative of the functor F(X) = Q(Map(K, X)_+).

The Oka principle for sections of subelliptic submersions

Let X and Y be complex manifolds. One says that maps from X to Y satisfy the Oka principle if the inclusion of the space of holomorphic maps from X to Y into the space of continuous maps is a weak homotopy equivalence. In 1957 H. Grauert proved the Oka principle for maps from Stein manifolds to complex Lie groups and homogeneous spaces, as well as for sections of fiber bundles with homogeneous fibers over a Stein base. In 1989 M. Gromov extended Grauert's result to sections of submersions over a Stein base which admit dominating sprays over small open sets in the base; for proof see [F. Forstneric and J. Prezelj: Oka's principle for holomorphic fiber bundles with sprays, Math. Ann. 317 (2000), 117-154, and the preprint math.CV/0101040]. In this paper we prove the Oka principle for maps from Stein manifolds to any complex manifold Y that admits finitely many sprays which together dominate at every point of Y (such manifold is called subelliptic). The class of subelliptic manifolds contains all the elliptic ones, as well as complements of closed algebraic subvarieties of codimension at least two in a complex projective space or a complex Grassmanian. We also prove the Oka principle for removing intersections of holomorphic maps with closed complex subvarieties A of the target manifold Y, provided that the source manifold is Stein and the manifolds Y and Y\A are subelliptic.

Oka's principle for holomorphic submersions with sprays

We prove a theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, 851-897, 1989) to the effect that sections of certain holomorphic submersions h from a complex manifold Z onto a Stein manifold X satisfy the Oka principle, meaning that the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. The Oka principle holds if the submersion admits a fiber-dominating spray over a small neighborhood of any point in X. This extends a classical result of Grauert (Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133, 450-472, 1957). Gromov's result has been used in the proof of the embedding theorems for Stein manifolds and Stein spaces into Euclidean spaces of minimal dimension (Y. Eliashberg and M. Gromov, Ann. Math. 136, 123-135, 1992; J. Schurmann, Math. Ann. 307, 381-399, 1997). For further extensions see the preprints math.CV/0101034, math.CV/0107039, and math.CV/0110201.

Function spaces and shape theories

The purpose of this paper is to provide a geometric explanation of strong shape theory and to give a fairly simple way of introducing the strong shape category formally. Generally speaking, it is useful to introduce a shape theory as a localization at some class of \equivalences. We follow this principle and we extend the standard shape category Sh(HoTop) to Sh(pro-HoTop) by localizing pro-HoTop at shape equivalences. Similarly, we extend the strong shape category of Edwards{Hastings to sSh(pro-Top) by localizing pro-Top at strong shape equivalences. A map f : X! Y is a shape equivalence if and only if the induced function f : (Y;P )! (X;P ) is a bijection for all P 2 ANR. A map f : X ! Y of k-spaces is a strong shape equivalence if and only if the induced map f : Map(Y;P ) ! Map(X;P ) is a weak homotopy equivalence for all P 2 ANR. One generalizes the concept of being a shape equivalence to morphisms of pro-HoTop without any problem and the only dicult y is to show that a localization of pro-HoTop at shape equivalences is a category (which amounts to showing that the morphisms form a set). Due to peculiarities of function spaces, extending the concept of strong shape equivalence to morphisms of pro-Top is more involved. However, it can be done and we show that the corresponding localization exists. One can introduce the concept of a super shape equivalence f : X! Y of topological spaces as a map such that the induced map f : Map(Y;P )! Map(X;P ) is a homotopy equivalence for all P 2 ANR, and one can extend it to morphisms of pro-Top. However, the authors do not know if the corresponding localization exists. Here are applications of our methods: Theorem. A map f : X! Y of k-spaces is a strong shape equivalence if and only if f idQ : X k Q! Y k Q is a shape equivalence for each CW complex Q. Theorem. Suppose f : X! Y is a map of topological spaces. (a) f is a shape equivalence if and only if the induced function f : (Y;M)! (X;M) is a bijection for all M = Map(Q;P ), where P 2 ANR and Q is a nite CW complex. (b) If f is a strong shape equivalence, then the induced function f : (Y;M)! (X;M) is a bijection for all M = Map(Q;P ), where P 2 ANR and Q is an arbitrary CW complex. (c) If X, Y are k-spaces and the induced function f : (Y;M)! (X;M) is a bijection for all M = Map(Q;P ), where P 2 ANR and Q is an arbitrary CW complex , then f is a strong shape equivalence.

Etude topologique de l'espace des homéomorphismes de Brouwer, II

In this paper, sequel to Le Roux (Etude topologique de l 'espace des homéomorphismes de Brouwer, I), we persue the study of the space B of the fixed point free orientation preserving homeomorphisms of the plane, and of the subspace B τ consisting of the homeomorphisms that are conjugate to an affine translation. We first give another derivation of the weak homotopy type of the pair ( B, B τ) ; and then show that the space B τ is an absolute neighbourhood retract (ANR), homeomorphic to the product of the circle and the separable Hilbert space.

Subspaces of knot spaces

The inclusion of the space of all knots of a prescribed writhe in a particular isotopy class into the space of all knots in that isotopy class is a weak homotopy equivalence.

SEMI-INFINITE FORMS AND TOPOLOGICAL VERTEX OPERATOR ALGEBRAS

Semi-infinite forms on the moduli spaces of genus-zero Riemann surfaces with punctures and local coordinates are introduced. A partial operad of semi-infinite forms is constructed. Using semi-infinite forms and motivated by a partial suboperad of the partial operad of semi-infinite forms, topological vertex partial operads of type k<0 and strong topological vertex partial operads of type k<0 are constructed. It is proved that the category of (locally-)grading-restricted (strong) topological vertex operator algebras of type k<0 and the category of (weakly) meromorphic ℤ×ℤ-graded algebras over the (strong) topological vertex partial operad of type k are isomorphic. As an application of this isomorphism theorem, the following conjecture of Lian-Zuckerman and Kimura-Voronov-Zuckerman is proved: A strong topological vertex operator algebra gives a (weak) homotopy Gerstenhaber algebra. These results hold in particular for the tensor product of the moonshine module vertex operator algebra, the vertex algebra constructed from a rank 2 Lorentz lattice and the ghost vertex operator algebra, studied in detail first by Lian and Zuckerman.