Non-trivial Homotopy Groups Research Articles

Homotopy groups of the moduli space of metrics of positive scalar curvature

We prove that for many degrees in a stable range the homotopy groups of the moduli space of metrics of positive scalar curvature on S^n and on other manifolds are non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov-Lawson to an exotic smooth families of spheres due to Hatcher. As described, this works for all manifolds of suitable dimension and for the quotient of the space of metrics of positive scalar curvature by the (free) action of the subgroup of diffeomorphisms which fix a point and its tangent space. We also construct special manifolds where the quotient of the space of metrcis of positive scalar curvature by the full diffeomorphism group has non-trivial higher homotopy groups.

Homology exponents for $H$-spaces

We say that a space X admits a \emph{homology exponent} if there exists an exponent for the torsion subgroup of H^*(X;\mathbb Z) . Our main result states that if an H -space of finite type admits a homology exponent, then either it is, up to 2 -completion, a product of spaces of the form B\mathbb Z/2^r , S^1 , \mathbb C P^\infty , and K(\mathbb Z,3) , or it has infinitely many non-trivial homotopy groups and k -invariants. Relying on recent advances in the theory of H -spaces, we then show that simply connected H -spaces whose mod 2 cohomology is finitely generated as an algebra over the Steenrod algebra do not have homology exponents, except products of mod 2 finite H -spaces with copies of \mathbb C P^\infty and K(\mathbb Z,3) .

The length of a shortest geodesic loop at a point

In this paper we prove that given a point p ∈ M n , where M n is a closed Riemannian manifold of dimension n, the length of a shortest geodesic loop lp(M n ) at this point is bounded above by 2nd, where d is the diameter of M n . Moreover, we show that on a closed simply connected Riemannian manifold M n with a nontrivial second homotopy group there either exist at least three geodesic loops of length less than or equal to 2d at each point of M n , or the length of a shortest closed geodesic on M n is bounded from above by 4d.

Generic section of a hyperplane arrangement and twisted Hurewicz maps

We consider a twisted version of the Hurewicz map on the complement of a hyperplane arrangement. The purpose of this paper is to prove surjectivity of the twisted Hurewicz map under some genericity conditions. As a corollary, we also prove that a generic section of the complement of a hyperplane arrangement has nontrivial homotopy groups.

Cellularization of classifying spaces and fusion properties of finite groups

One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their Bℤ/p-cellularization. In the easiest cases this is a classifying space of a finite group (always a finite p-group). If not, we show that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears and compute explicitly the cellularization.

Relating Postnikov pieces with the Krull filtration: a spin-off of Serre's theorem

We characterize H-spaces which are p-torsion Postnikov pieces of finite type by a cohomological property together with a necessary acyclicity condition. When the mod p cohomology of an H-space is finitely generated as an algebra over the Steenrod algebra we prove that its homotopy groups behave like those of a finite complex. In particular, a p-complete infinite loop space has a finite number of non-trivial homotopy groups if and only if its mod p cohomology satisfies this finiteness condition.

Some analogs of Zariski’s Theorem on nodal line arrangements

For line arrangements in P^2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal basis of the abelianization. We consider higher dimensional analogs of the above situation. For these analogs, we give purely combinatorial complete descriptions of the following topological invariants (over an arbitrary field): the twisted homology of the complement, with arbitrary rank one coefficients; the homology of the associated Milnor fiber and Alexander cover, including monodromy actions; the coinvariants of the first higher non-trivial homotopy group of the Alexander cover, with the induced monodromy action.

Equivariant chain complexes, twisted homology and relative minimality of arrangements

We show that the π-equivariant chain complex ( π=π 1(M( A)) ), C •( X) , associated to a Morse-theoretic minimal CW-structure X on the complement M( A) of an arrangement A , is independent of X. The same holds for all scalar extensions, C •( X)⊗ Zπ KZ , K a field, where X is an arbitrary minimal CW-structure on a space M. When A is a section of another arrangement A , we show that the divisibility properties of the first Betti number of the Milnor fiber of A obstruct the homotopy realization of M( A) as a subcomplex of a minimal structure on M( A ) . If A is aspherical and A is a sufficiently generic section of A , then H ∗(M( A);L) may be described in terms of π, L and χ(M( A)) , for an arbitrary local system L; explicit computations may be done, when A is fiber-type. In this case, explicit KZ -presentations of arbitrary abelian scalar extensions of the first non-trivial higher homotopy group of M( A) , π p ( M), may also be obtained. For nonresonant abelian scalar extensions, the CZ -rank of π p(M)⊗ Zπ CZ is combinatorially determined.

Compact open topology and CW homotopy type

The class W of spaces having the homotopy type of a CW complex is not closed under formation of function spaces. In 1959, Milnor proved the fundamental theorem that, given a space Y∈ W and a compact Hausdorff space X, the space Y X of continuous functions X→ Y, endowed with the compact open topology, belongs to W . P.J. Kahn extended this in 1982, showing that Y X∈ W if X has finite n-skeleton and π k ( Y)=0, k> n. Using a different approach, we obtain a further generalization and give interesting examples of function spaces Y X∈ W where X∈ W is not homotopy equivalent to a finite complex, and Y∈ W has infinitely many nontrivial homotopy groups. We also obtain information about some topological properties that are intimately related to CW homotopy type. As an application we solve a related problem concerning towers of fibrations between spaces of CW homotopy type.

Yukawa couplings in SO(10) heterotic M-theory vacua

We demonstrate the existence of a class of N=1 supersymmetric nonperturbative vacua of Hořava–Witten M-theory compactified on a torus fibered Calabi–Yau 3-fold Z with first homotopy group π 1(Z)= Z 2 , having the following properties: (1) SO(10) grand unification group, (2) net number of three generations of chiral fermions in the observable sector, and (3) potentially viable matter Yukawa couplings. These vacua correspond to semistable holomorphic vector bundles V Z over Z having structure group SU(4) C , and generically contain M5-branes in the bulk space. The nontrivial first homotopy group allows Wilson line breaking of the SO(10) symmetry. Additionally, we propose how the 11-dimensional Hořava–Witten M-theory framework may be used to extend the perturbative calculation of the top quark Yukawa coupling in the realistic free-fermionic models to the nonperturbative regime. The basic argument being that the relevant coupling couples twisted–twisted–untwisted states and can be calculated at the level of the Z 2× Z 2 orbifold without resorting to the full three generation models.

Standard models from heterotic M-theory

We present a class of N = 1 supersymmetric models of particle physics, derived directly from heterotic M-theory, that contain three families of chiral quarks and leptons coupled to the gauge group SU(3)C×SU(2)L×U(1)Y. These models are a fundamental form of “brane-world” theories, with an observable and hidden sector each confined, after compactification on a Calabi–Yau threefold, to a BPS three-brane separated by a five dimensional bulk space with size of the order of the intermediate scale. The requirement of three families, coupled to the fundamental conditions of anomaly freedom and supersymmetry, constrains these models to contain additional five-branes wrapped around holomorphic curves in the Calabi–Yau threefold. These five-branes “live” in the bulk space and represent new, non-perturbative aspects of these particle physics vacua. We discuss, in detail, the relevant mathematical structure of a class of torusfibered Calabi–Yau threefolds with non-trivial first homotopy groups and construct holomorphic vector bundles over such threefolds, which, by including Wilson lines, break the gauge symmetry to the standard model gauge group. Rules for constructing phenomenological particle physics models in this context are presented and we give a number of explicit examples. CERN-TH/99-407, UPR-853T December 1999 ∗current address

D-sphalerons and the topology of string configuration space

We show that unstable D-branes play the role of ``D-sphalerons'' in string theory. Their existence implies that the configuration space of Type II string theory has a complicated homotopy structure, similar to that of an infinite Grassmannian. In particular, the configuration space of Type IIA (IIB) string theory on $\R^{10}$ has non-trivial homotopy groups $\pi_k$ for all $k$ even (odd).

Representation theory of homotopy types with at most two non-trivial homotopy groups localized at a prime

Representation theory of homotopy types with at most two non-trivial homotopy groups localized at a prime

Homotopical localizations of spaces

For a map f of spaces, Dror Farjoun and the author have constructed an f -localization functor, where a space Y is called f -local when map( f, Y ) is an equivalence. This very general construction gives all known idempotent homotopy functors of spaces. The main theorem of this paper shows that f -localization functors always preserve fiber sequences of connected H-spaces up to small error terms. For instance, the localization with respect to the n th Morava K -theory preserves such fiber sequences up to error terms with at most three nontrivial homotopy groups. This implies, for example, that a K (1)-homology equivalence of H -spaces must induce an isomorphism of v 1 -periodic homotopy groups. Results are also obtained on the A -nullification or A -periodization functors, which are just the f -localization functors for the maps f from spaces A to points. Two spaces are said to have the same nullity when they give the same nullification functors, and it is shown that arbitrary sets of nullity classes have both least upper bounds and greatest lower bounds. The A -nullifications of nilpotent Postnikov spaces are completely determined.

Spaces with finitely many non-trivial homotopy groups all of which are finite

It is well-known that the homotopy type of a connected CW-space can be faithfully represented by a simplicial group [1]. Under the assumption that the space has only finitely many non-trivial homotopy groups, all of which are finite, we show that this simplicial group may be chosen such that its group of n-simplices is finite for each n ⩾ 0.

Duality on Noncompact Manifolds and Complements of Topological Knots

Let $\Sigma$ be the image of a topological embedding of ${S^{n - 2}}$ into ${S^n}$. In this paper the homotopy groups of the complement ${S^n} - \Sigma$ are studied. In contrast with the situation in the smooth and piecewise-linear categories, it is shown that the first nonstandard homotopy group of the complement of such a topological knot can occur in any dimension in the range 1 through $n - 2$. If the first nonstandard homotopy group of the complement occurs above the middle dimension, then the end of ${S^n} - \Sigma$ must have a nontrivial homotopy group in the dual dimension. The complement has the proper homotopy type of ${S^1} \times {\mathbb {R}^{n - 1}}$ if both the complement and the end of the complement have standard homotopy groups in every dimension below the middle dimension. A new form of duality for noncompact manifolds is developed. The duality theorem is the main technical tool used in the paper.

Duality on noncompact manifolds and complements of topological knots

Let Σ \Sigma be the image of a topological embedding of S n − 2 {S^{n - 2}} into S n {S^n} . In this paper the homotopy groups of the complement S n − Σ {S^n} - \Sigma are studied. In contrast with the situation in the smooth and piecewise-linear categories, it is shown that the first nonstandard homotopy group of the complement of such a topological knot can occur in any dimension in the range 1 through n − 2 n - 2 . If the first nonstandard homotopy group of the complement occurs above the middle dimension, then the end of S n − Σ {S^n} - \Sigma must have a nontrivial homotopy group in the dual dimension. The complement has the proper homotopy type of S 1 × R n − 1 {S^1} \times {\mathbb {R}^{n - 1}} if both the complement and the end of the complement have standard homotopy groups in every dimension below the middle dimension. A new form of duality for noncompact manifolds is developed. The duality theorem is the main technical tool used in the paper.

Rational Hopf G-spaces with two nontrivial homotopy group systems

Rational Hopf G-spaces with two nontrivial homotopy group systems

Nontopological global field dynamics.

We use arguments based on Derrick's theorem to show that the property of collapse which is the key feature of global texture appears in several field theory models with broken global O(N) symmetry. Such models do not necessarily have nontrivial third homotopy group of the vacuum manifold but may give rise to collapsing global field configurations with properties similar to textures. It is verified that configurations with planar and cylindrical geometries do not collapse. The existence of critical parameters for collapse of spherically symmetric global field configurations is verified both analytically and numerically.

The group of homotopy self-equivalences of non-simply-connected spaces using Postnikov decompositions

SynopsisWe give here a group extension sequence for calculating, for a non-simply-connected space X, the group of self-homotopy-equivalence classes which induce the identity automorphism of the fundamental group, that is the kernel of the representation → aut (π1(X)). This group extension sequence gives in terms of , where Xn is the n-th stage of a Postnikov decomposition. As special cases, we calculate for non-simply-connected spaces having at most two non-trivial homotopy groups, in dimensions 1 and n, as the unit group of a semigroup structure on ; and we calculate up to extension for non-simply-connected spaces having at most three non-trivial homotopy groups. The group is, for nice spaces, isomorphic to the groups and of self-homotopy-equivalence classes of X in the categories top*M and top M, respectively, where X→M = K1(π1(X)) is a top fibration which determines an isomorphism of the fundamental group; and our results are obtained initially in topM.