Let W-n,W-k be the Stiefel manifold U(n)/U(n - k). For odd primes p and for k <= (p - 1)(p - 2), we give a homotopy decomposition of the based loop space Omega W-n,W-k as a product of p - 1 factors, each of which is the based loops on a finite H-space. Similar decompositions are obtained for Sp(n)ISp(n - k) and O(n)/O(n - k) and upper bounds on the homotopy exponents are obtained.

Different constructions by Cooke, Harper and Zabrodsky and by Cohen and Neisendorfer produce torsion free finite p-local H-spaces of rank l < p − 1. The first construction goes through when l = p − 1 and we show the second does as well. However, the space produced need not be an H-space. We give a criterion for when an H-space is obtained. In the special case of rank 2 mod-3 H-spaces, we also give a practical test for when the criterion holds, and use this to give many new examples of finite H-spaces.

The concept of C-k-spaces is introduced, situated at an intermediate stage between H-spaces and T-spaces. The C-k-space corresponds to the k-th Milnor-Stasheff filtration on spaces. It is proved that a space X is a C-k-space if and only if the Gottlieb set G(Z, X) = [Z, X] for any space Z with cat Z <= k, which generalizes the fact that X is a T-space if and only if G(Sigma B, X) = [Sigma B, X] for any space B. Some results on the C-k-space are generalized to the C-k(f)-space for a map f: A -> X. Projective spaces, lens spaces and spaces with a few cells are studied as examples of C-k-spaces, and non-C-k-spaces.

For any odd prime $p$, we prove that the induced homomorphism from the mod $p$ cohomology of the classifying space of a compact simply-connected simple connected Lie group to the Weyl group invariants of the mod $p$ cohomology of the classifying space of its maximal torus is an epimorphism except for the case $p=3$, $G=E_8$.

We calculate the Weyl group invariants with respect to a maximal torus of the exceptional Lie group $E_6$.

We study the structure of the $E_2$-term of the Rothenberg-Steenrod spectral sequence converging to the mod 3 cohomology of the classifying space of the compact, connected, simply connected, exceptional Lie group of rank 6.

Lusternik–Schnirelmann category of a manifold gives a lower bound of the number of critical points of a differentiable map on it. The purpose of this paper is to show how to construct cone-decompositions of manifolds by using functions of class C 1 and their gradient flows, where cone-decompositions are used to give an upper bound for the Lusternik–Schnirelmann category which is a homotopy invariant of a topological space. In particular, the Morse–Bott functions on the Stiefel manifolds considered by Frankel (1965) are effectively used to construct the conedecompositions of Stiefel manifolds and symmetric Riemannian spaces to determine their Lusternik–Schnirelmann categories.

We classify rational homotopy types of elliptic spaces with homotopy Euler characteristic zero for dim < 8.

We estimate the number of homotopy types of the gauge groups of Sp(2) and SU(3).

In this paper, by making use of the Cartan models, we will construct cellular decompositions of some symmetric Riemannian spaces such as Sp(n)/U(n), U(n)/O(n), U(2n)/Sp(n), O(2n)/U(n), SU(n)/SO(n), SU(2n)/Sp(n), SO(2n)/U(n).

We compute the cotorsion product of the mod 2 cohomology of spinor group spin(n), which is the E_2-term of the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of the classifying space of the spinor group spin(n). As a consequence of this computation, we show the non-collapsing of the Rothenberg-Steenrod spectral sequence for n > 16.

We determine the Lusternik-Schnirelmann category of the irreducible, symmetric Riemann spaces SU(n)/SO(n) and SU(2n)/Sp(n) of type AI and AII respectively.

where T (X) is the fat wedge and T n+1(X)→∏X is the inclusion map. The weak Lusternik-Schnirelmann category wcatX is the least integer n such that the reduced diagonal map ∆̄n+1 : X → ∧n+1X is trivial. Then it is easy to see that wcatX ≤ catX , since ∧n+1X = ∏X/T (X). The strong Lusternik-Schnirelmann category CatX is the least integer n such that there exist a space X ′ which is homotopy equivalent to X and is covered by (n+1) open subsets contractible in themselves. CatX is closely related with catX , and Ganea and Takens [14] showed that catX ≤ CatX ≤ catX + 1.

Urban development rapidly expanding from lowland to surrounding hills and mountains poses increasing risks in geo-hazards, including liquefaction during earthquakes, and failure of artificial and natural slopes. Objective of this study is to develop methodologies for assessing vulnerability to these hazards, and technologies for improving the performance of geotechnical works in urban areas. This paper summarizes the results of the study with respect to (1) performance of retaining walls at waste fill in waterfront areas, (2) hazard mapping of natural slope failures through monitoring using laser scanners, and (3) hazard assessment of residential areas in valley fills and hills.

We determine the Stiefel-Whitney classes of the second exterior representation and the spin representation of Spin(15), which are useful to calculate the mod 2 cohomology of the classifying space of the exceptional Lie group E_8.

We show that the Rothenberg--Steenrod spectral sequence converging to the mod 3 cohomology of the classifying space of the exceptional Lie group E_8 does not collapse at the E_2-level.

We describe Mui invariants in terms of Milnor operations and give a simple proof for Mui's theorem on rings of invariants of polynomial tensor exterior algebras with respect to the action of finite general linear groups. Moreover, we compute some rings of invariants of Weyl groups of maximal non-toral elementary abelian p-subgroups of exceptional Lie groups.

Introduction The mod 2 cohomology of $BLSO(n)$ The mod 2 cohomology of $BLG$ for $G=Spin(n)\ (7\leq n\leq 9)$ The mod 2 cohomology of $BLG$ for $G=G_2,F_4$ A multiplication on a twisted tensor product The twisted tensor product associated with $H^*(Spin(N) \mathbb{Z}/2)$ A manner for calculating the homology of a DGA The Hochschild spectral sequence Proof of Theorem 1.6 Computation of a cotorsion product of $H^*(Spin(10) \mathbb{Z}/2)$ and the Hochschild homology of $H^*(BSpin(10) \mathbb{Z}/2)$ Proof of Theorem 1.7 Proofs of Proposition 1.9 and Theorem 1.10 Appendix Bibliography.