For an action of a compact torus $T$ on a smooth compact manifold~$X$ with isolated fixed points the number $\frac{1}{2}\dim X-\dim T$ is called the complexity of the action. In this paper we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that $\mathbb{H}P^2/T^3\cong S^5$ and $S^6/T^2\cong S^4$, for the homogeneous spaces $\mathbb{H}P^2=Sp(3)/(Sp(2)\times Sp(1))$ and $S^6=G_2/SU(3)$. Here the maximal tori of the corresponding Lie groups $Sp(3)$ and $G_2$ act on the homogeneous spaces by the left multiplication. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of 8-dimensional manifolds with the action of $T^3$, generalizing $\mathbb{H}P^2$. We prove that their orbit spaces are homeomorphic to $S^5$ as well. We link this result to Kuiper--Massey theorem and some of its generalizations.
In this paper we prove, that none of the three combinatorial 8- manifolds on 15 vertices constructed by Brehm and Kuhnel, each of which is a cohomology quaternionic projective plane, can be combinatorially embedded in the Euclidean 12-space, though they have tight polyhedral embeddings in Euclidean 14-space. This extends a similar method already known for the nonembeddings of real and complex projective planes.
Spaces in the genus of infinite quaternionic projective space which admit essential maps from infinite complex projective space are classified. In these cases the sets of homotopy classes of maps are described explicitly. These results strengthen the classical theorem of McGibbon and Rector on maximal torus admissibility for spaces in the genus of infinite quaternionic projective space. An interpretation of these results in the context of Adams-Wilkerson embedding in integral $K$-theory is also given.
It is shown that only countably many spaces in the genus of $\hpinfty$, the infinite quaternionic projective space, can admit essential maps from $\cpinfty$, the infinite complex projective space. Examples of countably many homotopically distinct spaces in the genus of $\hpinfty$ which admit essential maps from $\cpinfty$ are constructed. These results strengthen a theorem of McGibbon and Rector which states that among the uncountably many homotopy types in its genus, $\hpinfty$ is the only one which admits a maximal torus.
Let $[ρ_{i_k},[ρ_{i_{k-1}},...,[ρ_{i_{1}}, ρ_{i_2}] ...]]$ be an iterated commutator of self-maps $ρ_{i_j} : Σ{\Bbb H}P^\infty \to Σ{\Bbb H}P^\infty, j = 1,2, ..., k$ on the suspension of the infinite quaternionic projective space. In this paper, it is shown that the image of the homomorphism induced by the adjoint of this commutator is both primitive and decomposable. The main result in this paper asserts that the set of all homotopy types of spaces having the same $n$-type as the suspension of the infinite quaternionic projective space is the one element set consisting of a single homotopy t…
The problem of computing the integral cohomology ring of the symmetric square of a topological space has been of interest since the 1930s, but limited progress has been made on the general case until recently. In this work we offer a solution for the complex and quaternionic projective spaces $KP^n$, by taking advantage of their rich geometrical structure. Our description is in terms of generators and relations, and our methods entail ideas that have appeared in the literature of quantum chemistry, theoretical physics, and combinatorics. We deal first with the case $KP^\infty$, and proceed by …
We prove the following statement motivated by the quaternionic linear algebra: there is no continuous map from the quaternionic Grassmannian to the quaternionic projective space such that the pull back of the first Pontryagin class of the tautological bundle over the projective space is equal to the first Pontryagin class of the tautological bundle over the Grassmannian.
We consider the manifold $Fl_n(\mathbb{H})=Sp(n)/Sp(1)^n$ of all complete flags in $\mathbb{H}^n$, where $\mathbb{H}$ is the skew-field of quaternions. We study its equivariant $K$-theory rings with respect to the action of two groups: $Sp(1)^n$ and a certain canonical subgroup $T:=(S^1)^n\subset Sp(1)^n$ (a maximal torus). For the first group action we obtain a Goresky-Kottwitz-MacPherson type description. For the second one, we describe the ring $K_T(Fl_n(\mathbb{H}))$ as a subring of $K_T(Sp(n)/T)$. This ring is well known, since $Sp(n)/T$ is a complex flag variety.
A finite $CW$-complex $X$ is $C$-trivial if for every complex vector bundle $ξ$ over $X$, the total Chern class $c(ξ)=1$. In this note we completely determine when each of the following spaces are $C$-trivial: suspensions of stunted real projective spaces, suspensions of stunted complex projective spaces and suspensions of stunted quaternionic projective spaces.
A torus manifold is a closed smooth manifold of dimension $2n$ having an effective smooth $T^n = (S^1)^n$-action with non-empty fixed points. Petrie \cite{petrie:1973} has shown that any homotopy equivalence between a complex projective space $\CP^n$ and a torus manifold homotopy equivalent to $\CP^n$ preserves their Pontrjagin classes. A \emph{generalized Bott manifold} is a closed smooth manifold obtained as the total space of an iterated complex projective space bundles over a point, where each fibration is a projectivization of the Whitney sum of a finite many complex line bundles. For ins…
Torus manifolds are topological generalization of smooth projective toric manifolds. We compute the rational cohomology ring of a class of smooth locally standard torus manifolds whose orbit space is a connected sum of simple polytopes.
We compute the integral torus-equivariant cohomology ring for weighted projective space for two different torus actions by embedding the cohomology in a sum of polynomial rings $\oplus_{i=0}^n \Z[t_1, t_2,..., t_n]$. One torus action gives a result complementing that of Bahri, Franz, and Ray. For the other torus action, each basis class for weighted projective space is a multiple of the basis class for ordinary projective space; we identify each multiple explicitly. We also give a simple formula for the structure constants of the equivariant cohomology ring of ordinary projective space in term…
In this paper, for any Milnor hypersurface we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalised Buchstaber-Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.
This note provides a reference for some properties of the Thom spectrum $Mξ$ over $ΩΣ\CPi$. Some of this material is used in recent work of Kitchloo and Morava. We determine the $Mξ$-cohomology of $\CPi$ and show that $Mξ^*(\CPi)$ injects into power series over the algebra of non-symmetric functions. We show that $Mξ$ gives rise to a commutative formal group law over the non-commutative ring $π_*Mξ$. We also discuss how $Mξ$ and some real and quaternionic analogues behave with respect to spectra that are related to these Thom spectra by splittings and by maps.
In part I of this work we studied the spaces of real algebraic cycles on a complex projective space P(V), where V carries a real structure, and completely determined their homotopy type. We also extended some functors in K-theory to algebraic cycles, establishing a direct relationship to characteristic classes for the classical groups, specially Stiefel-Whitney classes. In this sequel, we establish corresponding results in the case where V has a quaternionic structure. The determination of the homotopy type of quaternionic algebraic cycles is more involved than in the real case, but has a simi…
In this paper, we show how to construct graph theoretical models of n-dimensional continuous objects and manifolds. These models retain topological properties of their continuous counterparts. An LCL collection of n-cells in Euclidean space is introduced and investigated. If an LCL collection of n-cells is a cover of a continuous n-dimensional manifold then the intersection graph of this cover is a digital closed n-dimensional manifold with the same topology as its continuous counterpart. As an example, we prove that the digital model of a continuous n-dimensional sphere is a digital n-sphere …
In this paper we continue to study (`strong') Nielsen coincidence numbers (which were introduced recently for pairs of maps between manifolds of arbitrary dimensions) and the corresponding minimum numbers of coincidence points and pathcomponents. We explore compatibilities with fibrations and, more specifically, with covering maps, paying special attention to selfcoincidence questions. As a sample application we calculate each of these numbers for all maps from spheres to (real, complex, or quaternionic) projective spaces. Our results turn out to be intimately related to recent work of D Gonca…
