We study co-H-maps from a suspension to the suspension of the projective plane and provide examples of non-suspension 3-cell co-H-spaces. These (infinitely many) examples are related to the homotopy groups of the 3-sphere. For each element of order 2 in $π_n(S^3)$, there is a corresponding non-suspension co-H-space of cells in dimensions 2, 3 and n+2.
Our ideas are to study Hopf invariants in combinatorial way by using the Cohen groups.
Let $X^k_{m,n}=Σ^k (\mathbb R\mathbb P^m/\mathbb R\mathbb P^n)$. In this note we completely determine the values of $k,m,n$ for which the total Stiefel-Whitney class $w(ξ)=1$ for any vector bundle $ξ$ over $X^k_{m,n}$.
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…
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.
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.
If $G$ is a compact connected Lie group and $T$ is a maximal torus, we give a wedge decomposition of $ΣG/T$ by identifying families of idempotents in cohomology. This is used to give new information on the self-maps of $G/T$.
We reformulate Milgram's model of a double loop suspension in terms of a preoperad of posets, each stage of which is the poset of all ordered partitions of a finite set. Using this model, we give a combinatorial model for the evaluation map and use it to study the Cohen representation for the group of homotopy classes of maps between double loop suspensions. Demonstrating the general theory, we recover Wu's shuffle relations and further provide a type of secondary relations in Cohen groups by using Toda brackets. In particular, we prove certain maps are null-homotopic by combining our relation…
Let X be a 1-connected compact space such that the algebra H*(X;Z/2) is generated by one single element. We compute the cohomology of the free loop space H*(LX;Z/2) including the Steenrod algebra action. When X is a projective space CP^n, HP^n, the Cayley projective plane CaP^2 or a sphere S^m we obtain a splitting result for integral and mod two cohomology of the suspension spectrum of LX_+. The splitting is in terms of the suspension spectrum of X_+ and the Thom spaces of the q-fold Whitney sums of the tangent bundle over X for non negative integers q.
We present a new short proof of the explicit formula for the group of links (and also link maps) in the 'quadruple point free' dimension. Denote by $L^m_{p,q}$ (respectively, $C^{m-p}_p$) the group of smooth embeddings $S^p\sqcup S^q\to S^m$ (respectively, $S^p\to S^m$) up to smooth isotopy. Denote by $LM^m_{p,q}$ the group of link maps $S^p\sqcup S^q\to S^m$ up to link homotopy. Theorem 1. If $p\le q\le m-3$ and $2p+2q\le 3m-6$ then \begin{equation*} L^m_{p,q}\cong π_p(S^{m-q-1})\oplusπ_{p+q+2-m}(SO/SO_{m-p-1})\oplus C^{m-p}_p\oplus C^{m-q}_q. \end{equation*} Theorem 2. If $p, q\le m-3$ and $…
Let M be one of the projective spaces CP^n, HP^n for n>1 or the Cayley projective plane OP^2, and let LM denote the free loop space on M. Using Morse theory methods, we prove that the suspension spectrum of (LM)_+ is homotopy equivalent to the suspension spectrum of M_+ wedge a family of Thom spaces of explicit vector bundles over the tangent sphere bundle of M.
We study the homotopy types of certain spaces closely related to the spaces of algebraic (rational) maps from the $m$ dimensional real projective space into the $n$ dimensional complex projective space for $2\leq m\leq 2n$ (we conjecture this relation to be a homotopy equivalence). In an earlier article we proved that the homotopy types of the terms of the natural degree filtration approximate closer and closer the homotopy type of the space of continuous maps and obtained bounds that describe the closeness of the approximation in terms of the degrees of the maps. Here we improve the estimates…
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.
In this note, we use Curtis's algorithm and the Lambda algebra to compute the algebraic Atiyah-Hirzebruch spectral sequence of the suspension spectrum of $\mathbb{R}P^\infty$ with the aid of a computer, which gives us its Adams $E_2$-page in the range of $t<72$. We also compute the transfer map on the Adams $E_2$-pages. These data are used in our computations of the stable homotopy groups of $\mathbb{R}P^\infty$ in [6] and of the stable homotopy groups of spheres in [7].
We show that the space $\tilde{A}_{d}(m,n)$ consisting of all real projective classes of $(n+1)$-tuples of real coefficients homogeneous polynomials of degree $d$ in $(m+1)$ variables, without common real roots except zero, has the same homology as the space $ \Map(\RP^m,\Bbb \RP^n)$ of continuous maps from the $m$-dimensional real projective space $\RP^m$ into the $n$ real dimensional projective space $\RP^n$ up to dimension %in dimensions smaller than $(n-m)(d+1)-1$. This considerably improves the main result of \cite{AKY1}.
For the model two-complex $K$ of the group presentation $\mathcal{P}=\langle x,y\,|\,x^{k+1}yxy \rangle$, with $k\geq1$ odd, we describe representatives for all free and based homotopy classes of maps from $K$ into the real projective plane and we classify the homotopy classes containing only surjective maps. With this approach we get an answer, for maps into the real projective plane, for a classical question in topological root theory, which is known so far, in dimension two, only for maps into the sphere, the torus and the Klein bottle. The answer follows by proving that for all $k\geq1$ od…
For maps from $S^3$ and $\RP^3$ into $S^2$ and $\RP^2$, we study the problem of minimizing the root set by deforming the maps through homotopies. After presenting the classification of the homotopy classes of such maps, we prove that the minimal root set for a non null-homotopic map is either a circle or the disjoint union of two circle, according its range is $S^2$ or $\RP^2$, respectively.
Let $t$ be a positive integer. Following work of D. M. Davis, we study the topology of complex-projective product spaces, i.e. quotients of cartesian products of odd dimensional spheres by the diagonal $S^1$-action, and of the $t$-torsion lens product spaces, i.e. the corresponding quotients when the action is restricted to the $t^{\mathrm{th}}$ roots of unity. For a commutative complex-oriented cohomology theory $h^*$, we determine the $h^*$-cohomology ring of these spaces (in terms of the $t$-series for $h^*$, in the case of $t$-torsion lens product spaces). When $h^*$ is singular cohomology…
We study the homotopy types of spaces of algebraic (rational) maps from real projective spaces into complex projective spaces. In a previous paper we have shown that the inclusion of the first space into the second one is a homotopy equivalence. In this paper we prove that the homotopy types of the terms of the natural "degree" filtration approximate closer and closer the homotopy type of the space of continuous maps and obtain bounds that describe the closeness of the approximation in terms of the degree. Moreover, we compute the homotopy groups of the spaces in low dimensions.
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.
We prove the non-existence of special generic maps on complex projective space as our extended new result. Simplest special generic maps are Morse functions with exactly two singular points on spheres, or Morse functions in Reeb's theorem, and canonical projections of unit spheres. Manifolds represented as connected sums of products of manifolds diffeomorphic to unit spheres admit such maps in considerable cases. Real and complex projective spaces have been shown to admit no such maps in most cases by the author. This gives a complete answer for complex projective spaces as a corollary to a mo…
For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even torsion lens spaces and compl…
We study the existence of essential phantom maps into co-H-spaces, motivated by Iriye's observation that every suspension space $Y$ of finite type with $H_i(Y;\QQ)\neq 0$ for some $i>1$ is the target of essential phantom maps. We show that Iriye's observation can be extended to the collection of nilpotent, finite type co-H-spaces. This work hinges on an enhanced understanding of the connections between homotopy decompositions of looped co-H-spaces and coalgebra decompositions of tensor algebras due to Grbic, Theriault, and Wu.
The primary goal of this paper is to find a homotopy theoretic approximation to moduli spaces of holomorphic maps Riemann surfaces into complex projective space. There is a similar treatment of a partial compactification of these moduli spaces of consisting of irreducible stable maps in the sense of Gromov-Witten theory. The arguments follow those from a paper of G. Segal on the topology of the space of rational functions.
We will give a geometric description of the nth transversal homotopy monoid of k-dimensional complex projective space, where we stratify by lower dimensional complex projective spaces in the usual way. Transversal homotopy monoids are defined as classes of based transversal maps into Whitney stratified spaces up to equivalence through such maps. We will show the nth transversal homotopy monoid of k-dimensional complex projective space is isomorphic to isotopy classes of certain filtrations of the n-sphere. The required filtrations are by nested closed subspaces $X_0 \subset ... \subset X_k=S^n…
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…
The problem of approximating the infinite dimensional space of all continuous maps from an algebraic variety $X$ to an algebraic variety $Y$ by finite dimensional spaces of algebraic maps arises in several areas of geometry and mathematical physics. An often considered formulation of the problem (sometimes called the Atiyah-Jones problem after \cite{AJ}) is to determine a (preferably optimal) integer $n_D$ such that the inclusion from this finite dimensional algebraic space into the corresponding infinite dimensional one induces isomorphisms of homology (or homotopy) groups through dimension $…
We prove the projective plane $\rp^2$ is an absolute extensor of a finite-dimensional metric space $X$ if and only if the cohomological dimension mod 2 of $X$ does not exceed 1. This solves one of the remaining difficult problems (posed by A.N.Dranishnikov) in extension theory. One of the main tools is the computation of the fundamental group of the function space $\Map(\rp^n,\rp^{n+1})$ (based at inclusion) as being isomorphic to either $\Z_4$ or $\Z_2\oplus\Z_2$ for $n\ge 1$. Double surgery and the above fact yield the proof.
This paper presents two algorithms. In their simplest form, the first algorithm decides the existence of a pointed homotopy between given simplicial maps f, g from X to Y and the second computes the group $[ΣX,Y]^*$ of pointed homotopy classes of maps from a suspension; in both cases, the target Y is assumed simply connected and the algorithms run in polynomial time when the dimension of X is fixed. More generally, these algorithms work relative to a subspace A of X, fibrewise over a simply connected B and also equivariantly when all spaces are equipped with a free action of a fixed finite gro…
We present in this paper a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundle…
It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions, Symm. We offer the cohomology of the loop space of the suspension of the infinite complex projective space as a topological model for the ring of quasisymmetric functions, QSymm. We exploit standard results from topology to shed light on some of the algebraic properties of QSymm. In particular, we reprove the Ditters conjecture. We investigate a product on the loop space that gives rise to an algebraic structure which generalizes the Witt vector structure in the co…
We survey and expand on the work of Segal, Milgram and the author on the topology of spaces of maps of positive genus curves into $n$-th complex projective space, $n\geq 1$ (in both the holomorphic and continuous categories). Both based and unbased maps are studied and in particular we compute the fundamental groups of the spaces in question. The relevant case when $n=1$ is given by a non-trivial extension which we fully determine.
In this paper we investigate Abel maps on normal surface singularities described in \cite{NNI}. We investigate the affine version of the class of the images of Abel maps on normal surface singularities. More precisely we consider the projective clousure of the image of an Abel map, its dual projective variety and we substract from its degree the multiplicity of the infinite hyperplane on the dual variety. In the case of generic singularities we prove explicit combinatorial formulas of this invariant, in the general case we prove an upper bound.
