We study phantom maps and homology theories in a stable homotopy category S via a certain Abelian category A. We express the group P(X,Y) of phantom maps X -> Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X,HB). We show that any composite of two phantom maps is zero, and use this to reduce Margolis's axiomatisation conjecture to an extension problem. We show that a certain functor S -> A is the universal example of a homology theory with values in an AB 5 category and compare this with some results of Freyd.
We generalize theorems of McGibbon-Roitberg, Iriye, and Meier on the relations between phantom maps and rational homotopy, and apply them to provide new calculational examples of the homotopy sets Ph(X, Y ) of phantom maps and the subsets SPh(X, Y ) of special phantom maps.
In the first part, we determine conditions on spectra X and Y under which either every map from X to Y is phantom, or no nonzero maps are. We also address the question of whether such all or nothing behaviour is preserved when X is replaced with V smash X for V finite. In the second part, we introduce chromatic phantom maps. A map is n-phantom if it is null when restricted to finite spectra of type at least n. We define divisibility and finite type conditions which are suitable for studying n-phantom maps. We show that the duality functor W_{n-1} defined by Mahowald and Rezk is the analog of B…
We generalize some results of Gray and McGibbon-Roitberg on relations between phantom maps and rational homotopy to relative phantom maps. Since the $\lim^1$ and the profinite completion techniques do not apply to relative phantom maps, we develop new techniques.
We study the Gray index of phantom maps, which is a numerical invariant of phantom maps. It is conjectured that the only phantom map with infinite Gray index between finite-type spaces is the constant map. We disprove this conjecture by constructing a counter example. We also prove that this conjecture is valid if the target spaces of phantom maps are restricted to simply connected finite complexes. As an application of the counter example we show that $\SNT^{\infty}(X)$ can be non-trivial for some space $X$ of finite type.
A successful generalization of phantom map theory to the equivariant case for all compact Lie groups is obtained in this paper. One of the key observations is the discovery of the fact that homotopy fiber of equivariant completion splits as product of equivariant Eilenberg-Maclane spaces which seems impossible at first sight by the example of Triantafillou.
We study the so-called Gray filtration on the set of phantom maps between two spaces. Using both its algebraic characterization and the Sullivan completion approach to phantom maps, we generalize some of the recent results of Le, McGibbon and Strom. We particularly emphasize on the set of phantom maps with infinite Gray index, describing it in an original algebraic way. We furthermore introduce and study a natural filtration on SNT-sets (that is sets of homotopy types of spaces having the same n-type for all n), which appears to have the same algebraic characterization of the Gray one on phant…
The de Bruijn-Erdős theorem states that the chromatic number of an infinite graph equals the maximum of the chromatic numbers of finite subgraphs. Such a determinativeness by finite subobjects appears in the definition of a phantom map which is classical in algebraic topology. The topological method in combinatorics connects these two, which leads us to define the relative version of a phantom map: a map $f\colon X\to Y$ is called a relative phantom map to a map $φ\colon B\to Y$ if the restriction of $f$ to any finite subcomplex of $X$ lifts to $B$ through $φ$, up to homotopy. There are two ki…
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.
In this note, we attack a question posed ten years ago by Tsukiyama about the injectivity of the so- called Forgetable map. We show that we can insert the Forgetable map in an exact sequence and that the problem can be reduced to the computation of the sequence which turns out unexpectedly to be related to the phantom map problem and the famous Halperin conjecture in rational homotopy theory.
A phantom map is a potentially nontrivial map which induces the zero map on every homology theory and on homotopy groups. Zabrodsky has shown that in the presence of particular finiteness conditions on spaces $X$ and $Y$ every map $X\to Y$ is a phantom map. More specifically, Zabrodsky essentially requires $Y$ to be a finite CW complex and $X$ to be a Postnikov space. We show Zabrodsky's observations hold under less restrictive finiteness conditions on the spaces $X$ and $Y$, making use of the Zabrodsky lemma and the machinery of resolving classes. As an application we identify, up to extensio…
A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$, $m<M$, or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps ${\mathbb R}^{m+1} \to {\mathbb R}^{M+1}$ of growing degrees. Also, we find the homology groups of spaces of ${\mathbb Z}_r$-equivariant maps of odd-dimensional spheres for any $r$. As a technical tool, we calculate the homology groups of configurat…
We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties, and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show…
Given pointed $CW$-complexes $X$ and $Y$, $\rmph(X, Y)$ denotes the set of homotopy classes of phantom maps from $X$ to $Y$ and $\rmsph(X, Y)$ denotes the subset of $\rmph(X, Y)$ consisting of homotopy classes of special phantom maps. In a preceding paper, we gave a sufficient condition such that $\rmph(X, Y)$ and $\rmsph(X, Y)$ have natural group structures and established a formula for calculating the groups $\rmph(X, Y)$ and $\rmsph(X, Y)$ in many cases where the groups $[X,Ω\widehat{Y}]$ are nontrivial. In this paper, we establish a dual version of the formula, in which the target is the t…
The current work is motivated by the papers $[B_3]$, $[B_6]$, $[Be]$, $[Be-Tu]$. In particular, using Theorem 3.7 of $[B_3]$ and methods developed in this paper, the spectral and strong homology groups of continuous maps were defined and studied $[B_6]$, $[Be]$, $[Be-Tu]$. In this paper we will show that strong homology groups of continuous maps are a homology type functor, which is a strong shape invariant and has the semi-continuous property. We will formulate the new axioms and the conjunction on the uniqueness of the constructed functor.
It is shown that a surjective monotone map $X\to Y$ between finite $T_0$-spaces induces a surjective map on homology. As such a map turns out to be a sequence of edge contractions in the Hasse diagram of $X$, followed by a homeomorphism, this leads to an explicit relation between the Betti numbers of $X$ to those of $Y$ and the cokernels of the edge contraction maps on the order complexes.
We show that in every codimension greater than one there exists a mod 2 homology class in some closed manifold (of sufficiently high dimension) which cannot be realized by an immersion of closed manifolds. The proof gives explicit obstructions (in terms of cohomology operations) for realizability of mod 2 homology classes by immersions. We also prove the corresponding result in which the word `immersion' is replaced by `map with some restricted set of multi-singularities'.
We prove an extension to the simplicial Nerve Lemma which establishes isomorphism of persistent homology groups, in the case where the covering spaces are filtered. While persistent homology is now widely used in topological data analysis, the usual Nerve Lemma does not provide isomorphism of persistent homology groups. Our argument involves some homological algebra: the key point being that although the maps produced in the standard proof of the Nerve Lemma do not commute as maps of chain complexes, the maps they induce on homology do.
It is known that neither immersions nor maps with a fixed finite set of multisingularities are enough to realize all mod 2 homology classes in manifolds. In this paper we define the notion of realizing a homology class up to cobordism; it is shown that for realization in this weaker sense immersions are sufficient, but maps with a fixed finite set of multisingularities are still insufficient.
We apply a version of the Chas-Sullivan-Cohen-Jones product on the higher loop homology of a manifold in order to compute the homology of the spaces of continuous and holomorphic maps of the Riemann sphere into a complex projective space. This product makes sense on the homology of maps from a co-H space to a manifold, and comes from a ring spectrum. We also build a holomorphic version of the product for maps of the Riemann sphere into homogeneous spaces. In the continuous case we define a related module structure on the homology of maps from a mapping cone into a manifold, and then describe a…
We study the singular homology (with field coefficients) of the moduli stack of stable n-pointed complex curves of genus g (the Deligne-Mumford compactification). Each of its irreducible boundary components determines via the Pontrjagin-Thom construction a map to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This implies the existence of many new torsion classes in the homology of the moduli stack.
The Leray spectral sequence of a map $f$ computes the homology of the domain of $f$ from the fibers of $f$. In this expository paper, we relate in full detail the Leray spectral sequence associated to a simplicial map $f$ to the Leray cosheaves of $f$. We then give applications to level set persistent homology and Reeb spaces.
Special generic maps are smooth maps between smooth manifolds with only definite fold points as their singularities. The problem of whether a closed $n$-manifold admits a special generic map into Euclidean $p$-space for $1 \leq p \leq n$ was studied by several authors including Burlet, de Rham, Porto, Furuya, Èliašberg, Saeki, and Sakuma. In this paper, we study rational homology $n$-spheres that admit special generic maps into $\mathbb{R}^{p}$ for $p<n$. We use the technique of Stein factorization to derive a necessary homological condition for the existence of such maps for odd $n$. We exami…
We consider the category of all locally Lipschitz contractible metric spaces and all locally Lipschitz maps, which is a wide class of metric spaces, including all finite dimensional Alexandrov spaces and all CAT spaces. We also consider the chain complex of normal currents with compact support in a metric space in the sense of Ambrosio and Kirchheim. In the present paper, its homology is proved to be a homotopy invariant on the category. To prove this result, we define a new topology on a space of Lipschitz maps between arbitrary metric spaces. This topology is proved to coincide with the usua…
Given a non-compact Riemannian manifold M and a submanifold N of codimension q, we will construct under certain assumptions on both M and N a wrong way map in uniformly finite homology. Using an equivariant version of the construction and applying it to universal covers, we will construct wrong way maps in homology of groups. As applications we discuss obstructions against positive scalar curvature metrics and against inessentialness.
The Reeb space of a continuous map is the space of all (elements representing) connected components of preimages endowed with the quotient topology induced from the natural equivalence relation on the domain. These objects are strong tools in (differential) topological theory of Morse functions, fold maps, which are their higher dimensional variants, and so on: they are in general polyhedra whose dimensions are same as those of the targets. In suitable cases Reeb spaces inherit topological information such as homology groups, cohomology rings, and so on, of the manifolds. This presents the fol…
For two differentiable maps between two manifolds of possibly different dimensions, the local and global coincidence homology classes are introduced and studied by Bisi- Bracci-Izawa-Suwa (2016) in the framework of Cech-de Rham cohomology. We take up the problem from the combinatorial viewpoint and give some finer results, in particular for the local classes. As to the global class, we clarify the relation with the cohomology coincidence class as studied by Biasi-Libardi-Monis (2015). In fact they introduced such a class in the context of several maps and we also consider this case. In particu…
Let NG0 denote the category of all pointed numerically generated spaces and continuous maps preserving base-points. In [SYH], we described a passage from bivariant functors to generalized homology and cohomology theories. In this paper, we construct a bivariant functor such that the associated cohomology is the Cech cohomology and the homology is the Steenrod homology (at least for compact metric spaces).
The work is motivated by the papers [Ba1], [Ba2], [Ba7], [Ba11], [Be] and [Be-Tu]. In particular, the strong homology groups of continuous maps were defined and studied in [Be] and [Be-Tu]. To show that given groups are homology type functor, it was required to construct the corresponding shape category. In this paper, we study this very problem. In particular, using the methods developed in [Ba7], [Ba-Ts]. the strong shape theory of continuous maps of compact metric spaces, so-called strong fiber shape theory is constructed.
In this note we study the homology of configuration spaces of closed manifolds. We sharpen the eventual periodicity results of Nagpal and Cantero-Palmer, prove integral homological stability for configuration spaces of odd-dimensional manifolds and introduce a stabilization map on the homology with $Z[1/2]$-coefficients of configuration spaces of odd-dimensional manifolds.
This paper aims to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. The key idea is to extend the definition of homology induced maps of correspondences using the framework of quiver representations. Our definition of homology induced maps is given by most persistent direct summands of representations, and the direct summands uniquely determine a persistent homology. We provide stability theorems of this process and show that the output persistent homology of the sampled map is the same as that of the u…
In this paper, we give a classification of orientation reversing periodic maps on closed surfaces which generalizes the theory of Nielsen for the orientation preserving periodic maps. On one hand, we give a group of data for each orientation reversing periodic map such that two periodic maps with the same data must be conjugate to each other. On the other hand, we give the criterion to judge when two different groups of data correspond to the same conjugacy class. As an application of the results of this paper, we shall show that a given orientation reversing periodic map on $Σ_g$ with period …
Developing an algorithm for computing the Betti numbers of semi-algebraic sets with singly exponential complexity has been a holy grail in algorithmic semi-algebraic geometry and only partial results are known. In this paper we consider the more general problem of computing the image under the homology functor of a semi-algebraic map $f:X \rightarrow Y$ between closed and bounded semi-algebraic sets. For every fixed $\ell \geq 0$ we give an algorithm with singly exponential complexity that computes bases of the homology groups $\mathrm{H}_i(X), \mathrm{H}_i(Y)$ (with rational coefficients) and…
For every strong coarse homology theory we construct a coarse assembly map as a natural transformation between coarse homology theories. We provide various conditions implying that this assembly map is an equivalence. These results generalize known results for the analytic coarse assembly map for K-homology to general coarse homology theories. Furthermore, we calculate the domain of the coarse assembly map explicitly in terms of locally finite homology theory.
The cell-attachment problem, perhaps first studied by J.H.C. Whitehead around 1940, asks one to describe the effect of attaching one or more cells, on the algebraic invariants of a topological space. This thesis studies the effect of cell attachments on loop space homology. It generalizes previous work by D. Anick on spherical 2-cones, and S. Halperin, J.-M. Lemaire and others on inert attaching maps. It is assumed that the loop space homology of the base space is torsion-free. Two classes of attaching maps are introduced: free and semi-inert attaching maps. For these cases, the loop space hom…
A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. In this paper, we use information about the degrees of maps between manifolds to construct new functorial semi-norms with interesting properties. In particular, we answer a question of Gromov by providing a functorial semi-norm that takes finite positive values on homology classe…
Transfer maps and projection formulas are undoubtedly one of the key tools in the development and computation of (co)homology theories. In this note we develop an unified treatment of transfer maps and projection formulas in the non-commutative setting of dg categories. As an application, we obtain transfer maps and projection formulas in algebraic K-theory, cyclic homology, topological cyclic homology, and other scheme invariants.
Such modern applications of topology as data analysis and digital image analysis have to deal with noise and other uncertainty. In this environment, topological spaces often appear equipped with a real valued function. Persistence is a measure of robustness of the homology classes of the filtration of the lower level sets of this function. In this paper we introduce the homology group of filtration as the product of the kernels of the homology maps of the inclusions. This group contains all possible homology classes in all elements of the filtration so that we can later pick the features that …
We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspace of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for Čech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive Čech complexes to the co…
We show that various flavors of Witt vectors are functorial with respect to multiplicative polynomial laws of finite degree. We then deduce that the $p$-typical Witt vectors are functorial in multiplicative polynomial maps of degree at most $p-1$. This extra functoriality allows us to extend the $p$-typical Witt vectors functor from commutative rings to $\mathbb{Z}/2$-Tambara functors, for odd primes $p$. We use these Witt vectors for Tambara functors to describe the components of the dihedral fixed-points of the real topological Hochschild homology spectrum at odd primes.
Let $N \subset M$ be a submanifold embedding of spin manifolds of some codimension $k \geq 1$. A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that $M$ does not admit a metric of positive scalar curvature if $k = 2$ and the Dirac operator of $N$ has non-trivial index, provided that suitable conditions are satisfied. In the cases $k=1$ and $k=2$, Zeidler and Kubota, respectively, established more systematic results: There exists a transfer $\mathrm{KO}_\ast(\mathrm{C}^{\ast} π_1 M)\to \mathrm{KO}_{\ast - k}(\mathrm{C}^\ast π_1 N)$ which maps the index class of…
Morita introduced in 2008 a 1-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His 1-cocycle contains all the "traces" of Johnson homomorphisms which he introduced fifteen years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita's 1-cocycle based on a simple and explicit construction. Our 1-cocycle is proved to satisfy several fundamental properties, including a connection with the Magnus representation and the LMO homomorphism. As an application, we show that the rational abelianization…
We use assembly maps to study $\mathbf{TC}(\mathbb{A}[G];p)$, the topological cyclic homology at a prime $p$ of the group algebra of a discrete group $G$ with coefficients in a connective ring spectrum $\mathbb{A}$. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism on homotopy groups. For infinite groups, we establish pro-isomorphism, (split) injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we…
As a branch of algebraic and differential topology of manifolds, the theory of Morse functions and their higher dimensional versions or fold maps and its application to algebraic and differential topology of manifolds is fundamental, important and interesting. This paper is on explicit construction of fold maps and homology groups and cohomology rings of their Reeb spaces: they are defined as the spaces of all connected components of preimages of the maps, and in suitable situations inherit some topological information such as homology groups and cohomology rings of the manifolds. Explicit con…
The purpose of this paper is to generalize a theorem of Segal from [Seg79] proving that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding space of continuous maps through a range of dimensions increasing with degree. We will address if a similar result holds when other almost complex structures are put on a projective space. For any compatible almost complex structure J on CP^2, we prove that the inclusion map from the space of J-holomorphic maps to the space of continuous maps induces a homology surjection through a …
In the context of orientable circuits and subcomplexes of these as representing certain singular spaces, we consider characteristic class formulas generalizing those classical results as seen for the Riemann-Hurwitz formula for regulating the topology of branched covering maps and that for monoidal transformations which include the standard blowing-up process. Here the results are presented as cap product pairings, which will be elements of a suitable homology theory, rather than characteristic numbers as would be the case when taking Kronecker products once Poincaré duality is defined. We fur…
For a countable discrete group $G$, we construct a new and concrete model for the equivariant topological $K$-homology theory of $G$, which is defined for all $G$-actions, not just for proper $G$-actions. The construction of our model combines some of the ideas from the localization algebra approach to the coarse Baum-Connes conjecture and the controlled algebra approach to the Farrell-Jones conjecture. It brings new ideas into the study of the Baum-Connes conjecture. First, as the model is defined for all $G$-actions, we are able to define relative assembly maps. We prove, by using an inducti…
Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under $\mathbb{Z}_2$ coefficients for a sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis. First, we observe that it is not hard to simulate simplicial maps by inclusion maps but not necessarily in a monotone direction. This, combined with the known algorithms for zigzag persistence, provide…
We study here some aspects of the topology of the space of smooth, stable, genus 0 curves in a Riemannian manifold $X$, i.e. the Kontsevich stable curves, which are not necessarily holomorphic. We use the Hofer-Wysocki-Zehnder polyfold structure on this space and some natural characteristic classes, to show that for $X=BU,$ the rational homology of the spherical mapping space injects into the rational homology of the space of stable curves. We also give here a definition of what we call $q$-complete symplectic manifolds, which roughly speaking means Gromov-Witten theory captures all informatio…
