Ordinary Homology Research Articles

The Hopf ring for Bockstein-nil homology of [formula omitted

Works by P.R. Turner et al. show that mod 2 ordinary homology of Q S n can be described as a coalgebraic ring in an elegant way. In this paper, we establish an odd prime analogue, using the Bockstein-Nil homology, dual to the Bockstein-Nil cohomology introduced by L.M. Ha and K. Lesh.

Jumping of the nef cone for Fano varieties

We construct Q \textbf {Q} -factorial terminal Fano varieties, starting in dimension 4, whose nef cone jumps when the variety is deformed. It follows that de Fernex and Hacon’s results on deformations of 3-dimensional Fanos are optimal. The examples are based on the existence of high-dimensional flips which deform to isomorphisms, generalizing the Mukai flop. We also improve earlier results on deformations of Fano varieties. Toric Fano varieties which are smooth in codimension 2 and Q \textbf {Q} -factorial in codimension 3 are rigid. The divisor class group is deformation-invariant for klt Fanos which are smooth in codimension 2 and Q \textbf {Q} -factorial in codimension 3. The Cox ring deforms in a flat family under deformation of a terminal Fano which is Q \textbf {Q} -factorial in codimension 3. A side result which seems to be new is that the divisor class group of a klt Fano variety maps isomorphically to ordinary homology.

The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)

AbstractWe show that there is a homotopy cofiber sequence of spectra relating Carlsson's deformation K-theory of a group G to its “deformation representation ring,” analogous to the Bott periodicity sequence relating connective K-theory to ordinary homology. We then apply this to study simultaneous similarity of unitary matrices.

Strong excision and strong shape invariance are equivalent for homology theories on the category of compact metric pairs

In 1960, Milnor gave an axiomatic characterization of Steenrod homology as an ordinary homology theory on the category of compact metric pairs satisfying the strong excision axiom and the cluster axiom. The subject of this paper is to explore the role of the strong excision axiom on its own. It is proved that for any homology theory on the category of compact metric pairs strong excision is equivalent to strong shape invariance.

Arbeitsgemeinschaft mit aktuellem Thema: Twisted K-Theory

Ordinary (co)homology theories always come along with twisted versions of themselves; the most basic example is cohomology twisted by the orientation bundle, which shows up when one discusses Poincar\'e duality (and more general push-forward maps) for non-orientable manifolds.

THE EXISTENCE OF CERTAIN (CO-)LIMITS IN THE CATEGORY OF CHAIN FUNCTORS

Chain functors A*, B* have been introduced for calculating generalized homology theories by using chains and cycles, as one is doing for ordinary homology theories by means of chain complexes. Like chain complexes these chain functors form a category ℭh displaying interesting properties by themselves. The present paper emphasizes more the algebraic aspescts of this category ℭh. Although not every morphism f ∈ ℭh(A*, B*) in the category of chain functors admits a kernel or a cokernel, it turns out that: (1) all cofibrations have a cokernel, (2) all regular fibrations have a kernel, (3) the pushout of a cofibration along a cofibration exists in ℭh, respectively the dual statement for fibrations, (4) there are interesting results about exact sequences involving (co-)fibrations.

Topological conformal field theories and Calabi–Yau categories

This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov–Witten invariants (at all genera). These Gromov–Witten type invariants depend on a Calabi–Yau A ∞ category, which plays the role of the target in ordinary Gromov–Witten theory. When we use an appropriate A ∞ version of the derived category of coherent sheaves on a Calabi–Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov–Witten invariants are recovered. The assumptions are that open-closed Gromov–Witten theory can be constructed for X, and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.

Transfers for ramified covering maps in homology and cohomology

Making use of a modified version, due to McCord, of the Dold-Thom construction of ordinary homology, we give a simple topological definition of a transfer for ramified covering maps in homology with arbitrary coefficients. The transfer is induced by a suitable map between topological groups. We also define a new cohomology transfer which is dual to the homology transfer. This duality allows us to show that our homology transfer coincides with the one given by L. Smith. With our definition of the homology transfer we can give simpler proofs of the properties of the known transfer and of some new ones. Our transfers can also be defined in Karoubi's approach to homology and cohomology. Furthermore, we show that one can define mixed transfers from other homology or cohomology theories to the ordinary ones.

Stratified Morse Theory: Past and Present

In 1974, Mark Goresky and Robert MacPherson began their development of intersection homology theory (see [24] in these volumes), and their first paper on this topic appeared in 1980; see [12]. At that time, they were missing a fundamental tool which was available for the study of smooth manifolds; they had no Morse Theory for stratified spaces. Goresky and MacPherson wished to have a Stratified Morse Theory to allow them to prove a Lefschetz hyperplane theorem for the intersection homology of complex singular spaces, just as ordinary Morse Theory yields the Lefschetz Hyperplane Theorem for ordinary homology of complex manifolds ([34], §7). The time was ripe for a stratified version of Morse Theory. In 1970, Mather had given a rigorous proof of Thom’s first isotopy lemma [33]; this result says that proper, stratified, submersions are locally-trivial fibrations. In 1973, Morse functions on singular spaces had been defined by Lazzeri in [25], and the density and stability of Morse functions under perturbations had been proved in [37]. We shall recall these definitions and results in Section 2. What was missing was the analog of the fundamental result of Morse Theory, a theorem describing how the topology of a space is related to the critical points of a proper Morse function. In [16], Goresky and MacPherson proved such a theorem for stratified spaces. Suppose that M is a smooth manifold, that X is a Whitney stratified subset ofM , and that f : X → R is a proper function which is the restriction of a smooth function on M . For all v ∈ R, let X≤v := f−1((−∞, v]). Suppose that a, b ∈ R, a < b, and f−1([a, b]) contains a single (stratified) critical point, p, which is non-degenerate (see Definition 2.3) and contained in the open set f−1((a, b)). Let S be the stratum containing p. Then, the Main Theorem of Stratified Morse Theory (see Theorem 2.16) says that the topological space X≤b is obtained from the space X≤a by attaching a space A to X≤a along a subspace B ⊆ A, where the pair (A,B), the Morse data, is the product of the tangential Morse data of f at p and the normal Morse data of f at p. This result is especially powerful in the complex analytic case, where the normal Morse data depends on the stratum S, but not on the point p or on the particular Morse function f . Detailed proofs of these results appeared in the 1988 book Stratified Morse Theory [16]; we present a summary of a number of these results in Section 2. Even before the appearance of [16], Goresky and MacPherson published two papers, Stratified Morse Theory [15] and Morse Theory and Intersection Homology Theory [14], which contained announcements of many of the fundamental definitions and results of Stratified Morse Theory. In addition, these two papers showed that Stratified Morse Theory has a number of important applications to complex analytic spaces, including homotopy results, the desired Lefschetz Hyperplane Theorem for intersection homology, and the first proof that the (shifted) nearby cycles of a perverse sheaf are again perverse. We shall discuss these results and others in Section 3.

Homology of linear groups via cycles in BG× X

For X a smooth separated scheme of finite type over a field k, and for G a linear algebraic group over k, we construct homology-type functors H i(X,G) from cycles in the simplicial scheme BG× X. When X=Spec( k) and k is algebraically closed, these groups are the ordinary homology groups H i ( G( k)) of the discrete group of k-rational points of G. We construct a spectral sequence beginning with our groups and converging to the étale cohomology of the simplicial scheme BG, thus relating this theory to conclude with some calculations in the case where X= Spec( R) .

A Note on the Homotopy Type of Wait-Free Atomic Snapshot Protocol Complexes

In the atomic snapshot system model, the processes of an asynchronous distributed system communicate by atomic write and atomic snapshot read operations on a shared memory consisting of single-writer multiple-reader registers. The processes may fail by crashing. It is shown that in this model, a wait-free full-information protocol complex is homotopy equivalent to the underlying input complex. A span in the sense of Herlihy and Shavit provides the homotopy equivalence. It follows that the protocol and input complexes are indistinguishable by ordinary homology or homotopy groups.

The 𝑅𝑂(𝐺)-graded equivariant ordinary homology of 𝐺-cell complexes with even-dimensional cells for 𝐺=ℤ/𝕡

The 𝑅𝑂(𝐺)-graded equivariant ordinary homology of 𝐺-cell complexes with even-dimensional cells for 𝐺=ℤ/𝕡

Iterated homology and decompositions of simplicial complexes

Kalai has conjectured that a simplicial complex can be partitioned into Boolean algebras at least as roughly, as a shifting-preserving collapse sequence of its algebraically shifted complex. In particular, then, a simplicial complex could (conjecturally) be partitioned into Boolean intervals whose sizes are indexed by its iterated Betti numbers, a generalization of ordinary homology Betti numbers. This would imply a long-standing conjecture made (separately) by Garsia and Stanley concerning partitions of Cohen-Macaulay complexes into Boolean intervals.

Lawson homology for varieties with small Chow groups and the induced filtration on the Griffiths groups

These groups form an intriguing set of invariants. Despite the vast literature on this subject, the groups have been calculated only in very few cases, such as for the projective spaces. One of the goals of this note is to understand these invariants in case the variety has “few cycles” in the sense that rational and homological equivalence coincide for all m-cycles up to a certain rank m = s. Actually, the results only deal with that part of Lawson homology that is not “visible” in ordinary homology in that it is in the kernel of the cycle class maps cm,k+2m : LmHk+2m(X)→ Hk+2m(X),

Iterated Homology of Simplicial Complexes

We develop an iterated homology theory for simplicial complexes. This theory is a variation on one due to Kalai. For Δ a simplicial complex of dimension i>d − 1, and each i>r e 0, …,i>d, we define i>rth iterated homology groups of Δ. When i>r e 0, this corresponds to ordinary homology. If Δ is a cone over Δ′, then when i>r e 1, we get the homology of Δ′. If a simplicial complex is (nonpure) shellable, then its iterated Betti numbers give the restriction numbers, i>hi>k,j, of the shelling. Iterated Betti numbers are preserved by algebraic shifting, and may be interpreted combinatorially in terms of the algebraically shifted complex in several ways. In addition, the depth of a simplicial complex can be characterized in terms of its iterated Betti numbers.

Bousfield localizations of classifying spaces of nilpotent groups

Let G G be a finitely generated nilpotent group. The object of this paper is to identify the Bousfield localization L h B G L_hBG of the classifying space B G BG with respect to a multiplicative complex oriented homology theory h ∗ h_{*} . We show that L h B G L_hBG is the same as the localization of B G BG with respect to the ordinary homology theory determined by the ring h 0 h_0 .

D n = 0: Generalized Homology

We study the generalized homology associated with a nilpotent endomorphism $d$ satisfying $d^N=0$. For simplicial modules we construct such nilpotent endomorphisms and we prove a general result relating the corresponding generalized homologies to the ordinary homology. We also discuss the generalization of the notion of graded differential algebra in this context.

On Towers Approximating Homological Localizations

Our object of study is the natural tower which, for any given map f:A→B and each space X, starts with the localization of X with respect to f and converges to X itself. These towers can be used to produce approximations to localization with respect to any generalized homology theory E∗, yielding, for example, an analogue of Quillen's plus-construction for E∗. We discuss in detail the case of ordinary homology with coefficients in Z/p or Z[1/p]. Our main tool is a comparison theorem for nullification functors (that is, localizations with respect to maps of the form f:A→pt), which allows us, among other things, to generalize Neisendorfer's observation that p-completion of simply-connected spaces coincides with nullification with respect to a Moore space M(Z[1/p], 1).

Computer Search for Nilpotent Complexes

The concept of nilpotency for a topological space is a generalization of simple connectivity. That it is a fruitful generalization was shown by Dror, Kan, Bousfield, Hilton, and others. In 1977 Brown and Kahn proved that the dimension of a nilpotent complex can be read from the ordinary homology groups, just as in the case of a simply connected complex. They also showed that if a nilpotent complex has finite and nontrivial fundamental group, its dimension must be at least 3. In 1985 Lewis showed that for any finite nilpotent group there is a (not necessarily finite) three-dimensional nilpotent complex with that fundamental group. The smallest finite nilpotent group for which it was unknown whether a finite threedimensional nilpotent complex exists was Z 2 ⊕ Z 6. The authors, together with a team of undergraduate students at Fordham University, used computers to search for threedimensional finite nilpotent complexes over groups of the form Z n ⊕ Z m. Such complexes were eventually found for Z 2 ⊕ Z 6, Z 2 ⊕ Z 10, and Z 3 ⊕ Z 6. This article describes the strategy for constructing nilpotent complexes of dimension three, and some of the issues in implementing the computer search. The main computational issues are “normalizing” matrices, especially to the Smith normal form, and mapping matrices over Z to matrices over Z p for various primes p. We conclude with a summary of the complexes discovered and open questions.

Stable homotopy as a triangulated functor

Let J-I be the homotopy category of all finite spectra, ~the homotopy category of all spectra and Y--[ 89 be the homotopy category of all prime-to-2 spectra. Let H: ~--y ~ (Abelian groups) be a homological functor (i.e. a generalized homology theory). A natural question is whether H can be lifted to a triangulated functor; is there a functor /-t: ~--y~D (Abelian groups) so that /~ composed with the usual homology functor on D(Abelian groups) is H. Here D (Abelian groups) = D(Z) is the derived category of Z-modules. If H is ordinary homology, the answer is clearly yes. After all, the ordinary homology of a space is given by the singular chain complex; almost by definition one has a lifting/~. By the Brown Representability Theorem, the general statement follows from the special case H = H . , the stable homotopy functor. Let us be a little more precise: