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Greg Friedman

Publications and source records attributed to Greg Friedman.

35 records · Page 2Linked to original sources
arXiv2008

On the chain-level intersection pairing for PL pseudomanifolds

James McClure recently showed that the domain for the intersection pairing of PL chains on a PL manifold $M$ is a subcomplex of $C_*(M)\otimes C_*(M)$ that is quasi-isomorphic to $C_*(M)\otimes C_*(M)$ and, more generally, that the intersection pairing endows $C_*(M)$ with the structure of a partially-defined commutative DGA. We generalize this theorem to intersection pairings of PL intersection chains on PL stratified pseudomanifolds and demonstrate the existence of a partial restricted commutative DGA structure. This structure is shown to generalize the iteration of the Goresky-MacPherson intersection product. As an application, we construct an explicit "roof" representation of the intersection homology pairing in the derived category of sheaves and verify that this sheaf theoretic pairing agrees with that arising from the geometric Goresky-MacPherson intersection pairing.

math.GT↗
arXiv2008

Intersection homology Kunneth theorems

Cohen, Goresky and Ji showed that there is a Kunneth theorem relating the intersection homology groups $I^{\bar p}H_*(X\times Y)$ to $I^{\bar p}H_*(X)$ and $I^{\bar p}H_*(Y)$, provided that the perversity $\bar p$ satisfies rather strict conditions. We consider biperversities and prove that there is a Künneth theorem relating $I^{\bar p,\bar q}H_*(X\times Y)$ to $I^{\bar p}H_*(X)$ and $I^{\bar q}H_*(Y)$ for all choices of $\bar p$ and $\bar q$. Furthermore, we prove that the Kunneth theorem still holds when the biperversity $p,q$ is "loosened" a little, and using this we recover the Kunneth theorem of Cohen-Goresky-Ji.

math.GT↗
arXiv2005

Some non-trivial PL knots whose complements are homotopy circles

We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities $S^{n-2}\subset S^n$, $n\geq 5$, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial.

math.GT↗
arXiv2004

Knot spinning

This is an introduction to the construction of higher-dimensional knots by spinning methods. Simple spinning of classical knots was introduced by E. Artin in 1926, and several generalizations have followed. These include twist spinning, superspinning or p-spinning, frame spinning, roll spinning, and deform spinning. We survey these constructions and some of their most important applications, as well as some newer hybrids due to the author. The exposition, meant to be accessible to a broad audience, emphasizes a geometric approach to visualizing these constructions.

math.GT↗
arXiv2004

Microreboot -- A Technique for Cheap Recovery

A significant fraction of software failures in large-scale Internet systems are cured by rebooting, even when the exact failure causes are unknown. However, rebooting can be expensive, causing nontrivial service disruption or downtime even when clusters and failover are employed. In this work we separate process recovery from data recovery to enable microrebooting -- a fine-grain technique for surgically recovering faulty application components, without disturbing the rest of the application. We evaluate microrebooting in an Internet auction system running on an application server. Microreboots recover most of the same failures as full reboots, but do so an order of magnitude faster and result in an order of magnitude savings in lost work. This cheap form of recovery engenders a new approach to high availability: microreboots can be employed at the slightest hint of failure, prior to node failover in multi-node clusters, even when mistakes in failure detection are likely; failure and recovery can be masked from end users through transparent call-level retries; and systems can be rejuvenated by parts, without ever being shut down.

cs.OS↗
arXiv2004

Singular chain intersection homology for traditional and super-perversities

We introduce a singular chain intersection homology theory which generalizes that of King and which agrees with the Deligne sheaf intersection homology of Goresky and MacPherson on any topological stratified pseudomanifold, compact or not, with constant or local coefficients, and with traditional perversities or superperversities (those satisfying p(2)>0). For the case p(2)=1, these latter perversitie were introduced by Cappell and Shaneson and play a key role in their superduality theorem for embeddings. We further describe the sheafification of this singular chain complex and its adaptability to broader classes of stratified spaces.

math.GT↗
arXiv2004

Superperverse intersection cohomology: stratification (in)dependence

Within its traditional range of perversity parameters, intersection cohomology is a topological invariant of pseudomanifolds. This is no longer true once one allows superperversities, in which case intersection cohomology may depend on the choice of the stratification by which it is defined. Topological invariance also does not hold if one allows stratifications with codimension one strata. Nonetheless, both errant situations arise in important situations, the former in the Cappell-Shaneson superduality theorem and the latter in any discussion of pseudomanifold bordism. We show that while full invariance of intersection cohomology under restratification does not hold in this generality, it does hold up to restratifications that fix the the top stratum.

math.GT↗
arXiv2004

Groups of locally-flat disk knots and non-locally-flat sphere knots

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the ``boundary knot'' that occurs around the singular set, the set of points at which the embedding is not locally-flat. If a knot has only point singularities, this is equivalent to studying the groups of a PL locally-flat disk knot and its boundary sphere knot; in this case, we obtain a complete classification of all such group pairs in dimension $\geq 6$. For more general knots, we also obtain complete classifications of these group pairs under certain restrictions on the singularities. Finally, we use spinning constructions to realize further examples of boundary knot groups.

math.GT↗
arXiv2004

Cobordism of disk knots

We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, and any two even-dimensional disk knots with isotopic boundary knots are cobordant rel boundary. However, the cobordism rel boundary theory of odd-dimensional disk knots is more subtle. Generalizing results of Levine on cobordism of sphere knots, we define disk knot Seifert matrices and show that two higher-dimensional disk knots with isotopic boundaries are cobordant rel boundary if and only if their disk knot Seifert matrices are algebraically cobordant. We also find necessary and sufficient conditions to realize a Seifert matrix cobordism class among the disk knots corresponding to a fixed boundary knot, assuming the boundary knot has no middle-dimensional 2-torsion. This classification is performed by relating the Seifert matrix of a disk knot to its Blanchfield pairing and by establishing a close connection between this Blanchfield pairing and the Farber-Levine torsion pairing of the boundary knot (in fact, for disk knots satisfying certain connectivity assumptions, the disk knot Blanchfield pairing will determine the boundary Farber-Levine pairing).

math.GT↗
arXiv2003

All frame-spun knots are slice

Frame-spun knots are constructed by spinning a knot of lower dimension about a framed submanifold of S^n. We show that all frame-spun knots are slice (null-cobordant).

math.GT↗
arXiv2003

Alexander polynomials of non-locally-flat knots

We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties. For knots with point singularities, we obtain a classification of these polynomials that is complete except for one special low-dimensional case. This classification extends existing classifications for PL locally-flat knots. For knots with higher-dimensional singularities, we further extend the necessary conditions on the invariants. We also construct several varieties of singular knots to demonstrate realizability of certain families of polynomials as generalized Alexander polynomials. These constructions, of independent interest, generalize known knot constructions such as frame spinning and twist spinning.

math.GT↗
arXiv2003

Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata

In this paper, we develop Leray-Serre-type spectral sequences to compute the intersection homology of the regular neighborhood and deleted regular neighborhood of the bottom stratum of a stratified PL-pseudomanifold. The E^2 terms of the spectral sequences are given by the homology of the bottom stratum with a local coefficient system whose stalks consist of the intersection homology modules of the link of this stratum (or the cone on this link). In the course of this program, we establish the properties of stratified fibrations over unfiltered base spaces and of their mapping cylinders. We also prove a folk theorem concerning the stratum-preserving homotopy invariance of intersection homology.

math.GT↗
arXiv2003

Intersection Alexander polynomials

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the intersection Alexander polynomials satisfy certain duality and normalization conditions analogous to those of ordinary Alexander polynomials, and we explore the relationships between the intersection Alexander polynomials and certain generalizations of the classical Alexander polynomials that are defined for non-locally-flat knots. We also investigate the relations between the intersection Alexander polynomials of a knot and the intersection and classical Alexander polynomials of the link knots around the singular strata. To facilitate some of these investigations, we introduce spectral sequences for the computation of the intersection homology of certain stratified bundles.

math.GT↗