On the homology of the Harmonic Archipelago

We calculate the singular homology and \v{C}ech cohomology groups of the Harmonic archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda's proof that the first singular homology groups of these spaces are isomorphic

On the singular homology of one class of simply-connected cell-like spaces

In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like Peano continua, called {\sl Snake space}. In the sequel we introduced the functor $SC(-,-)$ defined on the category of all spaces with base points and continuous mappings. For the circle $S^1$, the space $SC(S^1, \ast)$ is a Snake space. In the present paper we study the higher-dimensional homology and homotopy properties of the spaces $SC(Z, \ast)$ for any path-connected compact spaces $Z$

Smooth free involution of HCP3H{\Bbb C}P^3 and Smith conjecture for imbeddings of S3S^3 in S6S^6

This paper establishes an equivalence between existence of free involutions on $H{\Bbb C}P^3$ and existence of involutions on $S^6$ with fixed point set an imbedded $S^3$, then a family of counterexamples of the Smith conjecture for imbeddings of $S^3$ in $S^6$ are given by known result on $H{\Bbb C}P^3$. In addition, this paper also shows that every smooth homotopy complex projective 3-space admits no orientation preserving smooth free involution, which answers an open problem [Pe]. Moreover, the study of existence problem for smooth orientation preserving involutions on $H{\Bbb C}P^3$ is completed.Comment: 10 pages, final versio

N-complexes as functors, amplitude cohomology and fusion rules

We consider N-complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology only vanishes on injective functors providing a well defined functor on the stable category. For left truncated N-complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive N-complexes is proved to be isomorphic to an Ext functor of an indecomposable N-complex inside the abelian functor category. Finally we show that for the monoidal structure of N-complexes a Clebsch-Gordan formula holds, in other words the fusion rules for N-complexes can be determined.Comment: Final versio

Syzygies in equivariant cohomology for non-abelian Lie groups

We extend the work of Allday-Franz-Puppe on syzygies in equivariant cohomology from tori to arbitrary compact connected Lie groups G. In particular, we show that for a compact orientable G-manifold X the analogue of the Chang-Skjelbred sequence is exact if and only if the equivariant cohomology of X is reflexive, if and only if the equivariant Poincare pairing for X is perfect. Along the way we establish that the equivariant cohomology modules arising from the orbit filtration of X are Cohen-Macaulay. We allow singular spaces and introduce a Cartan model for their equivariant cohomology. We also develop a criterion for the finiteness of the number of infinitesimal orbit types of a G-manifold.Comment: 28 pages; minor change

Hodge Theory on Metric Spaces

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. The appendix by Anthony Baker provides a separable, compact metric space with infinite dimensional \alpha-scale homology.Comment: appendix by Anthony W. Baker, 48 pages, AMS-LaTeX. v2: final version, to appear in Foundations of Computational Mathematics. Minor changes and addition

On Mason's rigidity theorem

Following an argument proposed by Mason, we prove that there are no algebraically special asymptotically simple vacuum space-times with a smooth, shear-free, geodesic congruence of principal null directions extending transversally to a cross-section of Scri. Our analysis leaves the door open for escaping this conclusion if the congruence is not smooth, or not transverse to Scri. One of the elements of the proof is a new rigidity theorem for the Trautman-Bondi mass.Comment: minor typos correcte

Endoscopic Versus Surgical Step-Up Approach for Infected Necrotizing Pancreatitis (ExTENSION):Long-term Follow-up of a Randomized Trial

Background & Aims: Previous randomized trials, including the Transluminal Endoscopic Step-Up Approach Versus Minimally Invasive Surgical Step-Up Approach in Patients With Infected Pancreatic Necrosis (TENSION) trial, demonstrated that the endoscopic step-up approach might be preferred over the surgical step-up approach in patients with infected necrotizing pancreatitis based on favorable short-term outcomes. We compared long-term clinical outcomes of both step-up approaches after a period of at least 5 years. Methods: In this long-term follow-up study, we reevaluated all clinical data on 83 patients (of the originally 98 included patients) from the TENSION trial who were still alive after the initial 6-month follow-up. The primary end point, similar to the TENSION trial, was a composite of death and major complications. Secondary end points included individual major complications, pancreaticocutaneous fistula, reinterventions, pancreatic insufficiency, and quality of life. Results: After a mean follow-up period of 7 years, the primary end point occurred in 27 patients (53%) in the endoscopy group and in 27 patients (57%) in the surgery group (risk ratio [RR], 0.93; 95% confidence interval [CI], 0.65–1.32; P = .688). Fewer pancreaticocutaneous fistulas were identified in the endoscopy group (8% vs 34%; RR, 0.23; 95% CI, 0.08–0.83). After the initial 6-month follow-up, the endoscopy group needed fewer reinterventions than the surgery group (7% vs 24%; RR, 0.29; 95% CI, 0.09–0.99). Pancreatic insufficiency and quality of life did not differ between groups. Conclusions: At long-term follow-up, the endoscopic step-up approach was not superior to the surgical step-up approach in reducing death or major complications in patients with infected necrotizing pancreatitis. However, patients assigned to the endoscopic approach developed overall fewer pancreaticocutaneous fistulas and needed fewer reinterventions after the initial 6-month follow-up. Netherlands Trial Register no: NL8571