The E_1-term of the (2-local) bo-based Adams spectral sequence for the sphere spectrum decomposes into a direct sum of a v_1-periodic part, and a v_1-torsion part. Lellmann and Mahowald completely computed the d_1-differential on the v_1-periodic part, and the corresponding contribution to the E_2-term. The v_1-torsion part is harder to handle, but with the aid of a computer it was computed through the 20-stem by Davis. Such computer computations are limited by the exponential growth of v_1-torsion in the E_1-term. In this paper, we introduce a new method for computing the contribution of the v_1-torsion part to the E_2-term, whose input is the cohomology of the Steenrod algebra. We demonstrate the efficacy of our technique by computing the bo-Adams spectral sequence beyond the 40-stem.

We show that the Hopf elements, the Kervaire classes, and the -family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the C 2-fixed points of the Real bordism spectrum. A subset of these families is detected by the C 2-fixed points of Real JohnsonWilson theory E R (n), depending on n. In the proof, we establish an isomorphism between the slice spectral sequence and the C 2-equivariant May spectral sequence of B P R.

We establish a differential d 2 (D 1)= h 0 2 h 3 g 2 in the 51-stem of the Adams spectral sequence at the prime 2, which gives the first correct calculation of the stable 51 and 52 stems. This differential is remarkable since we know of no way to prove it without recourse to the motivic Adams spectral sequence. It is the last undetermined differential in the range of the first author's detailed calculations of the n-stems for n< 60 [6]. This note advertises the use of the motivic Adams spectral sequence to obtain information about classical stable homotopy groups.

We construct a cohomology theory of transitive digraphs and use it to give a new proof of a theorem of Gerstenhaber and Schack about isomorphism between simplicial cohomology and Hochschild cohomology of a certain algebra associated with the simplicial complex.

The path complex and its homology were defined in the previous papers of authors. The theory of path complexes is a natural discrete generalization of the theory of simplicial complexes and the homology of path complexes provide homotopy invariant homology theory of digraphs and (nondirected) graphs. In the paper we study the homology theory of path complexes. In particular, we describe functorial properties of paths complexes, introduce notion of homotopy for path complexes and prove the homotopy invariance of path homology groups. We prove also several theorems that are similar to the results of classical homology theory of simplicial complexes. Then we apply obtained results for construction homology theories on various categories of hypergraphs. We describe basic properties of these homology theories and relations between them. As a particular case, these results give new homology theories on the category of simplicial complexes.

To a digraph with a choice of certain integral basis, we construct a CW complex, whose integral singular cohomology is canonically isomorphic to the path cohomology of the digraph as introduced in\cite {GLMY}. The homotopy type of the CW complex turns out to be independent of the choice of basis. After a very brief discussion of functoriality, this construction immediately implies some of the expected but perhaps combinatorially subtle properties of the digraph cohomology and homotopy proved very recently\cite {GLMY2}. Furthermore, one gets a very simple expected formula for the cup product of forms on the digraph. On the other hand, we present an approach of using sheaf theory to reformulate (di) graph cohomologies. The investigation of the path cohomology from this framework, leads to a subtle version of Poincare lemma for digraphs, which follows from the construction of the CW complex.

In this work we study the following cutting pattern problem. Given a triangulated surface (i.e. a two-dimensional simplicial complex), assign each triangle with a triple of 1, one integer per edge, such that the assignment is both complete (i.e. every triangle has integers of both signs) and consistent (i.e. every edge shared by two triangles has opposite signs in these triangles). We show that this problem is the major challenge in converting a volumetric mesh consisting of prisms into a mesh consisting of tetrahedra, where each prism is cut into three tetrahedra. In this paper we provide a complete solution to this problem for topological disks under various boundary conditions ranging from very restricted one to the most flexible one. For each type of boundary conditions, we provide efficient algorithms to compute valid assignments if there is any, or report the obstructions otherwise. For all the proposed algorithms

We use a differential form cohomology theory on transitive digraphs to give a new proof of a theorem of Gerstenhaber and Schack about isomorphism between simplicial cohomology and Hochschild cohomology of a certain algebra associated with the simplicial complex.

In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed graphs (digraphs). We obtain a new notion of the path homology and cohomology of a digraph.

In [4] the authors observed that the topological methods in the theory of three-dimensional manifolds can be modified to settle some old problems in the classical theory of minimal surfaces in euclidean space (see also [1],[12]). In [4] and [5] we found that we could use the theory of minimal surfaces to extend the theorems of Papakriakopoulous, Whitehead and Shapiro, Stalling and Epstein on the Dehn's lemma, loop theorem and sphere theorem. The key point to our approach to these topological theorems is the following: Given a certain family of maps of the disk or sphere into our three-dimensional manifold M, we minimize the area of the maps (with respect to the pulled back metric) in this family and prove the existence of the minimal map. Then by using the area minimizing property of the map and the tower construction in topology, we prove that any area minimizing map in the family is an embedding. In this way

In this paper we consider a purely algebraic result. Then given a circle or cyclic group of prime order action on a manifold, we will use it to estimate the lower bound of the number of fixed points. We also give an obstruction to the existence of \mathbb {Z} _p action on manifolds with isolated fixed points when \mathbb {Z} _p is a prime.

Kawakubo and Uchida showed that, if a closed oriented 4k-dimensional manifold M admits a semi-free circle action such that the dimension of the fixed point set is less than 2k, then the signature of M vanishes. In this note, by using G-signature theorem and the rigidity of the signature operator, we generalize this result to more general circle actions. Combining the same idea with the remarkable WittenTaubesBott rigidity theorem, we explore more vanishing results on spin manifolds admitting such circle actions. Our results are closely related to some earlier results of ConnerFloyd, LandweberStong and HirzebruchSlodowy.

This paper contains several results concerning circle action on almost-complex and smooth manifolds. More precisely, we show that, for an almost-complex manifold M^{2mn} (resp. a smooth manifold M^{2mn} ), if there exists a partition M^{2mn} of weight M^{2mn} such that the Chern number M^{2mn} (resp. Pontrjagin number M^{2mn} ) is nonzero, then\emph {any} circle action on M^{2mn} (resp. M^{2mn} ) has at least M^{2mn} fixed points. When an even-dimensional smooth manifold M^{2mn} admits a semi-free action with isolated fixed points, we show that M^{2mn} bounds, which generalizes a well-known fact in the free case. We also provide a topological obstruction, in terms of the first Chern class, to the existence of semi-free circle action with\emph {nonempty} isolated fixed points on almost-complex manifolds. The main ingredients of our proofs are Bott's residue formula and rigidity theorem.

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant between the classical LS cocategory and the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to characterize finite homotopy nilpotent loop spaces in the spirit of Hubbuck’s Torus Theorem, and obtain corresponding results for p-compact groups and p-Noetherian groups.