-betti Numbers
نویسنده
چکیده
The Atiyah conjecture predicts that the L-Betti numbers of a finite CW -complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for positive 1-relator groups with torsion free abelianization. Putting everything together we establish a new class of groups for which the Atiyah conjecture holds, which contains all free groups and in particular is closed under taking subgroups, direct sums, free products, extensions with elementary amenable quotient, and under direct and inverse limits of directed systems. MSC: 55N25 (homology with local coefficients), 16S34 (group rings, Laurent rings), 46L50 (non-commutative measure theory)
منابع مشابه
On a special class of Stanley-Reisner ideals
For an $n$-gon with vertices at points $1,2,cdots,n$, the Betti numbers of its suspension, the simplicial complex that involves two more vertices $n+1$ and $n+2$, is known. In this paper, with a constructive and simple proof, wegeneralize this result to find the minimal free resolution and Betti numbers of the $S$-module $S/I$ where $S=K[x_{1},cdots, x_{n}]$ and $I$ is the associated ideal to ...
متن کاملShifting Operations and Graded Betti Numbers
The behaviour of graded Betti numbers under exterior and symmetric algebraic shifting is studied. It is shown that the extremal Betti numbers are stable under these operations. Moreover, the possible sequences of super extremal Betti numbers for a graded ideal with given Hilbert function are characterized. Finally it is shown that over a field of characteristic 0, the graded Betti numbers of a ...
متن کاملTHE LIMIT OF Fp-BETTI NUMBERS OF A TOWER OF FINITE COVERS WITH AMENABLE FUNDAMENTAL GROUPS
We prove an analogue of the Approximation Theorem of L-Betti numbers by Betti numbers for arbitrary coefficient fields and virtually torsionfree amenable groups. The limit of Betti numbers is identified as the dimension of some module over the Ore localization of the group ring.
متن کاملExtremal Betti Numbers of Rips Complexes
Upper bounds on the topological Betti numbers of Vietoris-Rips complexes are established, and examples of such complexes with high Betti
متن کاملOptimal Betti Numbers of Forest Ideals
We prove a tight lower bound on the algebraic Betti numbers of tree and forest ideals and an upper bound on certain graded Betti numbers of squarefree monomial ideals.
متن کاملBetti numbers in multidimensional persistent homology are stable functions
Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector-valued functions, called filtering functions. As is well known, in the case of scalar-valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, i.e. the dimensions of the images of the homomorphisms induced by the inclusions o...
متن کامل