25 research outputs found
Representations of Spaces
We explain how the notion of homotopy colimits gives rise to that of mapping
spaces, even in categories which are not simplicial. We apply the technique of
model approximations and use elementary properties of the category of spaces to
be able to construct resolutions. We prove that the homotopy category of any
monoidal model category is always a central algebra over the homotopy category
of Spaces.Comment: Final version, almost as it will appear in "Algebraic and Geometric
Topology"; 30 page
Homotopy exponents for large H-spaces
We show that H-spaces with finitely generated cohomology, as an algebra or as
an algebra over the Steenrod algebra, have homotopy exponents at all primes.
This provides a positive answer to a question of Stanley.Comment: 4 page
Algorithmic decomposition of filtered chain complexes
We present an algorithm to decompose filtered chain complexes into sums of
interval spheres. The algorithm's correctness is proved through principled
methods from homotopy theory. Its asymptotic runtime complexity is shown to be
cubic in the number of generators, e.g. the simplices of a simplicial complex,
as it is based on the row reduction of the boundary matrix by Gaussian
elimination. Applying homology to a filtered chain complex, one obtains a
persistence module. So our method also provides a new algorithm for the barcode
decomposition of persistence modules. The key differences with respect to the
state-of-the-art persistent homology algorithms are that our algorithm uses row
rather than column reductions, it intrinsically adopts both the clear and
compress optimisation strategies, and, finally, it can process rows according
to any random order
A topological data analysis based classification method for multiple measurements
HR was partly supported by a collaboration agreement between the University of Aberdeen and EPFL. WC was partially supported by VR 2014-04770 and Wallenberg AI, Autonomous System and Software Program (WASP) funded by Knut and Alice Wallenberg Foundation, Göran Gustafsson Stiftelse. JT is fully funded by the Wenner-Gren Foundation. JH is partially supported by VR K825930053. RR is partially supported by MultipleMS. The collaboration agreement between EPFL and University of Aberdeen played a role in the design of the neuron spiking analysis and in providing the data required, i.e. the neuronal network and the spiking activity. Open access funding provided by Karolinska Institute.Peer reviewedPublisher PD
Cellular properties of nilpotent spaces
We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield-Kan homology completion tower zkX whose terms we prove are all X–cellular for any X. As straightforward consequences, we show that if X is K–acyclic and nilpotent for a given homology theory K, then so are all its Postnikov sections PnX , and that any nilpotent space for which the space of pointed self-maps map .X; X/ is “canonically” discrete must be aspherical.Göran Gustafsson StiftelseFondo Europeo de Desarrollo RegionalMinisterio de Economía y Competitivida