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  • 1.
    Blomgren, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Chachólski, Wojciech
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the classification of fibrations2015In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 367, no 1, p. 519-557Article in journal (Refereed)
    Abstract [en]

    We identify the homotopy type of the moduli of maps with a given homotopy type of the base and the homotopy fiber. A new model for the space of weak equivalences and its classifying space is given.

  • 2.
    Blomgren, Martin
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Chachólski, Wojciech
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Farjoun, Emmanuel D.
    Hebrew University of Jerusalem.
    Segev, Yoav
    Ben-Gurion University.
    Idempotent deformations of finite groupsManuscript (preprint) (Other academic)
  • 3.
    Blomgren, Mats
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Chacholski, Wojciech
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Farjoun, E. D.
    Segev, Y.
    Idempotent transformations of finite groups2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 233, no 1, p. 56-86Article in journal (Refereed)
    Abstract [en]

    We describe the action of idempotent transformations on finite groups. We show that finiteness is preserved by such transformations and enumerate all possible values such transformations can assign to a fixed finite simple group. This is done in terms of the first two homology groups. We prove for example that except special linear groups, such an orbit can have at most 7 elements. We also study the action of monomials of idempotent transformations on finite groups and show for example that orbits of this action are always finite.

  • 4.
    Chacholski, Wojciech
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Covers of groups2014In: ALPINE EXPEDITION THROUGH ALGEBRAIC TOPOLOGY, KTH, Dept Math, S-10044 Stockholm, Sweden., 2014, p. 109-131Conference paper (Refereed)
    Abstract [en]

    We discuss constructions and categorical properties of various cellular approximation functors in the category of groups.

  • 5.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Damian, E.
    Farjoun, E. D.
    Segev, Y.
    The A-core and A-cover of a group2009In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 321, no 2, p. 631-666Article in journal (Refereed)
    Abstract [en]

    This paper provides a comprehensive investigation of the cellular approximation functor cell(A) G, in the category of groups. approximating a group G by a group A. We also study related notions such as A-injection, A-generation and A-constructibility of a group G and we find several interesting connections with the Schur multiplier H-2(G, Z). Our constructions are direct and are given in a slow and detailed manner.

  • 6.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Dwyer, W. G.
    Intermont, M.
    nu(*)-torsion spaces and thick classes2006In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 336, no 1, p. 13-26Article in journal (Refereed)
  • 7. Chacholski, Wojciech
    et al.
    Dwyer, W. G.
    Intermont, M.
    The A-complexity of a space2002In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 65, p. 204-222Article in journal (Refereed)
    Abstract [en]

    Suppose that A is a pointed CW-complex. The paper looks at how difficult it is to construct an A-cellular space B from copies of A by repeatedly taking homotopy colimits, this is determined by an ordinal number called the complexity of B. Studying the complexity leads to an iterative technique, based on resolutions, for constructing the A-cellular approximation CWA(X) of an arbitrary space X.

  • 8.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Farjoun, Emmanuel Dror
    Flores, Ramon
    Scherer, Jerome
    Cellular properties of nilpotent spaces2015In: Geometry and Topology, ISSN 1465-3060, E-ISSN 1364-0380, Vol. 19, no 5, p. 2741-2766Article in journal (Refereed)
    Abstract [en]

    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 z(k) X 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 P-n X, and that any nilpotent space for which the space of pointed self-maps map(*) (X, X) is "canonically" discrete must be aspherical.

  • 9. Chacholski, Wojciech
    et al.
    Libman, A.
    Tower techniques for cosimplicial resolutions2004In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 247, no 2, p. 385-407Article in journal (Refereed)
    Abstract [en]

    Let M be a simplicial model category and J : M --> M a simplicial coaugmented functor. Given an object X, the assignment n bar right arrow J(n+1)X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define J(s)X = tot(s)([n] bar right arrow J(n+1)X). An object X is called J-injective if it is a retract of JX in Ho(M) via the natural map. We show that certain homotopy limits of J-injective objects are J(s)-injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) --> X which admits a left contraction gives rise to a pro-weak equivalence of towers {X(-1)}(sgreater than or equal to0)-->{tot(S)X}(s greater than or equal to 0).

  • 10.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lundman, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Ramanujam, Ryan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Scolamiero, Martina
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Öberg, Sebastian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Multidimensional Persistence and NoiseManuscript (preprint) (Other academic)
  • 11.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Neeman, Amnon
    Australian Natl Univ, Math Sci Inst, Ctr Math & Its Applicat, Canberra, ACT 0200, Australia..
    Pitsch, Wolfgang
    Univ Autonoma Barcelona, Dept Matemat, Bellaterra 08193, Cerdanyola Del, Spain..
    Scherer, Jerome
    Ecole Polytech Fed Lausanne, Inst Math, Stn 8, CH-1015 Lausanne, Switzerland..
    Relative Homological Algebra Via Truncations2018In: Documenta Mathematica, ISSN 1431-0635, E-ISSN 1431-0643, Vol. 23, p. 895-937Article in journal (Refereed)
    Abstract [en]

    To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spal-tenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category A and fix a class of "injective objects" I. We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category D(A;I), we need more: the split error term must vanish. This is the case when I is the class of all injective R-modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4*-n axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.

  • 12.
    Chacholski, Wojciech
    et al.
    KTH, Superseded Departments, Mathematics.
    Parent, Paul-Eugene
    KTH, Superseded Departments, Mathematics.
    Stanley, D.
    Cellular generators2004In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 132, no 11, p. 3397-3409Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is twofold. On the one hand, we show that the kernelof the Bousfield periodization functor P-A is cellularly generated by a space B, i.e., we construct a space B such that the smallest closed class C( B) containing B is exactly C( A). On the other hand, we show that the partial order (Spaces, much greater than) is a complete lattice, where B much greater than A if B is an element of C(A). Finally, as a corollary we obtain Bousfield's theorem, which states that (Spaces, >) is a complete lattice, where B > A if B is an element of.

  • 13.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Pitsch, Wolfgang
    KTH.
    Scherer, Jerome
    KTH.
    Homotopy pull-back squares up to localization2006In: Alpine Anthology of Homotopy Theory / [ed] Arlettaz, D; Hess, K, PROVIDENCE: AMER MATHEMATICAL SOC , 2006, Vol. 399, p. 55-72Conference paper (Refereed)
    Abstract [en]

    We characterize the class of homotopy pull-back squares by means of elementary closure properties. The so called Puppe theorem which identifies the homotopy fiber of certain maps constructed as homotopy colimits is a straightforward consequence. Likewise we characterize the class of squares which are homotopy pull-backs "up to Bousfield localization". This yields a generalization of Puppe's theorem which allows us to identify the homotopy type of the localized homotopy fiber. When the localization functor is homological localization this is one of the key ingredients in the group completion theorem.

  • 14.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Pitsch, Wolfgang
    Scherer, Jerome
    Injective Classes Of Modules2013In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 12, no 4, p. 1250188-Article in journal (Refereed)
    Abstract [en]

    We study classes of modules over a commutative ring which allow to do homological algebra relative to such a class. We classify those classes consisting of injective modules by certain subsets of ideals. When the ring is Noetherian the subsets are precisely the generization closed subsets of the spectrum of the ring.

  • 15.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Pitsch, Wolfgang
    Scherer, Jerome
    Stanley, Don
    Homotopy Exponents for Large H-Spaces2008In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247Article in journal (Refereed)
    Abstract [en]

    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.

  • 16.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Scherer, Jerome
    Representations of spaces2008In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 8, no 1, p. 245-278Article in journal (Refereed)
    Abstract [en]

    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.

  • 17.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Scherer, Jerome
    Werndli, Kay
    HOMOTOPY EXCISION AND CELLULARITY2016In: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 66, no 6, p. 2641-2665Article in journal (Refereed)
    Abstract [en]

    Consider a push-out diagram of spaces C <- A -> B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy pull-back. This difference is measured in terms of the homotopy fibers of the original maps. Restricting our attention to the connectivity of these maps, we recover the classical Blakers-Massey Theorem.

  • 18.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Scolamiero, Martina
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Vaccarino, Francesco
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Combinatorial presentation of multidimensional persistent homologyManuscript (preprint) (Other academic)
    Abstract [en]

    A multifiltration is a functor indexed by Nr that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural Nr-graded R[x1,…,xr]-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in particular that the Nr-graded R[x1,…,xr]-modules that can occur as R-spans of multifiltrations of sets are the direct sums of monomial ideals.

  • 19.
    Chachólski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Farjoun, E. D.
    Göbel, R.
    Segev, Y.
    Cellular covers of divisible abelian groups2009In: ALPINE PERSPECTIVES ON ALGEBRAIC TOPOLOGY / [ed] Ausoni C; Hess K; Scherer J, 2009, Vol. 504, p. 77-97Conference paper (Refereed)
    Abstract [en]

    We determine all the possible values of the cellular approximation functor c(A) : cell(A)E -> E, where A is an arbitrary group and E is a divisible abelian group.

  • 20.
    Chachólski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Scolamiero, M.
    Vaccarino, F.
    Combinatorial presentation of multidimensional persistent homology2017In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 221, no 5, p. 1055-1075Article in journal (Refereed)
    Abstract [en]

    A multifiltration is a functor indexed by Nr that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural Nr-graded R[x1,…,xr]-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and R-modules. We prove in particular that the Nr-graded R[x1,…,xr]-modules that can occur as R-spans of multifiltrations of sets are the direct sums of monomial ideals.

  • 21.
    Chachólski, Wojtek
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Farjoun, E. D.
    Flores, R.
    Scherer, J.
    Idempotent functors and nilpotent spaces2016In: Trends in Mathematics, Birkhäuser Verlag, 2016, p. 63-67Conference paper (Refereed)
    Abstract [en]

    We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages and, in particular, classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield–Kan homology completion tower zk X whose terms are all X-cellular for any X.

  • 22.
    Manouchehrinia, A.
    et al.
    Karolinska Inst, Stockholm, Sweden..
    Chachólski, Wojciech
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Royal Inst Technol, Stockholm, Sweden..
    Hillert, J.
    Karolinska Inst, Stockholm, Sweden..
    Ramanujam, Ryan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Karolinska Inst, Stockholm, Sweden.;Royal Inst Technol, Stockholm, Sweden..
    Topological data analysis to identify subgroups of multiple sclerosis patients with faster disease progression2018In: Multiple Sclerosis, ISSN 1352-4585, E-ISSN 1477-0970, Vol. 24, p. 342-343Article in journal (Other academic)
  • 23.
    Scolamiero, Martina
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Chachólski, Wojciech
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lundman, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Ramanujam, Ryan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Öberg, Sebastian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Multidimensional Persistence and Noise2017In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 17, no 6, p. 1367-1406Article in journal (Refereed)
    Abstract [en]

    In this paper, we study multidimensional persistence modules (Carlsson and Zomorodian in Discrete Comput Geom 42(1):71–93, 2009; Lesnick in Found Comput Math 15(3):613–650, 2015) via what we call tame functors and noise systems. A noise system leads to a pseudometric topology on the category of tame functors. We show how this pseudometric can be used to identify persistent features of compact multidimensional persistence modules. To count such features, we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For one-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.

1 - 23 of 23
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