Lecture 1: Obstruction Theory, Eilenberg-Maclane Spaces, Postnikov Towers
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Persistent Obstruction Theory for a Model Category of Measures with Applications to Data Merging
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 8, Pages 1–38 (February 2, 2021) https://doi.org/10.1090/btran/56 PERSISTENT OBSTRUCTION THEORY FOR A MODEL CATEGORY OF MEASURES WITH APPLICATIONS TO DATA MERGING ABRAHAM D. SMITH, PAUL BENDICH, AND JOHN HARER Abstract. Collections of measures on compact metric spaces form a model category (“data complexes”), whose morphisms are marginalization integrals. The fibrant objects in this category represent collections of measures in which there is a measure on a product space that marginalizes to any measures on pairs of its factors. The homotopy and homology for this category allow measurement of obstructions to finding measures on larger and larger product spaces. The obstruction theory is compatible with a fibrant filtration built from the Wasserstein distance on measures. Despite the abstract tools, this is motivated by a widespread problem in data science. Data complexes provide a mathematical foundation for semi- automated data-alignment tools that are common in commercial database software. Practically speaking, the theory shows that database JOIN oper- ations are subject to genuine topological obstructions. Those obstructions can be detected by an obstruction cocycle and can be resolved by moving through a filtration. Thus, any collection of databases has a persistence level, which measures the difficulty of JOINing those databases. Because of its general formulation, this persistent obstruction theory also encompasses multi-modal data fusion problems, some forms of Bayesian inference, and probability cou- plings. 1. Introduction We begin this paper with an abstraction of a problem familiar to any large enterprise. Imagine that the branch offices within the enterprise have access to many data sources. -
Obstructions, Extensions and Reductions. Some Applications Of
Obstructions, Extensions and Reductions. Some applications of Cohomology ∗ Luis J. Boya Departamento de F´ısica Te´orica, Universidad de Zaragoza. E-50009 Zaragoza, Spain [email protected] October 22, 2018 Abstract After introducing some cohomology classes as obstructions to ori- entation and spin structures etc., we explain some applications of co- homology to physical problems, in especial to reduced holonomy in M- and F -theories. 1 Orientation For a topological space X, the important objects are the homology groups, H∗(X, A), with coefficients A generally in Z, the integers. A bundle ξ : E(M, F ) is an extension E with fiber F (acted upon by a group G) over an space M, noted ξ : F → E → M and it is itself a Cech˘ cohomology element, arXiv:math-ph/0310010v1 8 Oct 2003 ξ ∈ Hˆ 1(M,G). The important objects here the characteristic cohomology classes c(ξ) ∈ H∗(M, A). Let M be a manifold of dimension n. Consider a frame e in a patch U ⊂ M, i.e. n independent vector fields at any point in U. Two frames e, e′ in U define a unique element g of the general linear group GL(n, R) by e′ = g · e, as GL acts freely in {e}. An orientation in M is a global class of frames, two frames e (in U) and e′ (in U ′) being in the same class if det g > 0 where e′ = g · e in the overlap of two patches. A manifold is orientable if it is ∗To be published in the Proceedings of: SYMMETRIES AND GRAVITY IN FIELD THEORY. -
PIECEWISE LINEAR TOPOLOGY Contents 1. Introduction 2 2. Basic
PIECEWISE LINEAR TOPOLOGY J. L. BRYANT Contents 1. Introduction 2 2. Basic Definitions and Terminology. 2 3. Regular Neighborhoods 9 4. General Position 16 5. Embeddings, Engulfing 19 6. Handle Theory 24 7. Isotopies, Unknotting 30 8. Approximations, Controlled Isotopies 31 9. Triangulations of Manifolds 33 References 35 1 2 J. L. BRYANT 1. Introduction The piecewise linear category offers a rich structural setting in which to study many of the problems that arise in geometric topology. The first systematic ac- counts of the subject may be found in [2] and [63]. Whitehead’s important paper [63] contains the foundation of the geometric and algebraic theory of simplicial com- plexes that we use today. More recent sources, such as [30], [50], and [66], together with [17] and [37], provide a fairly complete development of PL theory up through the early 1970’s. This chapter will present an overview of the subject, drawing heavily upon these sources as well as others with the goal of unifying various topics found there as well as in other parts of the literature. We shall try to give enough in the way of proofs to provide the reader with a flavor of some of the techniques of the subject, while deferring the more intricate details to the literature. Our discussion will generally avoid problems associated with embedding and isotopy in codimension 2. The reader is referred to [12] for a survey of results in this very important area. 2. Basic Definitions and Terminology. Simplexes. A simplex of dimension p (a p-simplex) σ is the convex closure of a n set of (p+1) geometrically independent points {v0, . -
Picard Groupoids and $\Gamma $-Categories
PICARD GROUPOIDS AND Γ-CATEGORIES AMIT SHARMA Abstract. In this paper we construct a symmetric monoidal closed model category of coherently commutative Picard groupoids. We construct another model category structure on the category of (small) permutative categories whose fibrant objects are (permutative) Picard groupoids. The main result is that the Segal’s nerve functor induces a Quillen equivalence between the two aforementioned model categories. Our main result implies the classical result that Picard groupoids model stable homotopy one-types. Contents 1. Introduction 2 2. The Setup 4 2.1. Review of Permutative categories 4 2.2. Review of Γ- categories 6 2.3. The model category structure of groupoids on Cat 7 3. Two model category structures on Perm 12 3.1. ThemodelcategorystructureofPermutativegroupoids 12 3.2. ThemodelcategoryofPicardgroupoids 15 4. The model category of coherently commutatve monoidal groupoids 20 4.1. The model category of coherently commutative monoidal groupoids 24 5. Coherently commutative Picard groupoids 27 6. The Quillen equivalences 30 7. Stable homotopy one-types 36 Appendix A. Some constructions on categories 40 AppendixB. Monoidalmodelcategories 42 Appendix C. Localization in model categories 43 Appendix D. Tranfer model structure on locally presentable categories 46 References 47 arXiv:2002.05811v2 [math.CT] 12 Mar 2020 Date: Dec. 14, 2019. 1 2 A. SHARMA 1. Introduction Picard groupoids are interesting objects both in topology and algebra. A major reason for interest in topology is because they classify stable homotopy 1-types which is a classical result appearing in various parts of the literature [JO12][Pat12][GK11]. The category of Picard groupoids is the archetype exam- ple of a 2-Abelian category, see [Dup08]. -
Math 527 - Homotopy Theory Obstruction Theory
Math 527 - Homotopy Theory Obstruction theory Martin Frankland April 5, 2013 In our discussion of obstruction theory via the skeletal filtration, we left several claims as exercises. The goal of these notes is to fill in two of those gaps. 1 Setup Let us recall the setup, adopting a notation similar to that of May x 18.5. Let (X; A) be a relative CW complex with n-skeleton Xn, and let Y be a simple space. Given two maps fn; gn : Xn ! Y which agree on Xn−1, we defined a difference cochain n d(fn; gn) 2 C (X; A; πn(Y )) whose value on each n-cell was defined using the following \double cone construction". Definition 1.1. Let H; H0 : Dn ! Y be two maps that agree on the boundary @Dn ∼= Sn−1. The difference construction of H and H0 is the map 0 n ∼ n n H [ H : S = D [Sn−1 D ! Y: Here, the two terms Dn are viewed as the upper and lower hemispheres of Sn respectively. 1 2 The two claims In this section, we state two claims and reduce their proof to the case of spheres and discs. Proposition 2.1. Given two maps fn; gn : Xn ! Y which agree on Xn−1, we have fn ' gn rel Xn−1 if and only if d(fn; gn) = 0 holds. n n−1 Proof. For each n-cell eα of X n A, consider its attaching map 'α : S ! Xn−1 and charac- n n−1 teristic map Φα :(D ;S ) ! (Xn;Xn−1). -
Arxiv:2101.06841V2 [Math.AT] 1 Apr 2021
C2-EQUIVARIANT TOPOLOGICAL MODULAR FORMS DEXTER CHUA Abstract. We compute the homotopy groups of the C2 fixed points of equi- variant topological modular forms at the prime 2 using the descent spectral sequence. We then show that as a TMF-module, it is isomorphic to the tensor product of TMF with an explicit finite cell complex. Contents 1. Introduction 1 2. Equivariant elliptic cohomology 5 3. The E2 page of the DSS 9 4. Differentials in the DSS 13 5. Identification of the last factor 25 6. Further questions 29 Appendix A. Connective C2-equivariant tmf 30 Appendix B. Sage script 35 References 38 1. Introduction Topological K-theory is one of the first examples of generalized cohomology theories. It admits a natural equivariant analogue — for a G-space X, the group 0 KOG(X) is the Grothendieck group of G-equivariant vector bundles over X. In 0 particular, KOG(∗) = Rep(G) is the representation ring of G. As in the case of non-equivariant K-theory, this extends to a G-equivariant cohomology theory KOG, and is represented by a genuine G-spectrum. We shall call this G-spectrum KO, omitting the subscript, as we prefer to think of this as a global equivariant spectrum — one defined for all compact Lie groups. The G-fixed points of this, written KOBG, is a spectrum analogue of the representation ring, BG 0 BG −n with π0KO = KOG(∗) = Rep(G) (more generally, πnKO = KOG (∗)). These fixed point spectra are readily computable as KO-modules. For example, KOBC2 = KO _ KO; KOBC3 = KO _ KU: arXiv:2101.06841v2 [math.AT] 1 Apr 2021 This corresponds to the fact that C2 has two real characters, while C3 has a real character plus a complex conjugate pair. -
Fibrations I
Fibrations I Tyrone Cutler May 23, 2020 Contents 1 Fibrations 1 2 The Mapping Path Space 5 3 Mapping Spaces and Fibrations 9 4 Exercises 11 1 Fibrations In the exercises we used the extension problem to motivate the study of cofibrations. The idea was to allow for homotopy-theoretic methods to be introduced to an otherwise very rigid problem. The dual notion is the lifting problem. Here p : E ! B is a fixed map and we would like to known when a given map f : X ! B lifts through p to a map into E E (1.1) |> | p | | f X / B: Asking for the lift to make the diagram to commute strictly is neither useful nor necessary from our point of view. Rather it is more natural for us to ask that the lift exist up to homotopy. In this lecture we work in the unpointed category and obtain the correct conditions on the map p by formally dualising the conditions for a map to be a cofibration. Definition 1 A map p : E ! B is said to have the homotopy lifting property (HLP) with respect to a space X if for each pair of a map f : X ! E, and a homotopy H : X×I ! B starting at H0 = pf, there exists a homotopy He : X × I ! E such that , 1) He0 = f 2) pHe = H. The map p is said to be a (Hurewicz) fibration if it has the homotopy lifting property with respect to all spaces. 1 Since a diagram is often easier to digest, here is the definition exactly as stated above f X / E x; He x in0 x p (1.2) x x H X × I / B and also in its equivalent adjoint formulation X B H B He B B p∗ EI / BI (1.3) f e0 e0 # p E / B: The assertion that p is a fibration is the statement that the square in the second diagram is a weak pullback. -
Notes on Spectral Sequence
Notes on spectral sequence He Wang Long exact sequence coming from short exact sequence of (co)chain com- plex in (co)homology is a fundamental tool for computing (co)homology. Instead of considering short exact sequence coming from pair (X; A), one can consider filtered chain complexes coming from a increasing of subspaces X0 ⊂ X1 ⊂ · · · ⊂ X. We can see it as many pairs (Xp; Xp+1): There is a natural generalization of a long exact sequence, called spectral sequence, which is more complicated and powerful algebraic tool in computation in the (co)homology of the chain complex. Nothing is original in this notes. Contents 1 Homological Algebra 1 1.1 Definition of spectral sequence . 1 1.2 Construction of spectral sequence . 6 2 Spectral sequence in Topology 12 2.1 General method . 12 2.2 Leray-Serre spectral sequence . 15 2.3 Application of Leray-Serre spectral sequence . 18 1 Homological Algebra 1.1 Definition of spectral sequence Definition 1.1. A differential bigraded module over a ring R, is a collec- tion of R-modules, fEp;qg, where p, q 2 Z, together with a R-linear mapping, 1 H.Wang Notes on spectral Sequence 2 d : E∗;∗ ! E∗+s;∗+t, satisfying d◦d = 0: d is called the differential of bidegree (s; t). Definition 1.2. A spectral sequence is a collection of differential bigraded p;q R-modules fEr ; drg, where r = 1; 2; ··· and p;q ∼ p;q ∗;∗ ∼ p;q ∗;∗ ∗;∗ p;q Er+1 = H (Er ) = ker(dr : Er ! Er )=im(dr : Er ! Er ): In practice, we have the differential dr of bidegree (r; 1 − r) (for a spec- tral sequence of cohomology type) or (−r; r − 1) (for a spectral sequence of homology type). -
ON TOPOLOGICAL and PIECEWISE LINEAR VECTOR FIELDS-T
ON TOPOLOGICAL AND PIECEWISE LINEAR VECTOR FIELDS-t RONALD J. STERN (Received31 October1973; revised 15 November 1974) INTRODUCTION THE existence of a non-zero vector field on a differentiable manifold M yields geometric and algebraic information about M. For example, (I) A non-zero vector field exists on M if and only if the tangent bundle of M splits off a trivial bundle. (2) The kth Stiefel-Whitney class W,(M) of M is the primary obstruction to obtaining (n -k + I) linearly independent non-zero vector fields on M [37; 0391. In particular, a non-zero vector field exists on a compact manifold M if and only if X(M), the Euler characteristic of M, is zero. (3) Every non-zero vector field on M is integrable. Now suppose that M is a topological (TOP) or piecewise linear (IX) manifold. What is the appropriate definition of a TOP or PL vector field on M? If M were differentiable, then a non-zero vector field is just a non-zero cross-section of the tangent bundle T(M) of M. In I%2 Milnor[29] define! the TOP tangent microbundle T(M) of a TOP manifold M to be the microbundle M - M x M G M, where A(x) = (x, x) and ~(x, y) = x. If M is a PL manifold, the PL tangent microbundle T(M) is defined similarly if one works in the category of PL maps of polyhedra. If M is differentiable, then T(M) is CAT (CAT = TOP or PL) microbundle equivalent to T(M). -
Symmetric Obstruction Theories and Hilbert Schemes of Points On
Symmetric Obstruction Theories and Hilbert Schemes of Points on Threefolds K. Behrend and B. Fantechi March 12, 2018 Abstract Recall that in an earlier paper by one of the authors Donaldson- Thomas type invariants were expressed as certain weighted Euler char- acteristics of the moduli space. The Euler characteristic is weighted by a certain canonical Z-valued constructible function on the moduli space. This constructible function associates to any point of the moduli space a certain invariant of the singularity of the space at the point. In the present paper, we evaluate this invariant for the case of a sin- gularity which is an isolated point of a C∗-action and which admits a symmetric obstruction theory compatible with the C∗-action. The an- swer is ( 1)d, where d is the dimension of the Zariski tangent space. − We use this result to prove that for any threefold, proper or not, the weighted Euler characteristic of the Hilbert scheme of n points on the threefold is, up to sign, equal to the usual Euler characteristic. For the case of a projective Calabi-Yau threefold, we deduce that the Donaldson- Thomas invariant of the Hilbert scheme of n points is, up to sign, equal to the Euler characteristic. This proves a conjecture of Maulik-Nekrasov- Okounkov-Pandharipande. arXiv:math/0512556v1 [math.AG] 24 Dec 2005 1 Contents Introduction 3 Symmetric obstruction theories . 3 Weighted Euler characteristics and Gm-actions . .. .. 4 Anexample........................... 5 Lagrangian intersections . 5 Hilbertschemes......................... 6 Conventions........................... 6 Acknowledgments. .. .. .. .. .. .. 7 1 Symmetric Obstruction Theories 8 1.1 Non-degenerate symmetric bilinear forms . -
We Begin with a Review of the Classical Fibration Replacement Construction
Contents 11 Injective model structure 1 11.1 The existence theorem . 1 11.2 Injective fibrant models and descent . 13 12 Other model structures 18 12.1 Injective model structure for simplicial sheaves . 18 12.2 Intermediate model structures . 21 11 Injective model structure 11.1 The existence theorem We begin with a review of the classical fibration replacement construction. 1) Suppose that f : X ! Y is a map of Kan com- plexes, and form the diagram I f∗ I d1 X ×Y Y / Y / Y s f : d0∗ d0 / X f Y Then d0 is a trivial fibration since Y is a Kan com- plex, so d0∗ is a trivial fibration. The section s of d0 (and d1) induces a section s∗ of d0∗. Then (d1 f∗)s∗ = d1(s f ) = f 1 Finally, there is a pullback diagram I f∗ I X ×Y Y / Y (d0∗;d1 f∗) (d0;d1) X Y / Y Y × f ×1 × and the map prR : X ×Y ! Y is a fibration since X is fibrant, so that prR(d0∗;d1 f∗) = d1 f∗ is a fibra- tion. I Write Z f = X ×Y Y and p f = d1 f∗. Then we have functorial replacement s∗ d0∗ X / Z f / X p f f Y of f by a fibration p f , where d0∗ is a trivial fibra- tion such that d0∗s∗ = 1. 2) Suppose that f : X ! Y is a simplicial set map, and form the diagram j ¥ X / Ex X q f s∗ f∗ # f ˜ / Z f Z f∗ p f∗ ¥ { / Y j Ex Y 2 where the diagram ˜ / Z f Z f∗ p˜ f p f∗ ¥ / Y j Ex Y is a pullback. -
Almost Complex Structures and Obstruction Theory
ALMOST COMPLEX STRUCTURES AND OBSTRUCTION THEORY MICHAEL ALBANESE Abstract. These are notes for a lecture I gave in John Morgan's Homotopy Theory course at Stony Brook in Fall 2018. Let X be a CW complex and Y a simply connected space. Last time we discussed the obstruction to extending a map f : X(n) ! Y to a map X(n+1) ! Y ; recall that X(k) denotes the k-skeleton of X. n+1 There is an obstruction o(f) 2 C (X; πn(Y )) which vanishes if and only if f can be extended to (n+1) n+1 X . Moreover, o(f) is a cocycle and [o(f)] 2 H (X; πn(Y )) vanishes if and only if fjX(n−1) can be extended to X(n+1); that is, f may need to be redefined on the n-cells. Obstructions to lifting a map p Given a fibration F ! E −! B and a map f : X ! B, when can f be lifted to a map g : X ! E? If X = B and f = idB, then we are asking when p has a section. For convenience, we will only consider the case where F and B are simply connected, from which it follows that E is simply connected. For a more general statement, see Theorem 7.37 of [2]. Suppose g has been defined on X(n). Let en+1 be an n-cell and α : Sn ! X(n) its attaching map, then p ◦ g ◦ α : Sn ! B is equal to f ◦ α and is nullhomotopic (as f extends over the (n + 1)-cell).