DOCSLIB.ORG

Spectral Sequences in Homotopy Type Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spectral Sequences in Homotopy Type Theory Spectral Sequences in Homotopy Type Theory Floris van Doorn Carnegie Mellon University June 4, 2018 Joint work with Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Egbert Rijke and Mike Shulman. Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 1 / 27 Outline Cohomology in HoTT Spectral sequences Atiyah-Hirzebruch and Serre spectral sequences for cohomology Future work: Spectral sequences for homology Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 2 / 27 Classical theorem: The Eilenberg-MacLane spaces K(A; n) classify cohomology. Recall. The Eilenberg-MacLane space K(A; n) is the unique pointed type X with one nontrivial homotopy group πn(X) ' A. Cohomology How do we define cohomology? The usual constructions of singular cohomology is not homotopy invariant. Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 3 / 27 Cohomology How do we define cohomology? The usual constructions of singular cohomology is not homotopy invariant. Classical theorem: The Eilenberg-MacLane spaces K(A; n) classify cohomology. Recall. The Eilenberg-MacLane space K(A; n) is the unique pointed type X with one nontrivial homotopy group πn(X) ' A. Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 3 / 27 Cohomology We define the reduced cohomology of a pointed type X with coefficients in an abelian group A to be n ∗ He (X; A) :≡ kX ! K(A; n)k0: The unreduced cohomology can be defined similarly for any (not necessarily pointed) type X: n n H (X; A) :≡ kX ! K(A; n)k0 = He (X + 1;A): The group structure comes from the equivalence K(A; n) '∗ ΩK(A; n + 1). Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 4 / 27 Long Exact Sequence of Homotopy Groups Given a pointed map f : X !∗ Y with fiber F . Then we have the following long exact sequence. : : π2(f) π2(F ) π2(X) π2(Y ) π2(p1) π1(δ) π1(f) π1(F ) π1(X) π1(Y ) π1(p1) π0(δ) π0(f) π0(F ) π0(X) π0(Y ) π0(p1) Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 5 / 27 Spectral Sequences p;q DefinitionA spectral sequence consists of a family Er of abelian groups for p; q : Z and r ≥ 2. For a fixed r this gives the r-page of the spectral sequence. ::: q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 27 Spectral Sequences p;q p;q p+r;q−r+1 Definition ::: with differentials dr : Er ! Er such that dr ◦ dr = 0 (this is cohomologically indexed) ::: q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 27 Spectral Sequences p;q p;q p;q Definition ::: and with isomorphisms αr : H (Er) ' Er+1 p;q p;q p−r;q+r−1 where H (Er) = ker(dr )=im(dr ). q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 27 Spectral Sequences p;q p;q p;q Definition ::: and with isomorphisms αr : H (Er) ' Er+1 p;q p;q p−r;q+r−1 where H (Er) = ker(dr )=im(dr ). The differentials of Er+1 are not determined by Er. q q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • p;q p p;q p E2 E3 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 6 / 27 And we can get information about the diagonals on the infinity page. Convergence of Spectral Sequences q • • • • • In many spectral sequences the p;q • • • • • pages converge to E1 . • • • • • • • • • • • • • • • D0 D1 D2 D3 D4 p p;q E1 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 7 / 27 Convergence of Spectral Sequences q • • • • • In many spectral sequences the p;q • • • • • pages converge to E1 . • • • • • And we can get information about the diagonals on the • • • • • infinity page. • • • • • D0 D1 D2 D3 D4 p p;q E1 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 7 / 27 p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q D is built up from E1 for n = p + q in the following way: We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 p;q if there exists a spectral sequence Er such that Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q D is built up from E1 for n = p + q in the following way: We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 p;q The spectral sequence converges to E1 ; n p;q D is built up from E1 for n = p + q in the following way: We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 n p;q D is built up from E1 for n = p + q in the following way: We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q D is built up from E1 for n = p + q in the following way: Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 Convergence of Spectral Sequences For a bigraded abelian group Cp;q and graded abelian group Dn we write p;q p;q p+q E2 = C ) D p;q if there exists a spectral sequence Er such that p;q p;q The second page is E2 = C ; p;q The spectral sequence converges to E1 ; n p;q D is built up from E1 for n = p + q in the following way: We have short exact sequences: 0;n n n;1 0 ! E1 !D ! D ! 0 : : p;q n;p n;p+1 0 ! E1 !D ! D ! 0 p+1;q−1 n;p+1 n;p+2 0 ! E1 !D ! D ! 0 : : n;0 n;n 0 ! E1 !D ! 0 Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 8 / 27 Serre Spectral Sequence (Special Case) Theorem. Suppose f : X ! B and b0 : B and let F :≡ fibf (b0). Suppose that B is simply connected and A is an abelian group. Then p;q p q p+q E2 = H (B; H (F; A)) ) H (X; A): This is only true for unreduced cohomology. Floris van Doorn (CMU) Spectral Sequences in HoTT June 4, 2018 9 / 27 Example: Cohomology of K(Z; 2) 1 We will compute the cohomology groups of B = K(Z; 2) (which is CP ). f We define the map 1 −! K(Z; 2) determined by the basepoint ? : K(Z; 2).
Load more

Recommended publications
  • The Leray-Serre Spectral Sequence
    THE LERAY-SERRE SPECTRAL SEQUENCE REUBEN STERN Abstract. Spectral sequences are a powerful bookkeeping tool, used to handle large amounts of information. As such, they have become nearly ubiquitous in algebraic topology and algebraic geometry. In this paper, we take a few results on faith (i.e., without proof, pointing to books in which proof may be found) in order to streamline and simplify the exposition. From the exact couple formulation of spectral sequences, we ∗ n introduce a special case of the Leray-Serre spectral sequence and use it to compute H (CP ; Z). Contents 1. Spectral Sequences 1 2. Fibrations and the Leray-Serre Spectral Sequence 4 3. The Gysin Sequence and the Cohomology Ring H∗(CPn; R) 6 References 8 1. Spectral Sequences The modus operandi of algebraic topology is that \algebra is easy; topology is hard." By associating to n a space X an algebraic invariant (the (co)homology groups Hn(X) or H (X), and the homotopy groups πn(X)), with which it is more straightforward to prove theorems and explore structure. For certain com- putations, often involving (co)homology, it is perhaps difficult to determine an invariant directly; one may side-step this computation by approximating it to increasing degrees of accuracy. This approximation is bundled into an object (or a series of objects) known as a spectral sequence. Although spectral sequences often appear formidable to the uninitiated, they provide an invaluable tool to the working topologist, and show their faces throughout algebraic geometry and beyond. ∗;∗ ∗;∗ Loosely speaking, a spectral sequence fEr ; drg is a collection of bigraded modules or vector spaces Er , equipped with a differential map dr (i.e., dr ◦ dr = 0), such that ∗;∗ ∗;∗ Er+1 = H(Er ; dr): That is, a bigraded module in the sequence comes from the previous one by taking homology.
    [Show full text]
  • Some Extremely Brief Notes on the Leray Spectral Sequence Greg Friedman
    Some extremely brief notes on the Leray spectral sequence Greg Friedman Intro. As a motivating example, consider the long exact homology sequence. We know that if we have a short exact sequence of chain complexes 0 ! C∗ ! D∗ ! E∗ ! 0; then this gives rise to a long exact sequence of homology groups @∗ ! Hi(C∗) ! Hi(D∗) ! Hi(E∗) ! Hi−1(C∗) ! : While we may learn the proof of this, including the definition of the boundary map @∗ in beginning algebraic topology and while this may be useful to know from time to time, we are often happy just to know that there is an exact homology sequence. Also, in many cases the long exact sequence might not be very useful because the interactions from one group to the next might be fairly complicated. However, in especially nice circumstances, we will have reason to know that certain homology groups vanish, or we might be able to compute some of the maps here or there, and then we can derive some consequences. For example, if ∼ we have reason to know that H∗(D∗) = 0, then we learn that H∗(E∗) = H∗−1(C∗). Similarly, to preserve sanity, it is usually worth treating spectral sequences as \black boxes" without looking much under the hood. While the exact details do sometimes come in handy to specialists, spectral sequences can be remarkably useful even without knowing much about how they work. What spectral sequences look like. In long exact sequence above, we have homology groups Hi(C∗) (of course there are other ways to get exact sequences).
    [Show full text]
  • The Grothendieck Spectral Sequence (Minicourse on Spectral Sequences, UT Austin, May 2017)
    The Grothendieck Spectral Sequence (Minicourse on Spectral Sequences, UT Austin, May 2017) Richard Hughes May 12, 2017 1 Preliminaries on derived functors. 1.1 A computational definition of right derived functors. We begin by recalling that a functor between abelian categories F : A!B is called left exact if it takes short exact sequences (SES) in A 0 ! A ! B ! C ! 0 to exact sequences 0 ! FA ! FB ! FC in B. If in fact F takes SES in A to SES in B, we say that F is exact. Question. Can we measure the \failure of exactness" of a left exact functor? The answer to such an obviously leading question is, of course, yes: the right derived functors RpF , which we will define below, are in a precise sense the unique extension of F to an exact functor. Recall that an object I 2 A is called injective if the functor op HomA(−;I): A ! Ab is exact. An injective resolution of A 2 A is a quasi-isomorphism in Ch(A) A ! I• = (I0 ! I1 ! I2 !··· ) where all of the Ii are injective, and where we think of A as a complex concentrated in degree zero. If every A 2 A embeds into some injective object, we say that A has enough injectives { in this situation it is a theorem that every object admits an injective resolution. So, for A 2 A choose an injective resolution A ! I• and define the pth right derived functor of F applied to A by RpF (A) := Hp(F (I•)): Remark • You might worry about whether or not this depends upon our choice of injective resolution for A { it does not, up to canonical isomorphism.
    [Show full text]
  • HE WANG Abstract. a Mini-Course on Rational Homotopy Theory
    RATIONAL HOMOTOPY THEORY HE WANG Abstract. A mini-course on rational homotopy theory. Contents 1. Introduction 2 2. Elementary homotopy theory 3 3. Spectral sequences 8 4. Postnikov towers and rational homotopy theory 16 5. Commutative differential graded algebras 21 6. Minimal models 25 7. Fundamental groups 34 References 36 2010 Mathematics Subject Classification. Primary 55P62 . 1 2 HE WANG 1. Introduction One of the goals of topology is to classify the topological spaces up to some equiva- lence relations, e.g., homeomorphic equivalence and homotopy equivalence (for algebraic topology). In algebraic topology, most of the time we will restrict to spaces which are homotopy equivalent to CW complexes. We have learned several algebraic invariants such as fundamental groups, homology groups, cohomology groups and cohomology rings. Using these algebraic invariants, we can seperate two non-homotopy equivalent spaces. Another powerful algebraic invariants are the higher homotopy groups. Whitehead the- orem shows that the functor of homotopy theory are power enough to determine when two CW complex are homotopy equivalent. A rational coefficient version of the homotopy theory has its own techniques and advan- tages: 1. fruitful algebraic structures. 2. easy to calculate. RATIONAL HOMOTOPY THEORY 3 2. Elementary homotopy theory 2.1. Higher homotopy groups. Let X be a connected CW-complex with a base point x0. Recall that the fundamental group π1(X; x0) = [(I;@I); (X; x0)] is the set of homotopy classes of maps from pair (I;@I) to (X; x0) with the product defined by composition of paths. Similarly, for each n ≥ 2, the higher homotopy group n n πn(X; x0) = [(I ;@I ); (X; x0)] n n is the set of homotopy classes of maps from pair (I ;@I ) to (X; x0) with the product defined by composition.
    [Show full text]
  • Arxiv:Math/0305187V1 [Math.AT] 13 May 2003 Oehn Aiir Ieapiigo Iglrchmlg G Th Subtleties Cohomology Are Singular There of Time
    MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II DANIEL DUGGER Contents 1. Introduction 1 2. Sign conventions in singular cohomology 1 3. Spectral sequences for filtered spaces 3 4. The Postnikov/Whitehead spectral sequence 8 5. Bockstein spectral sequences 13 6. Thehomotopy-fixed-pointspectralsequence 15 7. Spectralsequencesfromopencoverings 16 References 19 1. Introduction This short paper is a companion to [D1]. Here the main results of that paper are used to establish multiplicative structures on a few standard spectral sequences. The applications consist of (a) applying [D1, Theorem 6.1] to obtain a pairing of spectral sequences, and (b) identifying the pairing on the E1- or E2-term with something familiar, like a pairing of singular cohomology groups. Most of the arguments are straightforward, but there are subtleties that appear from time to time. Originally the aim was just to record a careful treatment of pairings on Post- nikov/Whitehead towers, but in the end other examples have been included because it made sense to do so. These examples are just the ones that I personally have needed to use at some point over the years, and so of course it is a very limited selection. arXiv:math/0305187v1 [math.AT] 13 May 2003 In this paper all the notation and conventions of [D1] remain in force. In particu- lar, the reader is referred to [D1, Appendix C] for our standing assumptions about the category of spaces and spectra, and for basic results about signs for bound- ary maps. The symbol ∧ denotes the derived functor of ∧, and W⊥ denotes an ‘augmented tower’ as in [D1, Section 6].
    [Show full text]
  • Rational Homotopy Theory: a Brief Introduction
    Contemporary Mathematics Rational Homotopy Theory: A Brief Introduction Kathryn Hess Abstract. These notes contain a brief introduction to rational homotopy theory: its model category foundations, the Sullivan model and interactions with the theory of local commutative rings. Introduction This overview of rational homotopy theory consists of an extended version of lecture notes from a minicourse based primarily on the encyclopedic text [18] of F´elix, Halperin and Thomas. With only three hours to devote to such a broad and rich subject, it was difficult to choose among the numerous possible topics to present. Based on the subjects covered in the first week of this summer school, I decided that the goal of this course should be to establish carefully the founda- tions of rational homotopy theory, then to treat more superficially one of its most important tools, the Sullivan model. Finally, I provided a brief summary of the ex- tremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme “Interactions between Homotopy Theory and Algebra.” I hoped to motivate the students to delve more deeply into the subject themselves, while providing them with a solid enough background to do so with relative ease. As these lecture notes do not constitute a history of rational homotopy theory, I have chosen to refer the reader to [18], instead of to the original papers, for the proofs of almost all of the results cited, at least in Sections 1 and 2. The reader interested in proper attributions will find them in [18] or [24]. The author would like to thank Luchezar Avramov and Srikanth Iyengar, as well as the anonymous referee, for their helpful comments on an earlier version of this article.
    [Show full text]
  • 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).
    [Show full text]
  • Etale Cohomology of Curves
    ETALE COHOMOLOGY OF CURVES SIDDHARTH VENKATESH Abstract. These are notes for a talk on the etale cohomology of curves given in a grad student seminar on etale cohomology held in Spring 2016 at MIT. The talk follows Chapters 13 and 14 of Milne ([Mil3]). Contents 1. Outline of Talk 1 2. The Kummer Exact Sequence 2 3. Cohomology of Gm 4 4. Cohomology of µn 7 4.1. For Projective Curves 7 4.2. Lefschetz Trace Formula and Rationality of Zeta Function 7 4.3. For Affine Curves 10 5. Compactly Supported Cohomology for Curves 10 6. Poincare Duality 11 6.1. Local Systems in the Etale Topology 11 6.2. Etale Fundamental Group 11 6.3. The Duality Statement 13 7. Curves over Finite Fields 17 7.1. Hochschild-Serre Spectral Sequence 17 7.2. Poincare Duality 17 References 18 1. Outline of Talk The main goal of this talk is to prove a Poincare duality result for smooth curves over algebraically closed fields. Eventually, we will want to get a similar result for higher dimensional varieties as well but for now, we only have the machinery to deal with curves. Let me begin by recalling how Poincare duality works for smooth varieties of dimension n over C. In the analytic topology, we have a perfect pairing r 2n−r 2n H (X; Z) × Hc (X; Z) ! H (X; Z) = Z: More generally, you can replace the sheaf Z with any locally constant sheaf F to get a perfect pairing r 2n−r H (X; F) × H (X; Fˇ(1)) ! Z where Fˇ(1) = Hom(F; Z): This cannot be done in the Zariski topology because the topolgy is contractible (if the variety is irreducible).
    [Show full text]
  • Computations with Spectral Sequences
    Computations with spectral sequences Eric Peterson 2 Contents 0.1 Foreword........................................5 0.2 Acknowledgements . .6 1 Spectral sequences in general 7 1.1 Homology theories . .7 1.2 Filtrations and spectral sequences . .8 1.3 Convergence and the endgame . 10 1.4 Grading conventions and multiplicative structures . 11 2 Atiyah-Hirzebruch-Serre 13 2.1 The Atiyah-Hirzebruch spectral sequence . 13 2.2 H ∗CP1 ......................................... 14 2.3 H ∗RP1 ......................................... 14 2.4 KU ∗BZ=2 ....................................... 14 2.5 The Serre spectral sequence . 14 2.6 H ∗CP1 redux..................................... 16 2.7 H ∗RP1 and Gysin sequences . 23 2.8 Unitary groups . 25 2.9 Loopspaces of spheres . 30 2.10 The Steenrod algebra . 30 2.11 H ∗(BU 6 ;F2) .................................... 30 2.12 Unstableh homotopyi groups of S3 ........................ 30 3 Eilenberg-Moore 31 3.1 Computing Tor with Tate resolutions . 31 2n+1 3.2 H ∗(ΩS ;Fp ) .................................... 32 3.3 Complex projective spaces . 32 3.4 The James construction . 32 3.5 H ∗BU 6 redux.................................... 32 h i 4 Co/simplicial objects 33 4.1 Mayer-Vietoris . 33 4.2 The bar spectral sequence and K( p ; ) ..................... 33 F ∗ 4.3 The descent spectral sequence . 33 3 4 CONTENTS 5 The Adams spectral sequence 35 5.1 The dual of the Steenrod algebra . 35 5.2 Resolution by Hopf algebra quotients . 35 5.3 π ko2^ .......................................... 35 ∗ 5.4 π ku2^ .......................................... 35 ∗ 5.5 ko∗M(2) ......................................... 36 5.6 Massey products and secondary operations . 36 5.7 Change of rings . 36 5.8 π t m f ........................................
    [Show full text]
  • A Spectral Sequence for Tangent Cohomology of Algebras Over Algebraic Operads
    A spectral sequence for tangent cohomology of algebras over algebraic operads JOSÉ M. MORENO-FERNÁNDEZ PEDRO TAMAROFF Operadic tangent cohomology generalizes the existing theories of Harrison cohomology, Chevalley–Eilenberg cohomology and Hochschild cohomology to address the deformation the- ory of general types of algebras through the pivotal gadgets known as a deformation complexes. The cohomology of these is in general very non-trivial to compute, and in this paper we comple- ment the existing computational techniques by producing a spectral sequence that converges to the operadic cohomology of a fixed algebra. Our main technical tool is that of filtrations arising from towers of cofibrations of algebras, which play the same role cell attaching maps and skeletal filtrations do for topological spaces. As an application, we consider the rational Adams–Hilton construction on topological spaces, where our spectral sequence gives rise to a seemingly new and completely algebraic description of the Serre spectral sequence, which we also show is multiplicative and converges to the Chas– Sullivan loop product. We also consider relative Sullivan–de Rham models of a fibration p, where our spectral sequence converges to the rational homotopy groups of the identity component of the space of self-fiber-homotopy equivalences of p. MSC 2020: 18M70; 13D03, 18G40, 55P62 1 Introduction Algebraic operads are an effective gadget to study different types of algebras through a common language. In particular, they provide us with tools to study the deformation theory of such algeb- ras. For example, the operads controlling Lie, associative and commutative algebras —collectively known as the three graces of J.-L.
    [Show full text]
  • Spectral Sequences and Applications Tuesday 14:00-16:00
    Spectral Sequences and applications Tuesday 14:00-16:00 Stefan Friedl, Gerrit Herrmann and Enrico Toffoli 1 10 April Julian Hannes Generalities on homotopy groups 2 17 April Daniel Gr¨unbaum Fibrations and fiber bundles 3 24 April Christoph Schießl The path space construction - 1 May - - 4 8 May Stefan Wolf Spectral sequences from exact couples 5 15 May Dima Shvetsov The Serre spectral sequence and first applications - 22 May - - 6 29 May Catherine-Leveke Lowzow Construction of the sequence 7 5 June Dannis Zumbil The Hurewicz theorem and other applications 8 12 June The Serre spectral sequence in cohomology 9 19 June Constantin Benninger More about the homotpy groups of spheres 10 26 June Felix Rupprecht Applications to group (co)homology 11 3 July Jakob Schubert Generalized cohomology theories 12 10 July Daniel Fauser The Atiyah-Hirzebruch spectral sequence Detailed plan 1. Generalities on homotopy groups([Ha1, Section 4.1]) Recall the definition of homotopy groups and define relative homotopy groups. Discuss the group struc- ture and when these are abelian groups. Moreover, give a proof of [Ha1, Theorem 4.3]. Define the Hurewicz map and discuss [Ha1, Proposition 4.36] and the com- mutativity of the diagram on [Ha1, p. 370]. 2. Fibrations and fiber bundles ([Ha1, Section 4.2]) Give the definition of a fiber bundle, the homotopy lifting problem and fibrations. You should prove [Ha1, Theorem 4.41] and [Ha1, Proposition 4.48]. During your talk you should discuss several examples. Sketch the prove of [Ha1, Proposition 4.61]. What happens in the special case that the path in the prove is a loop and the fibration is a fiber bundle? 3.
    [Show full text]
  • Guide to the Serre Spectral Sequence in 10 Easy Steps!
    Guide to the Serre Spectral Sequence in 10 easy steps! Jane Tan January 2020 These notes are intended to serve as a practical and painless introduction to the Serre spectral sequence, based on a lecture by Andr´eHenriques. Through 10 step-by-step examples, we'll see that a lot can be deduced from just the co- homology of spheres and some well-known fibrations. There will be no proofs or even precise statements made; the aim is simply to get a feel for the calculations involved and pick up some basic properties of the spectral sequence along the way. To this end, instead of a definition, we begin by describing what a spectral sequence looks like. The data of a spectral sequence can be naturally organised into pages Er for r ≥ 1. Each page consists of a 2-dimensional lattice of objects, as well as maps of a specified bidegree between any two objects for which the degrees match up. These form a collection of chain complexes. The spectral sequence that we consider is first-quadrant. This means that if we view our groups as being located at points of the integer lattice on the Cartesian plane, everything below the horizontal axis or right of the vertical axis is zero. It will also be cohomologically graded, meaning differentials dr on Er have bidegree (r; 1 − r). s;t The group at position (s; t) on the rth page is denoted Er . Differentials on s;t s+r;t+1−r the rth page are therefore of the form Er ! Er .
    [Show full text]