Math 8360, Spring 2011

Topics in Topology

The goal of this course is to introduce methods from modern homotopy theory. Some specific topics include the homotopy theory of topological spaces, simplicial objects, derived categories, and rational homotopy theory. The common theme underlying the course is homotopy theory, and specifically the theory of model categories.

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Summary

The main goal of this webpage is to track the topics covered in the course on a day-to-day basis. This is partially for my own benefit as I don't have time to produce coherent lecture notes this semester. Please contact me if there is any other information you think might be appropriate here, or if you have any questions.

January 19

Motivation. Algebraic topology as a process of forgetting information. Homotopy category. Progressive refinements of Euler characteristic. Singular chains establish a functor between homotopy theories. Resulting division of labor between the homotopy theory and the algebra of chain complexes. The homotopy category as a localization. Example where localization may make the category too large for its current universe.

January 21

Background in homotopy theory. Basic operating principle: homotopy equivalent objects can and will be swapped out for each other. Certain types of objects and maps are better behaved than others. Basic definitions of the category of chain complexes of R-modules, R a commutative ring. Cⁿ=C_-n. Homology groups. Chain homotopy. Tensor product of chain complexes. Unit interval object I. A chain homotopy is the same as a map I⊗C→D. Shift operators. Brief discussion about sign conventions and the lecturer's objection to the notation C[1] versus ΣC.
Suggested exercises: Show that chain homotopy defines an equivalence relation, and respects composition and tensor product. Show that chain homotopic maps induce the same map on homology. Give examples of a map of chain complexes that is an isomorphism on homology, but not a chain homotopy equivalence. Give an example of two maps of chain complexes that give the same map on homology, but are not chain homotopic.

January 24

Class cancelled due to instructor falling on his head in the mailroom.
Suggested exercises: Avoid head injuries.

January 26

Quasi-isomorphisms. Chain homotopy equivalence implies quasi-isomorphism. Mapping cones, cell attachment. Analogy with topology. Homology groups H_k(C) are homotopy classes of maps Σ^kR → C. Chain complexes that admit a cell structure, such as bounded-below complexes of free modules. Whitehead theorem, proof (in the bounded below case) to be completed later.
Suggested exercises: Show that a mapping cylinder is always chain homotopy equivalent to the range. Show that two chain-homotopic maps have chain-homotopy-equivalent cones. Give a definition of new homology groups using maps from Σ^kP for an R-module P. Show that these are preserved by chain homotopy equivalence, but not preserved by quasi-isomorphism unless P is projective. Use maps out to define cohomology groups with coefficients in a module Q.

January 28

Completion of proof of Whitehead theorem for bounded-below complexes. Discussion of unbounded case. Analogy with topological version: one dimension at a time, push cells into the equivalent subcomplex and use the homotopy extension property to extend to the rest. Abstracting the general setup. Axioms for a model category: complete-cocomplete, 2-out-of-3, retract, lifting, and factorization. Examples: chain complexes, topological spaces.
Suggested exercises: Give an example of a map C → D of unbounded chain complexes of projective modules which is a weak equivalence but not a chain homotopy equivalence. Show that cofibrant, bounded-below chain complexes are precisely complexes of projectives. Describe all model categories where all weak equivalences are isomorphisms.

January 31

Cofibrant replacement. For nonnegatively graded complexes: iteratively step up, one degree at a time, and build a quasi-isomorphic complex which is levelwise projective by explicitly building in the correct homology groups. (Missing step: Make it levelwise surjective.) For unbounded complexes: Do the above for each connective cover and form a "mapping telescope". Proving factorization axioms in model categories usually require some form of these proofs. Goal next is to construct the homotopy category of a model category. In a model category, a weak equivalence X → Y, with fibrant source and cofibrant target, has a "homotopy inverse".
Suggested exercises: Given a map of complexes C → D, construct a factorization C → C' → D into an acyclic cofibration followed by a fibration. (Hint: Start by choosing, for each n, a surjective map P_n → D_n with projective domain.) If you see the lecturer accidentally teaching to certain audience members, give a discrete signal like banging on the desk.

February 2

Reminder of the model category axioms. Factorization implies every object has a weak equivalence from a cofibrant object, a weak equivalence to a fibrant one. In homotopy category, it suffices to define homotopy classes of maps between cofibrant-fibrant objects and check that they are independent of choice. Cylinder objects. Any two cylinder objects are equivalent. Left-homotopy relation between maps. Dually, path objects and right-homotopy relation between maps. Start of proof that, with cofibrant source, left homotopy implies right homotopy.
Suggested exercises: Define a canonical path object for chain complexes such that right homotopic maps are chain homotopic. Show that an acyclic cofibration i: X → Y with X fibrant admits a retraction r: Y → X. Show that the opposite category of a model category is a model category.

February 4

Proof that any two among cofibrations, fibrations, and weak equivalences determine the third. If X is cofibrant, then the map X → X ∐ X is a cofibration. Using this, showing that if f and f' are left homotopic to g and the source is cofibrant, then f and f' are right homotopic. Dual result has a dual proof. If source is cofibrant and target is fibrant, left and right homotopy coincide and give an equivalence relation. Example computing Hom(M, Σ^kN) in the homotopy category of chain complexes.
Suggested exercises: Carry out Goodwillie's exercise and show that there are exactly nine model category structures on the category of sets. Compute all homotopy classes of chain maps between suspensions of either R or R/(x) when R = k[x]/x^n. Give examples where left and right homotopy do not coincide, and give examples where they do not define an equivalence relation.

February 7

Homotopy relation respects composition on cofibrant-fibrant objects. Homotopy category of cofibrant-fibrant objects. Using cofibrant and fibrant replacement, we can "define" homotopy classes of maps between any two objects. Any two replacements give a unique choice of compatible isomorphism. Issue remains: choices must be made to define the underlying set of morphisms. Three possible solutions. Strengthen the model category axioms to include functorial factorization. Strengthen the axioms of set theory to include a global choice operator, such as Hilbert's ε or Bourbaki's τ. Use an anafunctor instead, by bloating up the object set of M to make a new model category M'with a fully faithful, essentially surjective map M' → M such that M' has choices of cofibrant/fibrant replacement for any object. Partially anticipated, but longer-than-expected discussion about the relative merits of doing something other than using Bourbaki's foundations for mathematics.
Suggested exercises: Determine how to "compose" anafunctors. For any small category I, functor I → C, and anafunctor C ← C' → D, construct a "lifted" composite I → D which is determined up to natural isomorphism. Avoid having any of Adrian Mathias' papers on your person.

February 9

Completing proof that we get a homotopy category. Showing that, for X → X' a weak equivalence between cofibrant objects, Y fibrant, [X',Y] → [X,Y] is an isomorphism. Use factorization to reduce to cases where the map is an acyclic cofibration (apply lifting) or an acyclic fibration (construct a retraction). Dual theorem for weak equivalences between fibrant objects. Chain complexes and the Yoneda pairing on Ext groups. Brief mention of Ext in terms of exact sequences. Hom-complex Hom(C,D) between two chain complexes, adjoint to the tensor product. Evaluation map Hom(C,D) ⊗ C → D, multiplication pairing. Hom(C,C) is a differential graded algebra. Examples.
Side note: Prof. Messing mentioned graded-commutativity. This does appear, but is usually specific to cases where the ground ring has a bialgebra structure.
Suggested exercises: Use "mapping cylinders" (and some more work) to show the isomorphism [X',Y] → [X,Y] without reducing to cases. If R and S are augmented k-algebras for a field k (i.e. there are maps k → R → k of k-algebras), show that Ext_R⊗kS(k,k) is isomorphic to Ext_R(k,k) ⊗_k Ext_S(k,k) as a graded ring. Compute the Ext-groups Ext(k,k) (with the Yoneda multiplication) with the base rings k[x], k[x]/xⁿ, and (harder) k[x,y]/(x², xy, y²).

February 11

Hom(C,C) is always a differential graded algebra. Expression in terms of diagrams. Things omitted: descending the tensor product down to the derived category, and triangulated structures. Topological spaces. Exponential law requires compact generation. In order to do homotopy theory, it is helpful to step away from spaces and their pathology. Definition of simplicial complexes. Definition of Δ-complexes (in Hatcher's terminology). Realization of both as topological spaces.
Suggested exercises: Give a proof, using only the adjunction properties, of the fact that the action map Hom(C,C) ⊗ C → C is associative. Define left and right modules over a differential graded algebra as a generalization of chain complexes over a ring. Give a construction of a 2-torus, a 2-sphere, and a surface of genus 2 using both simplicial complexes and Δ-complexes. Give a preliminary definition of a map of simplicial complexes or Δ-complexes.

February 14

Elaboration on geometric realization for Δ-complexes. Category Δ^inj consisted of finite, nonempty ordered sets and order-preserving injections. Δ-complexes are contravariant functors F: Δ^inj → Sets so we don't need to remember the identities for the face operators. Δ-complexes don't have enough maps because they can't squish simplices. Category Δ consisting of finite, nonempty ordered sets and monotone maps. Simplicial sets are contravariant functors F: Δ → Sets. Indicated expression in terms of face and degeneracy operators. Generally much larger. Example of the unit interval. Useful property: cartesian products are handled exceptionally easily.
Suggested exercises: Explicitly determine the identities satisfied by the face and degeneracy operators, or find a place to look them up. Describe a way to realize the standard n-simplex Δ[n] as a simplicial complex, a Δ-complex, and a simplicial set (in the latter two cases, using the "functor" description). Define simplicial objects in other categories, such as groups, abelian groups, rings, et cetera.

February 16

Geometric realization of simplicial sets. Cosimplicial object representing the standard simplices. Degenerate simplices don't contribute points or relations to the realization. Construction of simplicial sets via the singular complex. Cosimplicial object in categories. Construction of simplicial sets via the nerve. The adjunction Hom(|K|,Z) = Hom(K,Sing(Z)) between simplicial sets and spaces.
Show that degenerate simplices are closed under boundaries. Show that every degenerate simplex is degenerate in a unique way; i.e. there exists a unique nondegenerate simplex and degeneracy map with the given simplex as image. Construct a cosimplicial object in chain complexes modeling the standard n-simplex, by analogy with the way we defined the unit interval object. Find three other examples of adjunctions between categories.

February 18

Quillen model structure on topological spaces. Weak equivalences are isomorphisms on homotopy groups. Fibrations are Serre fibrations. Cofibrations are retracts of relative cell inclusions. Cofibrations are always closed under retracts, pushouts, and transfinite composition, so it suffices to say that the maps Sⁿ → Dⁿ⁺¹ generate all the cofibrations. Similarly, the fibrations are determined by having the lifting property with respect to Dⁿ → Dⁿ⁺¹; these are the "generating acylic cofibrations". Verification of the first few model category axioms.
Suggested exercises: If K → L is an inclusion of simplicial sets, show |K| → |L| is a cofibration. Show that any map |K| → Z lifts to a map |K| → |Sing(Z)|. Conclude that the map |Sing(Z)| → Z is always an acyclic fibration.

February 21

Thematically similar proofs: CW-approximation, Postnikov towers, connective covers, existence of Eilenberg-Mac Lane spaces, Brown representability. Start with a partial answer, and attach cells to fix problems. The small object argument: given a set I of inclusion maps with compact domain, construction of a factorization of a general map X → Y into an I-cellular inclusion followed by a map which has the right lifting property with respect to I. Application to factorizing maps into acyclic cofibrations followed by fibrations. Application to factorizing maps into cofibrations followed by maps with the right lifting property with respect to Sⁿ → Dⁿ⁺¹. Such maps have the right lifting property with respect to all inclusions of CW-complexes, and hence they are acyclic fibrations.
Suggested exercises: Show that if I and J are two sets of maps, and all the maps in J are I-cellular inclusions, any map with the right lifting property with respect to I automatically has it with respect to J. If C is a finite category in which every set of maps Hom(x,y) has only zero or one elements, determine when C is complete and cocomplete, and describe the result of carrying out the small object argument with respect to a set I of maps.

February 23

Obstructions to extending maps from spheres to discs. There is an obstruction in π_n, and if that obstruction vanishes the possible choices are parametrized up to homotopy by π_n+1. Apply to show both lifting axioms for the category of topological spaces. Finally, topological spaces form a model category.
Suggested exercises: Carry out the proof more explicitly in the specific case where we have a fibration X → Y and we are trying to lift a point of Y to X. Show that if A → B is a CW-inclusion, the map of function spaces Map(B,X) → Map(A,X) is a fibration. Show that if A is a CW-complex and X → Y is a fibration, the map of function spaces Map(A,X) → Map(A,Y) is a fibration. Show that in either case, we get an acyclic fibration if the given map is a weak equivalence.

February 25

For an object A in a model category M, the under-category M_A/ inherits a model category structure. Weak equivalences, cofibrations, and fibrations are determined by the underlying map in M. Most axioms follow directly from the definitions, except for cocompleteness: colimits are different in M_A/. Example: pointed spaces. Dually, the over-category M_/X always inherits a model structure. Example: topological spaces over X. Fibrant objects are fibrations Y → X, cofibrant objects are cofibrant on the underlying space, and the homotopy relation is fibrewise homotopy. Projective and injective model structures on the category of maps ("pairs"). General nonsense doesn't handle objects with an action of a group G. Brief mention of Reedy model structures.
Suggested exercises: Give a proof that the projective and injective model structures on a category of maps are actually model structures. Generalize one of these to infinite sequences of maps X₁ → X₂ → X₃ → …. Explain why the other one doesn't work.

February 28

No class today (instructor is visiting MIT).

March 2

Simplicial sets. Difficulty with extracting homotopy-theoretic information in general, specifically with the fundamental group. Paths aren't composable, homotopy isn't an equivalence relation, inverses don't exist. Some simplicial sets are better for mapping into than others. Weak equivalences of simplicial sets are defined on geometric realization. Cofibrations are inclusions. Definitions of simplicial sets Δ[n], the boundary ∂Δ[n], and the horns Λ_k[n]. Geometric realizations of these. Fibrations have the left lifting property with respect to inclusions Λ_k[n] → Δ[n]. Acyclic fibrations have the left lifting property with respect to ∂Δ[n] → Δ[n].
Suggested exercises: If K is a simplicial set with only one zero-simplex, use the Seifert-Van Kampen theorem to find a presentation for the fundamental group of |K| with generators and relations in terms of the simplices of K. Similarly, if L is a simplicial set, describe π₀|L| in terms of simplices. Show that any category C is determined by its nerve NC, and describe the simplicial sets K that can be the nerve of a category in terms of lifting properties. Give a proof, using the functor description, that the boundary and horns are actually subcomplexes of Δ[n].

March 4

Horns and boundaries are unions of their face subcomplexes. Corresponding description of maps out in terms of compatible maps from (n-1)-simplices. A map has the left lifting property with respect to the inclusions ∂Δ[n] → Δ[n] if and only if it has the left lifting property with respect to all inclusions. Verification of some model category axioms. Small object argument implies that we can factor maps into: inclusion followed by "acyclic fibration," or anodyne map followed by fibration. Are "acyclic fibrations," which have the right lifting property with respect to horn inclusions, genuinely weak equivalences? Theorem of Quillen: the geometric realization of a fibration is a fibration.
Suggested exercises: Show that every simplex in Δ[n] can be expressed as φ^*σ for some σ and φ where the latter is surjective. If K → L is an inclusion of simplicial sets and σ is a simplex of L whose boundary is in K, show that there is a subcomplex consisting of the union of K with all elements φ^*σ for φ surjective. By considering K/L, show that for any simplex of the form φ^*σ, φ is uniquely determined. Anodyne maps are the smallest class of maps containing the inclusions Λ_k[n] → Δ[n] and closed under pushouts and transfinite composition: show that for any inclusion A → B, the map (A × I) ∪ (B × 0) → B × I is anodyne.

March 7

Nerves of categories. Maps into the nerve of a category are uniquely determined by the simplices in dimensions below two, and if an extension to the 2-simplices exists the map exists. Nerves are fibrant if and only if the category is a groupoid. Example of BG for G a group. Studying specifically [Y,NBG] for Y a simplicial set. Incomplete proof that such maps are some form of cocycles mod coboundary (incomplete because the instructor forgot that the next step was "plug in the identity"). First cohomology of a group with coefficients in a G-module A in terms of homotopy classes of sections of a map B(A⋊G) → BG on nerves.
Suggested exercises: For categories C and D, show that functors F: C → D are the same as functions NC → ND of simplicial sets, and that natural transformations are the same as homotopies NC × I → ND. In the proof in class, plug in σ = e or τ = e into the identities to show that two maps Y → NBG, corresponding to assignments g_σ and h_σ, are homotopic if and only if there exists an element s of G such that sg_σs^-1 = h_σ for all σ. Conclude that homotopy classes of maps NBG → NBH are conjugacy classes of group homomorphisms G → H. Carry out the full calculation identifying H¹(G;A) with the set of homotopy classes of sections as mentioned above.

March 9

No class today.

March 11

Definition of homotopy groups of a fibrant simplicial set (Kan complex). Making the definition rigorous requires proving that we have an equivalence relation and that multiplication operations exist, are well-defined, associative, unital, and have inverses. Proof that we have an equivalence relation in the case of π₀. Using diagrams to track the combinatorics of simplicial identities, horns, and their fillers. In the case of π₁, proof that we have an equivalence relation, that mult and part of the proof that multiplication exists and is well-defined.
Suggested exercises: Show diagrammatically that the multiplication in π₁ is well-defined, unital, associative, and has inverses (as many of these proofs as you can stand). Translate a few into proofs using the simplicial identities; do at least one (e.g. associativity) with the simplicial identities alone. Show that the relation defining π_n is an equivalence relation. Look up the definition of a quasicategory, alias ∞-category, alias (∞,1)-category (nlab, Wikipedia, n-category cafe post, HTT, find someone who can get you Joyal's unpublished preprint, etc etc).

March 14-18

Spring Break! But Jacob Lurie is visiting and you should really go see him speak if you can. His colloquium is on Tuesday, March 15 at 3:30pm.

March 21

Special properties of the adjoint pair of functors between simplicial sets and spaces. Functors between model categories aren't like what you'd necessarily expect; most naive versions either preserve too much or too little structure. Quillen adjoint pairs: the left adjoint preserves cofibrations and acyclic cofibrations, or equivalently the right adjoint preserves fibrations and acyclic fibrations. Both functors in a Quillen pair induce functors on the homotopy category (modulo the same set-theoretic issues as in defining the homotopy category!), called the derived functors. The resulting derived functors are still adjoint.
Suggested exercises: Show that a composite of two left Quillen adjoints is still a left Quillen adjoint. If M is a model category, and * is the trivial model category with one object and no nonidentity maps, decide when the unique functor M → * or an arbitrary functor * → M are left or right Quillen adjoints, and describe the adjunction.

March 23

Examples of Quillen adjoint pairs. For X → Y, Quillen adjoint pair of functors between the model categories M_/X and M_/Y. For a ring map R → S, adjunction between the forgetful functor and the "tensoring up" functor S⊗_R(-). Necessary and sufficient condition for an adjoint pair of functors to be an inverse equivalence of categories: the unit and counit must always be isomorphisms. Corresponding definition of Quillen equivalences. Simplicial sets and topological spaces are Quillen equivalent. Simplicial objects in an arbitrary category.
Suggested exercises: If C is a "perfect" chain complex (one which consists in total of finitely many nonzero R-modules which are finitely generated), show that C⊗_R(-) and Hom_R(C,-) are a Quillen adjoint pair. For X → Y a map in a model category M, give necessary and sufficient conditions for the induced Quillen functors between M_/X and M_/Y to be a Quillen equivalence. Investigate the two model structures on the category of maps in M and the two natural forgetful functors from this category to the category M itself; determine some Quillen adjoint pairs built from these.

March 25

Simplicial abelian groups and the Dold-Kan correspondence. Associated chain complexes. Subcomplexes of normalized chains and degenerate chains. Decomposition of each level n in a simplicial abelian group as a direct sum of isomorphic images of normalized chains, indexed by the surjections from [n] to other ordered sets. Sketch of proof that we get an equivalence of categories. Sketch of proof that the degenerate subcomplex is contractible, based on filtering by images of degeneracy maps and using chain contractions.
Suggested exercises: Complete the sketched proof that the degenerate subcomplex is always quasi-isomorphic to 0. Generalize the proof of the Dold-Kan correspondence to abelian categories, if you know what these are. Using this, describe an equivalence between nonnegatively graded cochain complexes and cosimplicial abelian groups (which are covariant functors from Δ to abelian groups).

March 28

More on simplicial abelian groups. Explicit description of normalized chains on the free abelian group on Δ[n]. Reverse functor in the Dold-Kan correspondence must be given by maps from these objects to the given chain complex. Homotopy groups. Proof that simplicial abelian groups always have fibrant underlying simplicial set, by translating lifting problem along standard adjunctions. Inclusion of a horn into a simplex gives rise, on normalized chains of the abelianization, to an acyclic cofibration. Translation of the definition of "homotopy groups" into the simplicial abelian context; it is directly related to the boundary map in the normalized chain complex.
If A is a simplicial abelian group, show that the group structure A x A → A induces a multiplication on the homotopy groups based at the identity element which coincides with the ordinary multiplication. If σ and τ are two elements in ∩ker(d_i), show that h gives a homotopy σ ~ τ in π_n if and only if h - s₀τ gives a homotopy between σ-τ and the trivial element. Use either of these to complete the proof that the Dold-Kan correspondence carries homotopy groups to homology groups. Take a general chain complex concentrated in degrees 0 and 1 and describe the associated simplicial abelian group it is taken to under the Dold-Kan correspondence. Describe the result if you take a chain complex not concentrated in nonnegative degrees, take the associated simplicial abelian group, and then apply normalized chains. Go see Joyal's lectures!

March 30

Singular homology. It factors: we take X, take its singular complex to remove pathology, "abelianize", and then convert from simplicial abelian groups to the more convenient category of chain complexes. Group objects in a general category C, expressed in terms of diagrams and in terms of lifts of the associated representable functor. Examples of abelianization: sets, groups, abelian groups, rings, topological spaces, based topological spaces. Abelianization converts cofiber sequences of topological spaces to short exact sequences of topological groups. Part of the Dold-Thom theorem: homotopy groups of the abelianization of a CW-complex X are the homology groups of X. Abelianization of (based) sphere Sⁿ gives an Eilenberg-Mac Lane space K(ℤ,n) - as a topological abelian group!
Suggested exercises: Determine the "abelianization" functor for nonunital rings. Show that abelianization of simplicial objects is levelwise abelianization. For an abelian group A, construct "Moore spaces" whose abelianizations are K(A,n). Describe a construction of the free ℤ/n-module on a topological space, and sketch an argument (assuming the Dold-Thom theorem) that the homotopy groups of the free ℤ/n module on a CW-complex X are the mod-n homology groups of X. Assuming that geometric realization of simplicial sets preserves finite products, show that it also preserves abelianization.

April 1

Derived functors. Left derived functors of additive functors on abelian categories using projective resolutions. Nonadditive functors don't preserve chain homotopies and chain homotopy equivalence. Dold-Kan correspondence and simplicial resolutions. Left derived functors for nonadditive functors are then applied levelwise. Analogy with abelianization. Example of the exterior square functor. Indications of further examples: derived group completion for topological monoids leads to algebraic K-theory, derived abelianization for topological groups leads to group homology (with a shift), derived abelianization for (augmented) simplicial rings is related to the cotangent complex.
Suggested exercises: Compute the value of the derived functors of the exterior square for finitely generated free groups and cyclic groups. Show that this derived functor 𝕃Λ takes A ⊕ B to 𝕃Λ(A) ⊕ A ⊗^𝕃 B ⊕ 𝕃Λ(B). Suppose that F → G → H are natural transformations of functors that take values in an abelian category such that, for any free object M, the resulting sequence 0 → F(M) → G(M) → H(M) →0 is exact; show that there is an induced long exact sequence of derived functors.

April 4

Unexpectedly long Q-and-A period on derived functors. Serre classes of abelian groups, or objects in an abelian category. Examples: finitely generated abelian groups, torsion groups, finite groups, finite groups of order only divisible by a fixed set of primes.
Suggested exercises: For a fixed R-module M, show that the set of modules N such that Tor^R(N,M) is zero in all degrees forms a Serre class. Determine which of the above classes have such an abelian group M.

April 6

C-epimorphisms, C-monomorphisms, and C-isomorphisms. Modules trapped in an exact sequence between objects of C are in C. Six-term kernel-cokernel exact sequence for a composite map and the consequences for C-theory. Statement of the C-theory five-lemma. Beginning of a crash course in spectral sequences. Exact couples, associated derived exact couple. Unrolled exact couples and the form of the "derived" exact couple (except I got the indices wrong, I'll correct that next time).
Suggested exercises: Prove the existence of the kernel-cokernel exact sequence for R-modules. For the r-fold derived exact couple, describe the maps in terms of the maps in the original exact couple. Given an unrolled exact couple, give direct definitions and proofs for the existence of "derived" unrolled exact couples that don't rely on taking infinite direct sums.

April 8

More on spectral sequences. Iterated "derived" exact couples. Example of an exact couple which starts with 0 and ends with an object we are computing: eventually the derived exact couple stops at a filtration of the object that we are computing. Internal gradings, bigraded spectral sequences. The Serre spectral sequence associated to a fibration with simply-connected base. Examples of S¹ → S³ → S² and the path-loop fibration ΩB → * → B.
Suggested exercises: Show that, given a chain complex C with a sequence of subcomplexes 0 < F⁰ < ... < Fⁿ = C, construct a spectral sequence starting with the homology groups of Fⁱ/F^i-1 and converging to the homology groups of C. Write down some small examples for yourself and compute the results! Compute the homology groups of ΩSⁿ for all n > 1.

April 11

Interaction between a Serre class C and spectral sequences. If a term in the spectral sequence is in C, so is the term in the same position on the next page. If the differentials out of and into a certain term come from objects in C, the term on the next page is C-isomorphic to the current one. If all entries on the final page are in C but one, the object being computed is C-isomorphic to that term. If all entries in columns below P are in C, the output in degree P is C-isomorphic to E²_P,0. If all entries in rows below Q are in C, the output in degree Q is C-isomorphic to E²_0,Q. For abelian groups, if A is finitely generated and B is in C, then A⊗B and Tor(A,B) are in C. Stronger result for saturated Serre classes. Application: For a fibration F → E → B with B simply-connected, if two of the spaces have finitely generated homology groups then so does the third. Spaces K(ℤ,n) all have finitely-generated homology groups.
Suggested exercises: Show that for a fibration as above, if two of the three spaces have homology groups which are all finite above degree 0 then so does the third. Similarly for torsion groups or groups which are finite and coprime to a set S of primes. Show that for such a fibration sequence, if B has finitely-generated homology groups and F has homology groups in a Serre class C, then the maps H_k(E) → H_k(B) are all C-isomorphisms. Similarly with the roles of B and F reversed. Show that if R is a Noetherian ring, then the class of finitely-generated R-modules is a Serre class. In this situation, if C is any Serre class of R-modules, M is a finitely-generated R-module, and N is in C, show that the Tor-groups Tor^R_k(M,N) are all in C for all k ≥ 0.

April 13

If A is a finitely generated abelian group, then an Eilenberg-Mac Lane space K(A,n) has finitely generated homology groups. Proof first for finite cyclic groups, then application of the Kunneth formula. Serre's result that for simply-connected spaces, finitely generated homology groups are equivalent to finitely generated homotopy groups. Counterexample in the non-simply-connected case of S¹ wedge S². Proof of Serre's theorem using Postnikov towers.
Suggested exercises: Using exercises from last time, show that if A is finite, then all the homology groups of K(A,m) in positive degrees are finite and annihilated by some power of |A|. Show that for a simply connected space X, the homotopy groups of X in degrees less than or equal to m are all finite and annihilated by powers of a fixed integer n if and only if the same is true for the homology groups. Find an element of π₂(S^1 v S^2) so that attaching a 3-cell along this map gives a new space so that the inclusion of S¹ is an isomorphism on homology groups but not an isomorphism on the second homotopy group.

April 15

Serre classes as applied to torsion groups. Notions of rational homology isomorphism and rational homotopy isomorphism (the latter only for simply-connected spaces). Commutative differential graded algebras. Ordinary cochain product is not strictly commutative, and Steenrod operations obstruct it being made so. Definition of the degree-k piecewise linear differential forms on a simplicial set. Examples.
Suggested exercises: Get a copy of Mosher and Tangora if you are interested in more about Steenrod operations and cup-i products. Give a careful definition of an obstruction to commutativity at the prime 3 using triple products and their natural chain homotopies. Compute the homology of the space K(ℚ,1) by expressing it as a union of smaller spaces.

April 18

More on the piecewise-linear rational de Rham complex of a simplicial set. Definition of the differential as exterior derivative of forms, making it into a chain complex. Wedge product makes it a commutative differential graded algebra. "Integration" map gives a map from this complex to the ordinary cochains with rational coefficients. Stokes' theorem implies that this is a chain map. Botched proof of the Poincare lemma, which (together with an inductive argument) shows that the integration map is always a quasi-isomorphism.
Suggested exercises: Repair the proof of the Poincare lemma. For a subcomplex of a simplicial set, define a relative PL de Rham complex that gives a long exact sequence. Show that this is compatible with the integration map. If A < K is a subcomplex so that the only nondegenerate simplices of K not in A are of a fixed dimension d, show that the relative PL de Rham complex is a direct product of copies of the complex for a d-simplex relative to its boundary. Use the Poincare lemma and this work to show that the PL de Rham complex is always quasi-isomorphic to the rational cochain complex.

April 20

Goal: We want a rational homotopy category and a homotopy category of commutative differential graded algebras, and the piecewise linear de Rham complex functor should give us a Quillen functor inducing an equivalence on "nice" objects. Fibrant objects should have homotopy and homology groups which are already rational vector spaces. Would like to understand K(ℚ,n) first. Construction of a K(ℚ,1) as a mapping telescope of multiplication-by-n maps on the circle. Computation of homotopy and homology groups. Suspension and the rational spheres Sⁿ_ℚ. Construction of "rational spaces" by attaching cells along maps from rational spheres; homology and homotopy are already rational vector spaces.
Suggested exercises: For a sequence of arbitrary maps of spaces X₀ → X₁ → ..., construct a mapping telescope and show that its homotopy and homology groups are the direct limits of the homotopy and homology groups in the system. Use this to construct spheres with an arbitrary family of primes inverted.

April 22

Homotopy theory for commutative DGAs. Weak equivalences are quasi-isomorphisms, fibrations are levelwise surjections. Cofibrations are generated by maps B → B[x] such that dx is in B; here a "polynomial" generator is either exterior or polynomial depending on whether it occurs in odd or even degree. Examples of computing homotopy classes of maps using cofibrant replacement.
Suggested exercises: Find a cofibrant replacement for ℚ[x,y]/(xy) when |x| and |y| are even.. Construct a (small) path object for a general commutative DGA and use it to characterize when two maps of CDGAs are homotopic in terms of "multiplicative chain homotopies".

April 25-29

Instructor is out of town.

May 2

Rational homotopy theory and the PL de Rham complex. Comparison between [X,Y] when Y is a K(ℚ,n); the PL de Rham complex of Y has a small, free DGA weakly equivalent to it. Such maps are equivalent to homotopy classes of maps between the piecewise-linear de Rham complexes. Indications for how to build a cofibrant replacement for the de Rham complex of a general Y using the Postnikov tower. Example of creating a small commutative DGA weakly equivalent to the wedge S³ v S³.
Suggested exercises: Assuming the equivalence between rational homotopy theory and the homotopy theory of commutative DGAs, find the rational homotopy classes of maps [Sⁿ,S^m] for all n, m > 1. Show that Y is weakly equivalent to a product of Eilenberg-Mac Lane spaces (with each homotopy group finitely generated) if and only if its cohomology ring is a free algebra. Find a commutative DGA weakly equivalent to the de Rham complex of S² v S² through dimension 5.

May 4-6

Lie bracket on homotopy groups, both in terms of the Whitehead product and the commutator on a topological group equivalent to the loop space. Induced (shifted) graded Lie algebra structure on rational homotopy groups. Interpretation of the homotopy of S³ v S³ as the free (shifted) graded Lie algebra on two generators (in general, wedge takes rational homotopy groups to the pushout of Lie algebras). Formula for the rank using the Poincare-Birkhoff-Witt theorem. Cofibrant replacements of DGAs for spheres, small exterior-algebra fibrant replacements. Applied to show that homotopy classes of maps from one CDGA to the CDGA of a sphere can be expressed in terms of "derived indecomposables": if I is the augmentation ideal, then one calculates the derived Hom-complex Hom(I/I²,ℚ).

...

Postmortem.

Contact information

By email: tlawson (at) math.umn.edu
By phone: 5-6802
By foot: Vincent Hall 323