Algebraic Topology 565 2016

February 21, 2016

The general format of the course is as in Fall quarter. We’ll use Hatcher Chapters 3-4, and we’ll start using selected part of Milnor’s Characteristic classes. One global notation change: From now on we fix a principal ideal domain R, and let H∗X denote H∗(X; R). This change extends in the obvious way to other notations: C∗X means RS·X (singular chains with coefficients in R), tensor products and Tor are over R, cellular homology is with coefficients in R, and so on. Course outline Note: Applications and exercises are still to appear. Some parts of the outline below won’t make sense yet, but still serve to give the general idea.

1. Homology of products. There are two steps to compute the homology of a product ∼ X × Y : (i) The Eilenberg-Zilber theorem that gives a chain equivalence C∗X ⊗ C∗Y = C∗(X × Y ), and (ii) the purely algebraic Kunneth theorem that expresses the homology of a tensor product of chain complexes in terms of tensor products and Tor’s of the homology of the individual complexes. I plan to only state part (ii) and leave the proof to the text. On the other hand, I’ll take a very different approach to (i), not found in Hatcher: the method of acyclic models. This more categorical approach is very slick and yields easy proofs of some technical theorems that are crucial for the cup product in cohomology later. There are explicit formulas for certain choices of the above chain equivalence and its inverse; in particular there is a very simple and handy formula for an inverse, known as the Alexander- Whitney map. In particular we will have an external product operation H∗X ⊗ H∗Y −→H∗(X × Y ) that is injective, is an isomorphism if one of H∗X, H∗Y is torsion-free (e.g. R is a field), and is associative and anti-commutative. If X is a topological group or even a “Hopf monoid” (a.k.a. “homotopy associative H-space”), this will yield a natural ring structure on H∗X. Moreover H∗X becomes a “Hopf algebra”, a property we will only discuss briefly here (but Hopf algebras and their topological manifestations would make a great topic for a presentation). Other related topics include homology of smash products (it behaves the same way, but using reduced homology) and cellular homology of products of CW-complexes. Note: The combination of Eilenberg-Zilber + Kunneth for C∗X, C∗Y is usually referred to simply as “the Kunneth theorem”. I’ll often follow this practice.

2. Cohomology. Cohomology is roughly the R-dual of homology, and is exactly the R- dual when the homology is R-free. Many of its basic properties follow directly from the

1 corresponding properties of the singular chain complex and homology; the proofs of these will be left to you (e.g. homotopy invariance, excision). The big diﬀerence is simply that cohomology is contravariant. This causes trouble in certain situations. For example, taking duals doesn’t commute with tensor products, which complicates the cohomology Kunneth theorem. Also, a direct system of spaces gives rise to an inverse system in cohomology, which can be problematic because in contrast to direct limits, inverse limits need not be exact. We’ll deal with these issues as they arise. On the other hand, the contravariance has the huge advantage that it yields a ring structure. The ring structure discussed in (1) only applies to Hopf monoids, and as we’ll see it is very rare for a space to admit the structure of Hopf monoid. Cohomology always has a ring structure, for the following simple reason: In both homology and cohomology we have an external product operation as described above. Since cohomology is contravariant, we can pull back the external product for X × X along the diagonal ∆ : X−→X × X, thereby obtaining a functorial anti-commutative ring structure on H∗X. This extra structure is very powerful; for instance, homotopy-equivalent spaces must have isomorphic cohomology rings. The cup product generalizes to a relative cup product

H∗(X,A) ⊗ H∗(Y,B)−→H∗(X × Y,A × Y ∪ X × B).

By a kind of formal adjunction we’ll also get the “cap product” that makes H∗X a module over H∗X. In order to make use of this ring structure, obviously we have to compute it; this is where life begins to get diﬃcult. There are two basic ways to at least get a foothold: (i) Poincar´eduality; and (ii) the Thom isomorphism and Gysin sequence. Then one can use the Kunneth theorem to get further computations. Our computational power will be enhanced a thousand-fold once we have spectral sequences, to be studied in Spring quarter. But to get started we’ll focus on (i), which is of great importance in its own right.

3. Poincaréduality. This comes in several increasingly general flavors. We’ll focus on the basic case first, namely the homology and cohomology of compact, boundaryless manifolds M. If dim M = n, the duality says that subject to a certain orientability condition, there k ∼ are isomorphisms H M = Hn−kM. This already puts restrictions on the homology of M, but the specific way in which the isomorphism comes about yields much more information: There is a “fundamental class” [M] ∈ HnM, and the isomorphism is given by cap product λ 7→ λ ∩ M. Since the cap product is defined in terms of the cup product, this will allow us to compute the ring structure in cohomology for real and complex projective spaces. Even if we are only interested in the compact case, we will be forced to consider non- compact manifolds as well. The point is that for our compact M we’d like to prove Poincaré duality by some kind of induction over a covering by charts, perhaps using the Mayer-Vietoris sequence, but the charts aren’t compact. The solution is to prove a more general theorem for non-compact manifolds, in which cohomology has to be replaced by cohomology with compact supports. Although efficient, the above approach to Poincaréduality is dry and uninspiring. Rest assured that we’ll bring it all back to the beautiful world of smooth manifolds and geom- etry, in particular showing how the duality is reflected in intersection theory (transversal

2 intersections of smooth embedded submanifolds). But for some of this we’ll need the next topic: 4. The Thom isomorphism, the Euler class and the Gysin sequence. The reference for this part is Milnor’s text. Let X be a compact Hausdorff space and let E be an n- dimensional vector bundle over X. We assume that either E is oriented or the coefficient ring R = F2. The Thom space T h(E) of E is just the one-point compactification of E. There k is a canonical Thom class uE ∈ H T h(E) such that (relative) cup product with uE gives an isomorphism H∗X−→H˜ ∗+nT h(E). One should think of this as a generalization of the suspension isomorphism, which corresponds to the case of a trivial bundle. The zero-section defines an inclusion i : X−→T h(E), and the Euler class is defined by ∗ e(E) = i uE. If the bundle is trivial then e(E) = 0, so this will give us (finally!) a sys- tematic way to show that certain bundles are non-trivial. In fact if the bundle even has a non-vanishing section then e(E) = 0. The Euler class is the first example of a characteristic class for vector bundles (to be studied in generality in spring). Euler’s name is attached to it because when X is a smooth compact manifold and τM is the tangent bundle, ∗ ∗ e(τM ) = χ(M)[M] , where χ is the Euler characteristic and [M] is the fundamental class in cohomology. Using cup product with the Euler class, we’ll get a certain long exact sequence called the Gysin sequence that will help us compute a few more cohomology rings. 5. Homotopy theory. 1. Based versus unbased homotopy classes, action of the fundamental groupoid on homotopy classes of maps (including homotopy groups). This material is scattered through Ch. 4 of Hatcher. It will be nice to have it settled at the beginning. 2. Compression lemma, weak equivalences, Whitehead’s theorem, cellular approximation. At this point we can easily construct K(G, 1)-spaces (i.e. spaces with fundamental group G and all higher homotopy groups trivial). 3. Then there is the fantastically useful homology version of Whitehead’s theorem. This says that if the CW-complexes involved are simply-connected (a hypothesis that can be weakened but not eliminated), then a map inducing an isomorphism on homology also in- duces an isomorphism on homotopy groups and hence is a homotopy equivalence by the basic Whitehead theorem. One approach to this, found in Hatcher, involves relative homotopy groups and the relative Hurewicz theorem. The downside of relative homotopy groups is that they are a bit messy; for example they are only groups for n ≥ 2 and only abelian for n ≥ 3. Another approach involves fibrations and the Serre spectral sequence. I prefer the latter approach because the payoff for time invested is a thousand times higher than what you get from relative homotopy groups. The downside is that we will have to postpone the proof (of homology Whitehead) until spring. 4. CW-approximation (this says that every space is weakly equivalent to a CW-complex). 5. Hurewicz theorem. I’ll avoid “homotopy excision” altogether, and give a different proof of the basic Hurewicz theorem (I’ll omit the relative Hurewicz theorem). The starting n ∼ point is the Brouwer-Hopf theorem πnS = Z, for which I’ll give the interesting classical proof. 6. Eilenberg-Maclane spaces (spaces with just one non-vanishing homotopy group). It turns out that these represent singular cohomology, at least on CW-complexes.