Toward Spectral Sequences 4: Spectra and Spectral Sequences

Topic
Spectra and representability of homology functors, spectral sequences and some example calculations. Part of the series outlined at Towards Spectral Sequences.

Transcript
 &lt;thermoplyae&gt; so i guess we'll get started? ChanServ has changed the topic to: Seminar in progress, type '!' and wait to be called if you have a                                  question. &lt;thermoplyae&gt; so last time we talked about CW-complexes and their homotopy properties, where we saw a             lot of long and boring theorems that gave us low-dim'l homotopy information and then failed for higher dimensions &lt;thermoplyae&gt; and we used that as motivation to consider less topological constructions, called homology theories, that behaved like homotopy except they didn't fail so miserably in             higher dimensions &lt;thermoplyae&gt; and then i abruptly left :) &lt;thermoplyae&gt; so there are a couple of things i need to talk about in the general context of homology             and then we're going to look at spectra which will hopefully give you some kind of              grasp on what homology &quot;really&quot; is, and then we'll begin doing spectral sequences &lt;thermoplyae&gt; so, to start, what i gave as the excision and exactness properties last time, can be              used via clever renaming of spaces to come up with a new l.e.s., called the              mayer-vietoris sequence &lt;thermoplyae&gt; given a CW complex X and two covering subcomplexes A, B (so a triad (X, A, B)), there's              a l.e.s of the form ... -&gt; H_n (A cap B) -&gt; H_n A (+) H_n B -&gt; H_n X -&gt; ... -&gt; 0 &lt;thermoplyae&gt; and this is pretty computationally useful.  for instance, we can use it to calculate the homology of a sphere with a slightly unintelligent cell structure on it (the             inductive one i gave, with two cells in each dimension) &lt;thermoplyae&gt; under that structure, the north hemisphere and the south hemisphere are subcomplexes that intersect at the equator. given that the homology groups of S^1 are (Z, Z, 0, 0, ...), these fit into an exact sequence 0 -&gt; H_3 X -&gt; 0 -&gt; 0 -&gt; H_2 X -&gt; Z             -&gt; 0 -&gt; H_1 X -&gt; Z -&gt; Z (+) Z -&gt; Z -&gt; 0, and then a mix of homological algebraic inference and some small amount of knowledge of what those maps actually are (which is             indicated in the proof of the sequence's exactness) yields H_* S^2 = (Z, 0, Z, 0, 0, ...) &lt;thermoplyae&gt; and, actually, a similar argument will work for any suspended space. recalling that SX = CX glued-along-X-to CX, we can cover SX with two (contractible) copies of CX and compute SX's homology in terms of CX &lt;thermoplyae&gt; which turns out to be exactly X's homology, shifted up by one dimension (exempting H_0             -- to deal with this formally, this is proven in the context of so-called &quot;reduced&quot;              homology, where H_0 = 0 for path-connected spaces) &lt;thermoplyae&gt; a similar sequence can also be constructed for cohomology, with the arrows running the other way &lt;thermoplyae&gt; now, suppose for a moment that our cohomology functor is actually representable, i.e.             H^n X = Toph^*(X, E_n) for some cell complex E_n &lt;thermoplyae&gt; by the adjointedness of the suspension and loop functors, this means that the sequence of spaces E_n must satisfy E_n = Loops E_{n+1}, which in no small part motivates the construction of spectra: &lt;thermoplyae&gt; a spectrum is a sequence of spaces E_n that satisfy SE_n includes into E_{n+1} as a             subcomplex &lt;thermoplyae&gt; a subspectrum of E_n is a family of subcomplexes F_n, one for each n, that is also a             spectrum &lt;thermoplyae&gt; we, of course, want to turn this into a category &lt;thermoplyae&gt; our first guess at how to make this happen is to associate a map f: E -&gt; F with a             sequence of maps f_n: E_n -&gt; F_n such that E_n -f_n-&gt; F_n --&gt; F_{n+1} is the same as              E_n --&gt; E_{n+1} -f_{n+1}-&gt; F_{n+1} is the same arrow &lt;thermoplyae&gt; which is to say that as we go farther up in the sequence of spaces, the cells belonging to earlier spaces are still acted upon in the same way &lt;thermoplyae&gt; for technical reasons, this is actually too strong. we weaken this to mean instead that a map of spectra is a partial function satisfying these properties such that the map is &quot;eventually&quot; defined on every cell &lt;thermoplyae&gt; more formally, a subspectrum is said to be cofinal if every cell in E_n is eventually contained in F_m, m &gt;= n, after some finite number of suspensions &lt;thermoplyae&gt; and a map of spectra E -&gt; F is a function in the first sense from G -&gt; F, where G is             cofinal in E &lt;thermoplyae&gt; we can construct a variety of spectra straight away. for instance, given a CW complex X, the sequence X, SX, SSX, SSSX, ... is obviously a spectrum &lt;thermoplyae&gt; given a spectrum E and a CW-complex X, we can defined (E smash X)_n to be E_n smash X.             associativity of the smash product shows that this is also a spectrum &lt;thermoplyae&gt; given our notion of a map being &quot;eventually&quot; defined, we're also free to shift our spectra around &lt;thermoplyae&gt; we can suspend a spectrum E by defining (SE)_n = E_{n+1}, shifting it down. we can also /desuspend/ a spectrum by definining (S^-1 E)_n = E_{n-1} and (S^-1 E)_0 = pt &lt;thermoplyae&gt; in fact, S S^-1 E = S^-1 S E = E, where it's worth emphasizing that &quot;=&quot; here is             equivalence of spectra &lt;thermoplyae&gt; almost straight from these definitions, we can demonstrate that, given a spectrum E, we             can associate the following homology and cohomology theories to it: &lt;thermoplyae&gt; E_n X = Spectra(S^n Sphere, E smash X), E^n = Spectra(X, S^n E) = Spectra(S^-n Sphere             smash X, E) &lt;thermoplyae&gt; let's talk about the homology for a moment first. by abuse of notation, we can think of this as pi_n(E smash X), where pi_n here means &quot;the limit of pi_{n+m}(E_m smash X)&quot; &lt;thermoplyae&gt; talking E_m to be the sphere spectrum, this is exactly the stable homotopy groups of X             discussed earlier (remember?  we calculated pi_3 S^2?) &lt;thermoplyae&gt; and so in some sense, homology /is/ a subfield of stable homotopy -- these theories, anyway, but we'll turn to that in a moment &lt;thermoplyae&gt; second, the definition for the cohomology theory associated to E may not be immediately transparent, but if we require that the map E_n -&gt; Loops E_{n+1} adjoint to SE_n -&gt; E_{n+1} is a homotopy equivalence, we find that this can be rewritten as E^n X = Toph^*(X, E_n), as per our initial motivation &lt;thermoplyae&gt; a spectrum satisfying this is called an Omega-spectrum &lt;thermoplyae&gt; now, given a homology theory that satisfies 1) weak homotopy equivalence is sent to             isomorphisms of homology groups 2) wedge sums of spaces are sent to coproducts of              homology groups 3) a slightly weaker form of the mayer-vietoris sequence, we can              construct a spectrum inductively that represents the homology functor &lt;thermoplyae&gt; and, for the most part, our interesting homology theories /do/ satisfy these things.              in particular, cellular / ordinary homology certainly does &lt;thermoplyae&gt; the spectrum that represents it turns out to be the spectrum of Eilenberg-Maclane              spaces.  H^n(X; G) = Toph^*(X, K(G, n)) &lt;thermoplyae&gt; now, we digress for a moment from spectra, talking about one last central point of              homology: &lt;thermoplyae&gt; it's fairly easy but lengthy to prove using straight homological algebra that H_n(X * Y) = (+)_{i + j = n} H_i(X) (x) H_j(Y) in the context of ordinary homology &lt;thermoplyae&gt; (up to some business with the Tor functor, of course) &lt;thermoplyae&gt; this formula is also true in other homology theories for other reasons, and is in             general called the 'Kunneth isomorphism' &lt;thermoplyae&gt; in particular, this means that there's a map (H_n X) * (H_m X) -&gt; H_{n+m} (X * X) &lt;thermoplyae&gt; so a sort of external product of homology classes &lt;thermoplyae&gt; a similar result also holds in cohomology, but there because we have the topological map X -&gt; X * X given by the diagonal, we can turn this into an internal cohomology product by postcomposing with the arrow H^{n+m} (X * X) -&gt; H^{n+m} X &lt;thermoplyae&gt; this product is called the cup product. it can be shown to be distributive and graded- commutative (i.e. x cup y = (-1)^{nm} y cup x, where x lives in H^n X and y lives in             H^m X) &lt;thermoplyae&gt; and so it turns ordinary cohomology into a ring (!) &lt;thermoplyae&gt; in the context of unreduced cohomology, where H^0 is a copy of the underlying ring, this can be seen as a unital algebra as well. something like H^* S^n can then be             expressed as &quot;the exterior algebra on one class of rank n&quot; &lt;thermoplyae&gt; and, as further application, this product [X, K(Z, n)] * [X, K(Z, m)] -&gt; [X, K(Z, n+m)] can be turned into a map K(Z, n) * K(Z, m) -&gt; K(Z, n + m) using Yoneda's lemma &lt;thermoplyae&gt; a spectrum with such a product on it is called a &quot;ring spectrum&quot;, because it turns the associated cohomology theory into a ring-valued functor &lt;thermoplyae&gt; now, i think that's plenty of homology. any questions before i move on to exact couples and spectral sequences? &lt;thermoplyae&gt; okay then &lt;thermoplyae&gt; so finally i think we've pushed enough symbols around to make spectral sequences at             least seem approachable &lt;thermoplyae&gt; again, usually the off-putting thing with spectral sequences is the extraordinary amount of indices (and thus the extraordinary amount of information), which makes the whole process quite confusing &lt;thermoplyae&gt; to start, an exact couple is a pair of modules D, E and a triple of maps i: D -&gt; D,             j: D -&gt; E, and k: E -&gt; D that's exact &lt;thermoplyae&gt; E has a differential on it, namely jk, since (jk)(jk) = j(kj)k = j0k = 0, and so given an exact couple we can construct a derived exact couple by replacing E with E's             homology wrt. this differential, D with im i \subset D &lt;thermoplyae&gt; i is replaced by its restriction to the new D, j is replaced with the map i(x) |-&gt; [j(x)], where [] denotes the homology class in E, and k with [y] |-&gt; k(y) &lt;thermoplyae&gt; it is six easy exercises in arrow chasing to show that this is again an exact couple &lt;thermoplyae&gt; now, let's construct an exact couple. take a space X and filter it into a bunch of             spaces X^n (i.e. X^n is contained in X^{n+1} and their colimit is X) &lt;thermoplyae&gt; using the exactness property of homology, X^{p-1} -&gt; X^p -&gt; X^p / X^{p-1} induces an              l.e.s in homology &lt;thermoplyae&gt; we take D and E to be bigraded groups with components given by D_{pq} = h_{p+q} X^p and E_{pq} = h_{pq}(X^p / X^{p-1}) &lt;thermoplyae&gt; i is defined componentwise, taking D_{pq} to D_{p+1, q-1} (corresponding to the first             map in the s.e.s. in Toph given above) &lt;thermoplyae&gt; j takes D_{pq} to E_{pq}, corresponding to the second map &lt;thermoplyae&gt; k takes E_{pq} to D_{p-1, q}, corresponding to the boundary map induced in the l.e.s. &lt;thermoplyae&gt; because E_{pq} is zero for p or q negative, it is easy to show that E stabilizes in the limit (i.e. each component of E stabilizes after we take homology finitely many times) &lt;thermoplyae&gt; the basic idea there is that the differential will each time have larger and larger degree, and eventually the sequences we'll be taking homology wrt. will look like 0 -&gt; E_{pq} -&gt; 0, and so E's homology there will just be E again &lt;thermoplyae&gt; it's also easy to show that E will in fact converge to h_*(X) (!!) &lt;thermoplyae&gt; in particular, if we pick X to be a cell complex filtered by its cell skeleta, the first derived homology will be cellular (!!!), i.e. E^2_{pq} = H_p(x; h_q pt), where h             is any nice homology theory and H is cellular/ordinary homology &lt;thermoplyae&gt; oh, we'll have to rewind in a moment, that there haven't been any flags thrown up is             not a good sign you guys are paying attention :P &lt;thermoplyae&gt; but anyway, this spectral sequence is called the atiyah-hirzebruch spectral sequence, and it states in words that given the ordinary homology of a space and the coefficient group of any other homology theory, we can compute the homology of the space in terms of that other homology theory &lt;thermoplyae&gt; now, the trouble i skipped over, is that i said it converged to h_* X without saying what this means. E^\infty_{pq} is still bigraded! &lt;thermoplyae&gt; it turns out that to finish this process off, h_n X is equal to an extension of the sequence of groups E^\infty_{pq} with p+q = n &lt;thermoplyae&gt; so if all but one of these groups vanish, h_n X will be equal to it. if there are two groups, it could be a direct product or a semidirect product. if there are more groups, it gets steadily more complicated &lt;thermoplyae&gt; if you're proving all this, it's easy to get excited at this point. you can compute so             much from so little information! &lt;thermoplyae&gt; we must not forget the incredible amount of work and unknowns here &lt;thermoplyae&gt; the differential on this structure is quite often nigh impossible to compute, as is the extension problem at the end &lt;thermoplyae&gt; there are nice cases in which it is possible -- for instance, if X has homology concentrated in even degrees and h_* pt is also concentrated in even degrees (or some             condition like this, i'm not thinking about it too hard), the spectral sequence will collapse immediately &lt;thermoplyae&gt; before we discard the notion of an exact couple, we ought to discuss another example: the homotopy exact couple &lt;thermoplyae&gt; set E_{pq} = pi_{p+q}(X^p, X^{p-1}) and D_{pq} = pi_{p+q}(X^p). the differentials, again, fall out of taking the relative homotopy l.e.s. of the pair (X^p, X^{p-1}) &lt;thermoplyae&gt; the hurewicz theorem (one of the things i told you to look up at the end of the last             lecture :P) gives that E_{pq} = 0 for q &lt; 0, which is also trivially true for p &lt; 0, and so this sequence will also stabilize for the same reason &lt;thermoplyae&gt; in fact, it'll converge to pi_* X, again after solving the same extension problem &lt;thermoplyae&gt; again restricting to the case when X is filtered by the cell skeleta, E^1 is isomorphic to chains on X (another construction of ordinary homology, called singular homology,             where in place of the cells themselves we take all possible mappings D^n -&gt; X and              construct differentials by restricting to the boundary) &lt;thermoplyae&gt; and the differential, in fact, is the same as singular homology's differential, so E^2 is H_* X. D^2 contains pi_n X because it's formed by im pi_n X^n -&gt; pi_n X^{n+1}, which is onto &lt;thermoplyae&gt; (and stable after that) &lt;thermoplyae&gt; a piece of the exact sequence belonging to the first derived couple looks like: &lt;thermoplyae&gt; ... -&gt; E^1_{p+1, 0} -&gt; D^2_{p, 0} -&gt; D^2_{p+1, -1} -&gt; E^2_{p, 0} -&gt; ... &lt;thermoplyae&gt; whoops, that should be E^2 &lt;thermoplyae&gt; replacing these groups with what i described above, we have ... -&gt; H_{p+1} X -&gt; Gamma_p X -&gt; pi_p X -&gt; H_p X -&gt; ..., where Gamma_p X = im j : pi_p X^{p-1} -&gt; pi_p X^p -&gt; D^2_{p, 0} &lt;thermoplyae&gt; a piece of this sequence is used to prove the hurewicz theorem, actually, and so it             shouldn't be a huge surprise it turns up here, since we used it in the construction &lt;thermoplyae&gt; and, by corollary of exactness, pi_{p+1} X -&gt; H_{p+1} X is onto when X is             (p-1)-connected &lt;thermoplyae&gt; the Hurewicz theorem itself states that pi_p X -&gt; H_p X is iso. when X is             (p-1)-connected, if you've forgotten &lt;thermoplyae&gt; so this is still very abstract, we haven't really computed anything, despite my usual penchant for avoiding proofs and replacing them with example computations. i'm going to sketch the proof of the serre spectral sequence, and then we'll do some honest computations &lt;thermoplyae&gt; so the first thing to notice is that the only reason we have D in our exact couple is             to keep track of where our differential is coming from. the majority of the information is actually contained in E('s homology), and given a formula for the differential on E at each step of the sequence (typically called each 'page' of the             sequence) we can do away with D &lt;thermoplyae&gt; and that's how we're going to approach the SSS. Serre's claim is that given a &quot;nice&quot; fibration E -&gt; B, there's a spectral sequence with E^2_{pq} given by H_p(B; h_q F) that converges to h_* E &lt;thermoplyae&gt; i'll roughly (/roughly/) sketch the central components of the existence proof: &lt;thermoplyae&gt; first, given a fibration E -&gt; B and a map f: X -&gt; B, we can construct the fibration f^*E -&gt; X given by taking a pullback (you remember what those are, right? ;) ) &lt;thermoplyae&gt; in fact, the fiber of the pullback fibration will be the same as the original fiber,             and so in some sense we're just taking the fiber over f(x) and placing it over x &lt;thermoplyae&gt; we'll say a map of fibrations (E -&gt; B) -&gt; (E' -&gt; B') is a pair of functions E -&gt; E',              B -&gt; B' such that E -&gt; E' -&gt; B' is the same as E -&gt; B -&gt; B' &lt;thermoplyae&gt; and a fiber homotopy is a homotopy X * I -&gt; E such that X * I -&gt; E -&gt; B is stationary,              i.e. the homotopy only moves things around in their fibers &lt;thermoplyae&gt; given a homotopy between f0, f1: X -&gt; B, there's a fiber map isomorphism between              f0^* E and f1^* E.  in addition, if two such homotopies /themselves/ are homotopic,              there's a fiber homotopy relating the two equivalences &lt;thermoplyae&gt; now, take path in B. this can be viewed as a homotopy of two maps pt -&gt; B, one for each endpoint, say x0 and x1. this gives rise to a fiber map E_{x0} -&gt; E_{x1} &lt;thermoplyae&gt; this, in turn, gives an action of pi_1 B on h_* F, since pi_1 B can be seen as             endomorphisms on F &lt;thermoplyae&gt; maps E_{x0} -&gt; E_{x1} induced in this way i'll call admissible &lt;thermoplyae&gt; a homotopy trivialization of a fibration E -&gt; B is an equivalence with B x F -&gt; B with an admissible fiber map between (B x F)_{b0} and E_{b0}. trivially, a contractible B             has a homotopy trivialization &lt;thermoplyae&gt; now, finally, we can get somewhere. take the relative CW-pair f: (D^p, S^{p-1}), and let k be the homology generator of H_p D^p &lt;thermoplyae&gt; whoops, a map relative CW-pairs f(D^p, S^{p-1}) -&gt; (X, Y), where (X, Y) satisfies Y             subset X subset B &lt;thermoplyae&gt; then there's a map h_q F -&gt; h_{p+q} ((D^p, S^{p-1}) * F) given by the external product on homology (i.e. kunneth map) &lt;thermoplyae&gt; D^p is contractible, so it admits a trivialization, so we can follow this map into h_{p+q}(f^* E, f^*E restricted to S^{p-1}) &lt;thermoplyae&gt; then we can follow f's induced map in homology into h_{p+q} (E restricted to X, E             restricted to Y) &lt;thermoplyae&gt; and so in this way, we can start approximating h_* E with cellular maps &lt;thermoplyae&gt; two homotopic f (say f0 and f1) yield equivalent maps constructed above up to a product by an admissible map phi. of course, if pi_1 B acts trivially on h_* F, then we don't             have to worry about this, and homotopic f yield the same induced map in homology, and this is the central condition for applying the SSS &lt;thermoplyae&gt; from here, it really is a cell-by-cell argument, culminating in the description E^2_{pq} = H_p(B; h_q F), converging to h_* E when h is a homology theory with products, weak homotopy equivalence, and wedge sum, and E -&gt; B is a fibration with trivial pi_1 B action &lt;thermoplyae&gt; there's a cohomology version of all this as well; H^p(B; h^q F) converges to h^* E &lt;thermoplyae&gt; the nice thing about the cohomology sequence is that it behaves well wrt the cup product introduced earlier &lt;thermoplyae&gt; given two fibrations E -&gt; B and E' -&gt; B', the map E x E' -&gt; B x B' is a fibration, and there's a map E_{pq}^r(1) (x) E_{st}^r(2) -&gt; E_{p+s,q+t}^r(3), where E(n) is the spectral sequence page belonging to 1) the first factor 2) the second factor 3) their             product &lt;thermoplyae&gt; the differential on this product page is given by d^r(3): (a (x) b) |-&gt;              d^r(1) a (x) b + (-1)^{|a|} a (x) d^r(2) b &lt;thermoplyae&gt; i.e. it is a derivation &lt;thermoplyae&gt; and so let's try to actually do something with these awfully complicated things &lt;thermoplyae&gt; let's do something easy to start.  we'll take the pathspace fibration PS^3 -&gt; S^3 with              fiber Loops S^3 = S^2, and we'll give ourselves that H^* S^2 = Lambda[x_2], the              exterior algebra on one class of degree 2 &lt;thermoplyae&gt; of course, the pathspace is contractible, and so its homology will vanish outside of              H_0 &lt;thermoplyae&gt; the goal here is to compute the cohomology of S^3, the only unknown out of the three spaces &lt;thermoplyae&gt; starting by filling things in, E^2_{pq} = H^p(B; H^q F). in particular, E_{pq} = H^0(B; H^q F) = H^q F, and so we know one whole column of groups &lt;thermoplyae&gt; actually, several rows too; since H^q(S^2) vanishes for q other than 0 and 2, rows other than 0 and 2 will be filled with zeros &lt;thermoplyae&gt; we know that we want this spectral sequence to end up with nothing on the page. in the second page, the differential d^2 runs from E_{pq} -&gt; E_{p+2, q-1}. our only interest at this point is how it acts on the generator in E_{02}, and because its codomain is             the trivial group it must do nothing &lt;thermoplyae&gt; taking homology then leaves everything we know about on this page alone &lt;thermoplyae&gt; we then try the next page, where the arrow now takes E_{02} to E_{30}. this kind of             differential, called a transgression, which runs from one 'edge' of the spectral sequence page to the other, is important because it means if we don't cause homology to             destroy groups now, they'll persist forever &lt;thermoplyae&gt; we need the group in E^3_{02} to vanish, and so it must be attached to E^3_{30} by an             isomorphism, which means E^4{02} will vanish &lt;thermoplyae&gt; since we get there by taking homology wrt. this differential

''Post-seminar observation: No, that's not it. I got confused and thought &quot;SS^2&quot; when I should have been thinking &quot;Loops S^2&quot;. This is the beginning of the H^* Loops S^2 calculation, but you need to continue from here by using the cup product on the two generators of positive degree, and then continue the cancellation process from there. Apologies.''

 &lt;thermoplyae&gt; and that's it. there's nothing else that could have happened, and the sequence collapses, E^\infty_{pq} is zero except when p = q = 0, and there it's Z, which is the homology of PS^2 &lt;thermoplyae&gt; what information did we get out of this? well, if we look at the row E_{p0}, we see the cohomology of S^3 with coefficients in Z, which is exactly Lambda[x_3] &lt;thermoplyae&gt; let's do something more complicated. i claim K(Z, 1) = S^1, which we proved using fibrations a couple lectures ago &lt;thermoplyae&gt; the pathspace fibration over K(Z, 2) has the form K(Z, 1) -&gt; pt -&gt; K(Z, 2), which we             can then fit into the SSS machinery &lt;thermoplyae&gt; anyway, we have E^2_{q0} = Z if q = 0 or 1 and vanishes for higher q. again, we need E^2_{10} to vanish, and the page 2 differential is transgressive, so it must hit a             Z-class in E^2_{02} &lt;thermoplyae&gt; their product is a class in E^2_{12}, say e_1 * x_2, and we can find its page 2 derivative using the product rule associated to d_2 &lt;thermoplyae&gt; d_2(e_1 * x_2) = d_2(e_1) * x_2 - e_1 d_2(x_2) = x_2^2, since d_2 x_2 = 0 &lt;thermoplyae&gt; we're already aware of the class x_2^2, since it comes from squaring a class we             already have, and this means that the differential E_{12} -&gt; E_{04} sends generators to              generators and is again an isomorphism &lt;thermoplyae&gt; this holds inductively as we go up p, each time yielding a generator in E^2_{2p, q}, each time equal to x_2^p &lt;thermoplyae&gt; and since the sequence collapses after that (since everything vanishes), there's             nothing left to do except extract the information about H^* K(Z, 2) &lt;thermoplyae&gt; which we find is a power series ring on one variable x_2 &lt;thermoplyae&gt; some easy homotopy theory can be used to show that K(Z, 2) = CP^\infty, and so now we             know H^* CP^\infty &lt;thermoplyae&gt; let's see if we can do one last example, something to show that this is again useful for more than just cohomology &lt;thermoplyae&gt; (but give me a moment to make sure i'm doing this right, i didn't plan much of this             talk :/ ) &lt;thermoplyae&gt; okay, i think i've got this right, someone correct me if i'm not :) &lt;thermoplyae&gt; let's take S^3's 3-connected cover, denoted S^3&lt;4&gt; &lt;thermoplyae&gt; this sits in a fibration K(Z, 2) -&gt; S^3&lt;4&gt; -&gt; S^3, as discussed last time in the             context of postnikov systems &lt;thermoplyae&gt; (i'm not sure that 2 is right, which is mostly what i'm hoping for confirmation on) &lt;thermoplyae&gt; now, we know the homology of S^3 and the homology of K(Z, 2) (as of a few seconds ago! :) ), and we want to get at the homology of S^3&lt;4&gt; because &lt;thermoplyae&gt; a) it sounds fun &lt;thermoplyae&gt; b) S^3&lt;4&gt; is 3-connected, which means its H^4 will be isomorphic to its pi_4 which is             isomorphic to S^3's pi_4 which is something we don't yet know &lt;thermoplyae&gt; so, to start, we know we'll have a class in E^2_00. recalling that E^2_pq = H^p(B ; H^q F), we'll also have classes in E^2_pq where q is even and p = 0 or 3. everywhere else, they ought to vanish &lt;thermoplyae&gt; we know that this will converge to S^3&lt;4&gt;, which is 3 connected, which means everything (except E^2_00) in the bottom 3 diagonals will have to vanish &lt;thermoplyae&gt; we can get to work on this beginning on the 3rd page, sending the generator of E^3_02 to the generator of E^3_30 &lt;thermoplyae&gt; then we're curious about what will happen to the other classes on the 3rd page. if             x_2 generated E^3_02, where will x_2^2 go? &lt;thermoplyae&gt; applying the derivation rule, d_3 x_2^2 = d_3 x_2 * x_2 + x_2 d_3 x^2 = e_3 x_2 + x_2 e_3. cup product is graded-commutative, and so this is 2 x_2 e_3, twice the generator of E^3_32 &lt;thermoplyae&gt; and so the homology there (i.e. E^4_32) will be Z / 2Z &lt;thermoplyae&gt; similarly, E^4_34 will be Z / 3Z and E^4_36 will be Z/4Z (up to me fucking up my             indices) &lt;thermoplyae&gt; and there the sequence stabilizes. for the first time, we might have to deal with that pesky extension problem &lt;thermoplyae&gt; but because each class in the left tower sits in a total even degree and each class in             the right tower sits in a total odd degree, nope, we got lucky, we don't have to &lt;thermoplyae&gt; and the correct way to express the cohomology's ring structure is as an exterior generator tensored with a divided power generator (again, up to me messing up a detail,             but i think i'm doing okay) &lt;thermoplyae&gt; of course, we only really care about H_4, which is Z_2, induced by E^4_04 &lt;thermoplyae&gt; and so by Hurewicz, pi_4 S^3 = Z_2 &lt;thermoplyae&gt; using some luck and a lot of backbreaking work, we can compute pi_5 S^3 in a similar way &lt;thermoplyae&gt; it turns out there's another spectral sequence, called the Adams spectral sequence, this is more suited to this work &lt;thermoplyae&gt; but we won't be going into it &lt;thermoplyae&gt; i think that's plenty; i've got two hours. &lt;thermoplyae&gt; i suggest that you all try to use this information to calculate H^* K(Z, 3), though perhaps only with Z_2-coefficients &lt;thermoplyae&gt; and given that H^* BU is a power series ring on countably many generators (one for each             even degree), maybe calculate H^* BU&lt;2k&gt; for k = 2, 3 &lt;thermoplyae&gt; i'll be in the channel all week, i can verify answers and regale you with related stories :) &lt;thermoplyae&gt; i'm going to walk over to the lab, but i've opened a screen session to catch any             questions; i'll be back online in a few to answer them &lt;thermoplyae&gt; and, of course, to accept your accolades &lt;thermoplyae&gt; bbiab