Toward Spectral Sequences 2: Homotopy and Fibrations

Topic
Homotopy and Fibrations. Part of the series outlined at Towards Spectral Sequences

Seminar
 [3:00pm] thermo: so it's 3:00 here, i guess we'll get started [3:01pm] kommodore: yeah... [3:01pm] Topic changed to &quot;Seminar in progress, type ! and wait to be called if you have a question.&quot; by chanserv. [3:01pm] thermoplyae: okay, so here's the second lengthy talk on algebraic topology [3:02pm] thermoplyae: last time we talked about the categories Top, Top*, and Toph, along with what what it meant for two functions to be homotopic and two spaces to be                     homotopically equivalent [3:03pm] thermoplyae: the plan today is to get as far as i can in the construction of (relative) homotopy groups, fibrations, cell complexes, and a bunch of homotopic facts about cell complexes [3:04pm] thermoplyae: the first thing to do is define a couple of functors that we'll see no end of: the (reduced) suspension and the (reduced) cone [3:04pm] thermoplyae: the (reduced) because sometimes other people call them reduced, but i won't                     bother. the adjective exists because there are non-pointed versions of all of                     these [3:06pm] thermoplyae: so, to start, we'll want three spaces to work with: I = [0, 1] with 0 the basepoint, S^0 = {+, -} with + the basepoint, and S^1 = {x in R^2 : |x| = 1} with (1, 0) the basepoint. &quot;the interval&quot;, &quot;the 0-sphere&quot;, and &quot;the 1-sphere&quot; or &quot;circle&quot; respectively [3:06pm] thermoplyae: our two functors are the cone of a space X, written CX, which is given by CX = X smash I and the suspension of a space, written SX, given by SX = S^1 smash X [3:08pm] thermoplyae: for example, the cone over S^1 can be constructed bit by bit. we start by                     taking I * S^1, which gives a cylinder [3:08pm] thermoplyae: then, we want to quotient together the subspace I that lies along the basepoint and the subspace S^1 that lies along the basepoint [3:09pm] thermoplyae: quotienting together the top end of the cylinder turns this into the familiar cone shape from grade school, quotienting a ray from the tip of the cone down to the bottom circle sort of &quot;flattens&quot; the cone but doesn't actually alter the topology [3:09pm] thermoplyae: and we're left with a space homeomorphic to D^2, &quot;the 2-disk&quot;, whose usual definition is {x in R^2 : |x| &lt;= 1} [3:10pm] thermoplyae: in general, i'll define D^n to be CS^(n-1) [3:10pm] thermoplyae: we can also construct higher spheres without mentioning any kind of ambient space. to start, let's suspend S^0, which begins life as S^0 * S^1 and looks like two copies of S^1 [3:12pm] thermoplyae: to start, the basepoint of one copy of S^1 and the single point in the other copy of S^1 that shares a coordinate with the basepoint will be quotiented together, giving a figure out [3:12pm] thermoplyae: we then collapse one of the circles to a point, leaving just S^1 behind, and so                     SS^0 = S^1 [3:13pm] thermoplyae: SS^1 is somewhat harder to picture; it starts life as S^1 * S^1, the torus, and then we do some funny quotienting [3:13pm] thermoplyae: ah, luckily i have a picture from an old discussion, ignore the triangulation information in this picture:  [3:13pm] thermoplyae: the important part is that we start with a rectangle, and then we glue each edge along its opposing edge [3:14pm] thermoplyae: which corresponds to taking I x I and quotienting along the extended relation i                     gave last time that we used to turn I into S^1 [3:14pm] thermoplyae: now, we want to quotient along a longitudinal circle and a latitudinal circle as part of our definition of smash product [3:15pm] thermoplyae: we can choose the boundary of the rectangle in the picture to perform this operation, which you should easily see to be homeomorphic to S^2 [3:16pm] thermoplyae: and, in general, SS^n = S^(n+1), which i'll take to be a definition but can also be shown to be equivalent to the usual S^n = {|x| in R^(n+1) : |x| = 1} definition [3:17pm] thermoplyae: it's worth noting that CX is contractable. in addition, this gives a relation between CX and SX; X is a subspace of CX, and so CX / X is homeomorphic to CX                     union-over-inclusion-of-X CX is homeomorphic to SX [3:18pm] thermoplyae: the cone can also be used to detect topological obstructions to homotopy of                      maps; a map f: X -&gt; Y is null-homotopic if and only if it admits an extension CX -&gt; Y that agrees with f when restricted to X [3:19pm] thermoplyae: and so there's some applications of the cone. the suspension is used to                     construct the homotopy groups of a space, and so that's where we'll use it                      heavily (of course, both continue to play central roles as we move along from                      there) [3:21pm] thermoplyae: to start, we'll need some more category theoretic things. we'll say a pointed object X is a group object if it has maps mu: X * X -&gt; X and phi: X -&gt; X such that the following diagrams commute:  [3:22pm] thermoplyae: when necessary, we'll work over pointed sets to illustrate [3:22pm] thermoplyae: the first diagram corresponds to associativity of mu. if we pick an element (a, b, c), following the bottom-left path gives (a, b, c) |-&gt; (a, bc) |-&gt; a(bc) and the upper-right path gives (a, b, c) |-&gt; (ab, c) |-&gt; (ab)c [3:23pm] thermoplyae: the second diagram corresponds to the basepoint selecting the identity [3:23pm] thermoplyae: and the last diagram corresponds to phi(x) acting as a left- and right-inverse for x [3:24pm] thermoplyae: if we dualize these diagrams (flip all the arrows, change all the limits to                     colimits), we get what i'll call a cogroup object:  [3:25pm] thermoplyae: this is likely to be a bit more alien, so it would be useful to come up with an                     example. luckily, we have an immediate one: for any space X the space SX                     admits an (H-)cogroup structure (i'll talk about what the H means in a moment) [3:26pm] thermoplyae: so this means that we'll want a map mu': SX -&gt; SX v SX. thinking back to what SX looks like, i claimed it was homeomorphic to CX glued to CX along the inclusion X -&gt; CX [3:27pm] thermoplyae: i also claimed that CX / X was homemorphic to SX. if we take CX glued along X                     to CX and quotient their shared copy of X to a point, i'll end up with two copies of CX / X glued together at a point, or SX v SX [3:27pm] thermoplyae: you should picture this as sort of 'pinching' the middle portion of a diamond figure, resulting in two diamond figures [3:28pm] thermoplyae: so, what's coassociativity mean in this context? i means i start with a                     diamond, pinch it, and then pinch the top diamond, that has to be the same as                      starting with a diamond, pinching it, then pinching the bottom diamond [3:29pm] thermoplyae: sure enough, you can construct stretching maps that make this work out (topology, as you'll recall, does not really care about size or distance in                     the metric sense) [3:29pm] thermoplyae: coidentity means that if we take our diamond, pinch it, and then collapse the top half, we'll end up with what we started with [3:32pm] thermoplyae: which brings us to coinversion, which i claim acts by flipping the diamond upside down [3:33pm] thermoplyae: and here's where that H- comes in -- the map that sends all of SX to the basepoint and the map that leaves the first half of SX alone and flips the other half upside down are not the same [3:33pm] thermoplyae: but they /are/ homotopic -- the second is null-homotopic [3:34pm] thermoplyae: and so SX admits an H-cogroup structure. so what's the point? [3:34pm] thermoplyae: the point is that representable functors take colimits to limits in the contravariant slot, and so Toph*(SX, Y) will have a group object structure on it [3:35pm] thermoplyae: group objects in the category of sets are the familiar groups, which means rather than having a functor Toph*(SX, -): Toph* -&gt; Set, it actually takes values in Grp [3:35pm] thermoplyae: this brings us to the homotopy groups of a space: pi_n X is defined to be                     Toph*(S^n, X) [3:36pm] thermoplyae: pi_n is group-valued for n &gt;= 1 because, as discussed SS^m = S^(m+1), and so                      S^n is the suspension of some space (namely S^(n-1)) for all n &gt;= 1 [3:37pm] thermoplyae: so let's start exploring what some of these are. pi_0 X has a particularly easy interpretation [3:37pm] thermoplyae: it corresponds to maps S^0 -&gt; X. one of the points of S^0 must go to the basepoint of X, and the other (because S^0 has the discrete topology) can go                     anywhere it pleases [3:38pm] thermoplyae: a homotopy of pointed maps S^0 -&gt; X corresponds to a path between the two 'free' parts of this map, and so the homotopy classes of S^0 -&gt; X correspond to the path components of X [3:41pm] _llll_: here's a question, pi_n=Toph*(S^n, X) and that S^n is an n-fold application of the S(uspension) functor, is that an instance of something more general? [3:42pm] thermoplyae: not that i've seen elsewhere [3:42pm] thermoplyae: well [3:43pm] thermoplyae: no, i was going to say something about relative homotopy, but it doesn't                     really address the question [3:44pm] thermoplyae: yeah, i have no idea. pressing on, let's look at what this group structure actually looks like for something like pi_1 [3:45pm] thermoplyae: so supposedly this group structure maps given two maps f, g: S^1 -&gt; X, we                     ought to be able to construct some other map (f . g): S^1 -&gt; X as given by                      this cogroup structure on S^1 [3:46pm] thermoplyae: specifically, S^1 --mu'-&gt; S^1 v S^1 --f v g-&gt; X, and so each function gets its own bit of S^1 to work on [3:48pm] thermoplyae: picturing S^1, again, as a diamond pinched in the middle, trying to                     parametrize this new path using the usual CCW-parameterization of S^1 would result in tracing out the top part of the pinched diamond first, then the bottom diamond [3:48pm] thermoplyae: which can be thought of as 'run along f first, then run along g' [3:48pm] thermoplyae: so the more familiar description of (f . g) that you budding algebraic topologists may have seen before might look more like (f . g)(t) = f(2t) for 0 &lt;= t &lt;= 1/2, g(2t - 1) for 1/2 &lt;= t &lt;= 1 [3:49pm] thermoplyae: this view of the homotopy groups is actually profitable (and generalizes                     readily to S^n) because it gives an easy proof that pi_n X is abelian for n &gt; 1 [3:50pm] thermoplyae: if we picture maps S^2 -&gt; X as maps D^2 -&gt; X where the action of the map on                     the boundary of D^2 sends everything to the basepoint, we can split D^2 up                      into two pieces, the first of which we use for f and the second for g [3:51pm] thermoplyae: we can then slide the first piece of the domain over the other and flip them (no, really, it is this easy, i challenge you to spend five minutes and                     construct the homotopy) [3:53pm] thermoplyae: of course, this doesn't work for S^1 -&gt; X; there's no space in D^1 to slide the domains up or down to make that proof work [3:53pm] thermoplyae: and, in fact, no proof will work; for example, pi_1 (S^1 v S^1) = Z free product Z [3:54pm] thermoplyae: proof of these sorts of calculational things will have to wait a moment; fibrations will be a powerful calculational tool, and we'll use them to come up with calculations for a few spaces [3:54pm] thermoplyae: one other thing before we move on to relative homotopy: some of you may be                     concerned as to much how much our choice of basepoint matters [3:55pm] thermoplyae: and the answer is that it doesn't; given pi_n (X, x_1), we can find an                     isomorphism with pi_n (X, x_0) so long as x_0 and x_1 lie within the same path component of X; the central point of the proof is to take n-sphere maps in X and 'distend' them into lightbulb shapes [3:56pm] thermoplyae: use the stem of the lightbulb to follow a path from x_0 to x_1, then use the bulb part to actually trace out the n-sphere map [3:57pm] thermoplyae: so, there, our first construction in algebraic topology. we managed to                     transform spaces into a sequence of groups [3:57pm] thermoplyae: of course, it would be nice to be able to compute some of these groups ),                      which in general is an incredibly difficult problem, but in some specific                      instances can be accomplished [3:58pm] thermoplyae: and to this end, we'll construct a couple other things: relative homotopy and                      fibrations, both of which induce very nice exact sequences on our homotopy                      groups, and then we can use homological algebra to squeeze information out                      about our spaces [3:59pm] thermoplyae: so, again, we want to think of maps D^n -&gt; X that send the boundary to the                      basepoint, rather than S^n -&gt; X [4:00pm] thermoplyae: and relative homotopy is summed up as picking a subspace A of X that we put                      the boundary of D^n -&gt; X into and say 'eh, close enough' [4:02pm] thermoplyae: we say two maps D^n -&gt; X whose boundaries lie in A are homotopic relative to A                     if there's a homotopy between them such H(x, t) is in A for all x on the boundary of D^n at all times t [4:02pm] thermoplyae: the group structure is constructed in basically the same way, split the domain D^n up into two pieces, one of which belongs to one map and the other map gets the rest [4:03pm] thermoplyae: (equiv. a pinch map CS^(n-1) -&gt; CS^(n-1) v CS^(n-1)) [4:03pm] thermoplyae: these relative homotopy groups are written pi_n (X, A, x_0) or pi_n (X, A) if                     we want to suppress the basepoint (and of course we do) [4:04pm] thermoplyae: if we pick A = {x0} the basepoint of X, we've recovered the old pi_n X [4:04pm] thermoplyae: given just this much, we can construct a l.e.s of homotopy groups [4:05pm] thermoplyae: given a subspace A -&gt; X, there's a l.e.s of the form ... -&gt; pi_{n+1} (X, A) -&gt; pi_n A -&gt; pi_n X -&gt; pi_n (X, A) -&gt; pi_{n-1) A -&gt; ... -&gt; pi_0 (X, A) -&gt; 0 [4:05pm] thermoplyae: this isn't really useful information without telling you 1) what the maps are 2) why it's exact [4:05pm] thermoplyae: i'll tell you half of 2 and along the way cover 1 :) [4:07pm] thermoplyae: let's pick an element of pi_n A. the map into pi_n X is given by inclusion, as is the map into pi_n (X, A) -- i said already that elements of pi_n X were special cases of elements of pi_n (X, A) [4:08pm] thermoplyae: since the element of pi_n A's image sits entirely inside A, it vanishes in                     pi_n (X, A) [4:09pm] thermoplyae: an element pi_n (X, A) is a D^n map into X with boundary in A.  the boundary of D^n is S^(n-1) by construction, and so restricting this map to the boundary of D^n gives an obvious choice of map S^(n-1) -&gt; A [4:09pm] thermoplyae: and that's the action of our map pi_n (X, A) -&gt; pi_{n-1} A [4:09pm] thermoplyae: so we pick an element pi_n X, include it into pi_n (X, A), and then restrict to the boundary [4:10pm] thermoplyae: because our map originally lived in pi_n X, its boundary sits entirely at the basepoint, which is the zero element of pi_{n-1} A [4:10pm] thermoplyae: finally, we pick an element pi_n (X, A), take its boundary in pi_{n-1} A and include it into pi_{n-1} X [4:11pm] thermoplyae: because it originally lived in pi_n (X, A), that original element corresponds to an extension of the map S^(n-1) -&gt; X to a map CS^(n-1) -&gt; X [4:11pm] thermoplyae: which corresponds to a null-homotopy [4:11pm] thermoplyae: tada [4:11pm] thermoplyae: so, questions before i go on to fibrations and we calculate some homotopy groups? [4:12pm] thermoplyae: i will take that as a no [4:12pm] thermoplyae: so we've been wiggling around homotopies for a while now, and fibrations explore a different aspect of this same idea [4:12pm] thermoplyae: given a (epimorphic) map E -&gt; B, when can a homotopy in B be lifted to a                     homotopy in E? [4:13pm] thermoplyae: terminology here: E is called the total space and B the base space. i'll call the map E -&gt; B by 'p' to avoid confusion (sometimes it's called pi, which is                     sort of stupid since we'll have homotopy groups to deal with too) [4:14pm] thermoplyae: so let's start by making this formal:  [4:14pm] thermoplyae: we have a surjective map p: E -&gt; B and a homotopy H: X * I -&gt; B [4:15pm] thermoplyae: given some partial lift of this homotopy into E (i.e. a map f: X -&gt; E such                     that X --inclusion at time zero-&gt; X * I --H-&gt; B is the same as                      X --f-&gt; E --p-&gt; B is the same map) [4:16pm] thermoplyae: we want to then be able to construct the remainder of the homotopy -- we want to guarantee the existence of an H^* such that the diagram commutes [4:16pm] thermoplyae: so what's an example of a fibration? any product space is a fibration, where p is given by projection onto any of the factors [4:17pm] thermoplyae: &quot;what's an example of something that /isn't/ a fibration?&quot; is a more interesting question [4:17pm] thermoplyae: let's take S^1 living in R^2. R^2 has a projection onto R, which gives a                     projection S^1 -&gt; D^1 [4:18pm] thermoplyae: is that a fibration? the map p: S^1 -&gt; D^1 is null-homotopic, and an initial lift of the projection map is to take S^1 -&gt; S^1 the identity map [4:19pm] thermoplyae: this reduces to the question &quot;is id: S^1 -&gt; S^1 null-homotopic?&quot; and the answer is no, but i'm not sure i've given you the tools to prove it yet. we'll certainly come back to this, and intuitively it should make sense [4:20pm] thermoplyae: in fact, in a proof that X * Y -&gt; X is a fibration you never use any kind of                     global structure of either space. all we /actually/ need is that E is locally homeomorphic to B * F for some space F, called the fiber [4:20pm] thermoplyae: and such a triple of spaces (along with a cover over B with local                     homeomorphisms and the projection map E -&gt; B) is called a fiber bundle [4:22pm] thermoplyae: running along this idea that we're losing some particular fiber's worth of                     information in our projection down to B, the preimage of B's basepoint by p is                      a subspace F of E also called the fiber (this coincides with the other F when                      E -&gt; B is a fiber bundle) [4:22pm] thermoplyae: since F is a subspace of E, we can consider the l.e.s of relative homotopy of                     the pair (E, F) [4:23pm] thermoplyae: what's this look like? substituting letters, it looks like ... -&gt; pi_{n+1} (E, F) -&gt; pi_n F -&gt; pi_n E -&gt; pi_n (E, F) -&gt; pi_{n-1} F -&gt; ... -&gt; 0 [4:23pm] thermoplyae: it's an easy lemma to prove that pi_n (E, F) is isomorphic to pi_n (B, {b0}) [4:24pm] thermoplyae: one half of the isomorphism is given by the projection map p, the other is                     given by lifting homotopies in B to homotopies in E [4:24pm] thermoplyae: and so we can rewrite this l.e.s as ... -&gt; pi_{n+1} B -&gt; pi_n F -&gt; pi_n E -&gt; pi_n B -&gt; pi_{n-1} F -&gt; ... -&gt; 0 [4:24pm] thermoplyae: called the l.e.s of the fibration E -&gt; B [4:25pm] thermoplyae: so, phew, let's compute some groups! (and you guys will probably be too                     tired -- bored, anyway -- to go on to cell complexes, so we'll stop there) [4:26pm] thermoplyae: so one example of a fibration (in fact a fibered space (in fact a 'covering' space, a special case of fibered spaces where the fiber is discrete)) is                     taking R to be a helix in R^3 and projecting onto a circle sitting in a plane [4:26pm] thermoplyae: this gives a map R -&gt; S^1 with fiber Z [4:27pm] thermoplyae: R is contractible, so its homotopy groups vanish, and so our l.e.s looks like ... -&gt; pi_2 Z -&gt; pi_2 R -&gt; pi_2 S^1 -&gt; pi_1 Z -&gt; pi_1 R -&gt; pi_1 S^1 -&gt; pi_0 Z                     -&gt; pi_0 R -&gt; pi_0 S^1 -&gt; 0 [4:32pm] thermoplyae: we identified pi_0 with path components earlier, so pi_0 S^1 = pi_0 R are 1-point sets and pi_0 Z is Z again [4:33pm] thermoplyae: pi_n R = 0 because it's contractible and pi_n Z = 0 because the path-component that contains Z's basepoint is also contractible [4:34pm] thermoplyae: and so our sequence actually looks like ... -&gt; 0 -&gt; 0 -&gt; pi_2 S^1 -&gt; 0 -&gt; 0 -&gt; pi_1 S^1 -&gt; Z -&gt; 0 -&gt; 0 -&gt; 0 [4:34pm] thermoplyae: a couple basic facts about l.e.s are that if 0 -&gt; A -&gt; 0 is exact then A = 0 and if 0 -&gt; A -&gt; B -&gt; 0 is exact then A is iso. to B [4:34pm] thermoplyae: this gives us the calculation that pi_n S^1 = Z if n = 1 and 0 otherwise [4:35pm] thermoplyae: it's worth noting that no complete classification (well, except in terms of                     braid groups for n = 2, which is neat but i'm not completely convinced it's                      useful yet -- they're also difficult to compute) of S^n exists n &gt; 1 [4:36pm] thermoplyae: we can, however, construct a fibration that tells us useful information about S^2 and S^3 [4:36pm] thermoplyae: surely some of you are familiar with complex projective space [4:37pm] thermoplyae: we can construct CP^1 is one of two ways: take C^2 and quotient by                     complex-linear subspaces (complex planes, in essence) or take C^1 and add a                      point that turns it into a compact space [4:38pm] thermoplyae: the quotient part of this can be seen to be a fibration S^3 -&gt; CP^1, and some identifying these two constructions as giving the same space gives CP^1 = S^2 [4:39pm] thermoplyae: the fiber of this map is S^1, and so we have an associated l.e.s of the form ... pi_3 S^2 -&gt; pi_2 S^1 -&gt; pi_2 S^3 -&gt; pi_2 S^2 -&gt; pi_1 S^1 -&gt; pi_1 S^3 -&gt; pi_1 S^2 -&gt; pi_0 S^1 -&gt; pi_0 S^3 -&gt; pi_0 S^2 -&gt; 0 [4:40pm] thermoplyae: since we calculated pi_n S^1, we have that pi_n S^3 = pi_n S^2 for n &gt; 2 [4:40pm] thermoplyae: in particular, some suspension calculations give that pi_3 S^3 = Z just like pi_1 S^1, and so pi_3 S^2 is nontrivial (!) [4:41pm] thermoplyae: the homotopy groups of S^1 are not all that special, but that pi_n S^m can be                     nontrivial for n &gt; m is a huge shock, and a complete analysis of this problem is a central object of research in algebraic topology and motivation for a lot of tools that have been invented [4:42pm] thermoplyae: one last example of a fibration, something to chew on because we'll use it                     next time: the map X^I -&gt; X given by evaluating point paths I -&gt; X at the far endpoint is a fibration [4:43pm] thermoplyae: and in fact, X^I is contractible; we can just not follow paths as far out to                     construct a deformation retract to the constant map to X's basepoint [4:44pm] thermoplyae: so what's the fiber of this map? well, all maps f in X^I satisfy f(0) = x_0, and if they're in the fiber that means f(1) = x_0 [4:44pm] thermoplyae: which means they're maps S^1 -&gt; X, or elements in Omega X, the loop space on X [4:45pm] thermoplyae: writing out the l.e.s associated to this fibration gives another proof that pi_n X = pi_{n-1} Omega X (you could have shown this earlier by using the                     exponential correspondence from last time and making S, Omega into an adjoint                      pair) [4:45pm] thermoplyae: let me skim my notes to make sure i'm not missing anything critical [4:46pm] thermoplyae: i don't think so. if you guys are looking for something easy to toy with for next time, what does the l.e.s tell us about pi_1 B when we have a fibration induced by a covering space? [4:46pm] thermoplyae: if you're looking for something hard, given a map A -&gt; B construct a pair of                     maps A -&gt; X -&gt; B such that A -&gt; X is a homotopy equivalence and X -&gt; B is a                      fibration [4:47pm] thermoplyae: or maybe prove that some of these examples i've given are, in fact, fibrations -- the fibered space definition is particularly worth your time [4:47pm] thermoplyae: or the other half of the exactness proof. i've left out all kinds of                     information :) [4:47pm] thermoplyae: and so i think i'm done, you guys can heckle now [4:48pm] _llll_: i forget, fibrations into B sometimes can eb classified by pullbacks against a                 universal one? or is that just for covering spaces [4:49pm] thermoplyae: just covering spaces; the universal covering space proof relies on the                      universal one being contractible and the covering space's fiber being discrete [4:51pm] FunctorSalad: example of a fibration that's not a fibre bundle? [4:51pm] thermoplyae: the pathspace fibration, which i don't think i gave a name to.  the X^I -&gt; X                      one [4:52pm] thermoplyae: your answer to the A -&gt; X -&gt; B question will also not be a fiber bundle [4:52pm] _llll_: ah, but fibre bundles can be classified (right?) [4:53pm] thermoplyae: i don't believe generic ones can, but i could be wrong. fiber bundles with a                     particular action on them can for sure [4:54pm] _llll_: yeah, that sounds about right [4:54pm] thermoplyae: it is also worth noting that what i called a fibration is sometimes called a                     'fibration in the sense of hurewicz' -- a 'fibration in the sense of serre' only requires that the fibration diagram commute when X is a CW-complex (to                     be defined next time) [4:57pm] _llll_: are we done? topic for next time? [4:58pm] thermoplyae: cell complexes and ordinary homology. spend a while talking about homotopy theorems for cell complexes, use that to motivate homology theories, construct ordinary homology, spectra if that somehow goes really, really quickly (really,                     really unlikely)