# Low Dimensional Topology

## May 3, 2010

### What is L-theory and why should I care: Part I

Filed under: Algebraic topology,knot concordance — dmoskovich @ 10:49 am

In the middle of February, Mark Powell was in Kyoto, where he taught me about algebraic surgery and other interesting topics. These are tools which were developed by high-dimensional algebraic topologists to tackle the kind of problems which they are interested in, and moreover they have a rather fierce reputation. So why should the low-dimensional topologist care about L-theory and suchlike? In this short series of posts, I’ll summarize the basics of what Mark taught me in February, and I’ll tell you why I care about L-theory and why I think you should care about it too (for those who can’t wait: because of Cochran-Orr-Teichner and because it provides a natural language to think about Blanchfield pairings).

The basic philosophical premise which underlies L-theory (and a good deal of algebraic topology) is that there are two ingredients which go into the topology of a manifold:

1. A chain complex over some ring $R$, considered up to chain homotopy. This tells you what the manifold looks like “from the inside”- how to build it up out of simplices or whatever, with coefficients in $R$.
2. Its Spivak normal fibration. This tells us how our manifold can embed locally in $\mathbb{R}^N$ for large enough $N$. So it tells us what the manifold looks like “from the outside”.

Today, I’ll tell you about a symmetric structure on a chain complex, where we ignore (2) altogether. This might or might not entail a loss of information, depending on whether or not 2 is invertible in $R$… but we don’t care, because either way investigating “topology of chain complexes” rather than “topology of manifolds” is a useful change of vantage point (I don’t care about the greater generality so much right now).Let’s see, then, how much topology we can do on a chain complex.
Fixing some notation, let
$C_\bullet(X)\overset{\text{def}}{=}C_n(X)\longrightarrow C_{n-1}(X) \longrightarrow\cdots\longrightarrow C_0(X)$
be a bounded chain complex over a ring $R$ with involution $r\mapsto \bar{r}$). The dual chain complex is
$C^\bullet(X)\overset{\text{def}}{=}C^0(X)\longrightarrow C^1(X) \longrightarrow\cdots\longrightarrow C^n(X)$
where $C^i(X)$ is defined to be $\mathrm{Hom}_R(C_i(X),R)$. The role of the involution is to allow us to make $\mathrm{Hom}(C^\bullet,R)$ from a left $R$-module to a right one, and vice versa. Recall the standard definition of the cap product $H_p(X)\times H^q(X)\longrightarrow H_{p-q}(X)$ as coming from the composition $C_\bullet(X)\overset{\Delta_0}{\longrightarrow} C_\bullet^{\mathrm{op}}\otimes C_\bullet\overset{\setminus}{\longrightarrow}\mathrm{Hom}_R(C^{\bullet},C_\bullet)$ (by the way, $C_\bullet^{\mathrm{op}}$ is the same thing as $C_\bullet$, except that I’m stressing that $R$ is acting on the right via its involution rather than on the left- this is usually suppressed in notation, but it’s the reason we need the involution, so I wanted to point it out).
The first map $\Delta_0: (C\otimes C)_k = \sum_{i+j =k} C_i \otimes C_j$ is called an approximation to the diagonal map $\Delta:X\longrightarrow X\times X$ (for reasons we’ll discuss below) and is induced by the diagonal map via the Eilenberg-Zilber Theorem. Morally, it corresponds to taking the “simplex coming from the front $i$ coordinates tensor the simplex coming from the back $k-i$ coordinates of a $k$-simplex”. The second map $\setminus$ is the slant product, sending $x\otimes y\in C_\bullet^{\mathrm{op}}\otimes C_\bullet$ to the map $\{f\mapsto \overline{f(x)}y\}$ (take note that this is not symmetric in x and y). Now choose a cycle $Y\in C_n$ (which will play the role of the fundamental class of a manifold or whatever). This is the only arbitrary choice we are going to make.Choosing $Y$ gives rise to a chain map $\varphi_0^\bullet$ which sends any $A\in C^i$ to $\setminus\Delta_0(Y)(A)\in C_{n-i}$, an expression which in other branches of algebraic topology would be called $Y\cap A$. If one wanted to descend to homology, the cap product with $[Y]$ would be sending a fundamental class $[Y]\in H_n(M)$ to the map $[Y]\cap -$ which would induce the Poincare duality isomorphism. But we don’t care about homology! This game is about the chain complexes themselves. Why chuck out information by passing to homology, for goodness sake!
Taking stock, what we have so far is a chain map $\varphi_0^\bullet$. Now’s where something spooky happens (and I’m curious how much of it is due to Steenrod, how much to Wall, how much to Ranicki, etc.). As its name suggests, the slant map is biased. The terms $\setminus (x\otimes y)$ and $\setminus(y\otimes x)$ are quite different. A much bigger embarassment, however, is that even the diagonal approximation $\Delta_0$ is biased- switch its formula by $x\otimes y\mapsto y\otimes x$, and you get a quite different map. This wouldn’t make a difference on the level of homology groups (because, as we’ll see, $\Delta_0$ is chain homotopic to its transpose), but on a chain complex, or if we want to define homology operations, it’s something which we really should worry about. It’s quite bad to have a diagonal approximation which isn’t symmetric with respect to reflecting through the diagonal. One would prefer to eliminate systemic biases from mathematical definitions. With motivation so noble, is it any wonder that we are about to be richly rewarded for our efforts?
Consider the transposition $T$ sending $x\otimes y\in C_\bullet^{\mathrm{op}}\otimes C_\bullet$ to $y\otimes x\in C_\bullet^{\mathrm{op}}\otimes C_\bullet$ (as always, my signs are rubbish). Let’s look at how $T$ acts on the diagonal approximation $\Delta_0$. In surgery theory, every chain complex longs to be a CW complex. The diagonal approximation longs to take a point $x$ in this CW complex to $x\times x$. It’s eternal yearning is to have as its image the dotted red line in the picture:

where $e_1,e_2,\ldots$ are chains in different dimensions. Is mapping chains to tensor products of chains all it can ever hope to achieve?

No! It can do more! Because $\Delta_0$ and $T\Delta_0$ are chain homotopic (which you can see using the Acyclic Carrier Theorem), a higher diagonal approximation map $\Delta_1$ can come to the diagonal approximation’s aid, mapping $e_k\in C_k$ to a $(k+1)$-chain in $(C\times C)_\bullet$. The relation between $\Delta_0$ and $\Delta_1$ is given by the expression $\Delta_0-T\Delta_0=d_\otimes\Delta_1 \pm \Delta_1 d$. The maps $\Delta_0$ and $\Delta_1$ work together to provide a better approximation to the diagonal $\Delta$ than either approximation could ever hope to achieve on its own:

The best thing of all is that $\Delta_1$ is completely free- we made no arbitrary choices and imposed no restrictions to get it. This means that in any setting where we consider $\Delta_0$, we should really also consider $\Delta_1$. They’re both part and parcel of the same conceptual picture.
The new diagonal approximation map $\Delta_1$ gives rise to a chain homotopy $\varphi_1^\bullet\overset{\text{def}}{=}\setminus\Delta_1(Y)$. The relationship between $\varphi_0^{\bullet}$ and $\varphi_1^{\bullet}$ is given by
$\varphi_0^\bullet-T\varphi_0^\bullet=d\varphi_1^\bullet-\varphi_1^\bullet d^\ast.$
Thus $\varphi_1^\bullet$ is the chain-level reason that the difference between $\varphi_0^\bullet$ and $T\varphi_0^\bullet$ doesn’t matter at the level of homology.
But we needn’t stop here! The higher diagonal $\Delta_1$ can get help from an even higher map $\Delta_2$, and so on and so on, to reach ever closer to that dotted red line which diagonal maps so long for. Each diagonal approximation $\Delta_i: C_\bullet \longrightarrow (C\times C)_{\bullet+i}$ gives rise to a chain homotopy $\varphi_i^\bullet\overset{\text{def}}{=}\setminus\Delta_i(Y)$. The map $\varphi_i^j$ maps each cochain group $C^j$ to the chain group $C_{n-j+i}$. For $i, each $\varphi_i^\bullet$ is related to the one above it by the formula
$\varphi_i^\bullet+(-1)^{i+1}T\varphi_i^\bullet=d\varphi_{i+1}^\bullet-\varphi_{i+1}^\bullet d^\ast.$
And up we go… up, up, up, until we reach $\Delta_n$, and with it $\phi_n^\bullet$, which turns out to be the $0$th Steenrod square and to equal $\pm1$ (I wish I understood this conceptually!).

So now we get to the main definition of this post.

A symmetric structure on a bounded finitely generated projective chain complex $C_\bullet$ is a collection of chain maps $\phi_0^\bullet,\ldots,\phi_n^{\bullet}$ constructed as above, considered up to chain homotopy, or, in other words, is an element of
$Q^n(C^\bullet)\overset{\text{def}}{=}H_n(\mathrm{Hom}_{\mathbb{Z}[\mathbb{Z}_2]}(W,\mathrm{Hom}_R(C^{\bullet},C_{\bullet}))).$
where
$W\overset{\text{def}}{=}\mathbb{Z}[\mathbb{Z}_2]\overset{1-T}{\longrightarrow} \mathbb{Z}[\mathbb{Z}_2]\overset{1+T}{\longrightarrow}\cdots$
is a chain complex for $S^\infty$, which is the double cover of the infinite dimensional real projective plane, and thus admits a CW structure with two cells in every dimension.

The second formulation, which defines Q-groups is rather sleek- the kind of definition which looks beautiful and natural after you understand it, and completely inpenetrable before.
And so, when all is said and done, a symmetric structure is really nothing more than a chain complex equipped with chain-complex-level Poincare duality structure, given by the $\varphi$-maps. But why should we care about a symmetric structure on a chain complex, when most applications of Poincare duality use only an isomorphism on homology?
The motivation for the definition of a symmetric structure was, in Wall’s words, as a “simple and satisfactory version of the whole setup [of algebraic surgery]”. This, then, is a language in which you can very easily set up chain-level cup products and Steenrod operations, much as we set up chain-level cap products above. That would actually be the next natural thing to do (see Chapter 3 Section 4 of Prasolov’s book Elements of homology for all this and for a much better exposition for higher diagonal approximations than I’ve given here). The purpose of this post was to demonstrate that the setup really isn’t all that painful (and to internalize what Mark Powell taught me)… and the idea of a symmetric structure really is useful, honest!
Next installment then: mapping cones, surgery on chain complexes, cobordism of chain complexes, and L-groups.

Mark Powell wrote to me to suggest another good way to think of Q-groups:

A nice interpretation of the Q-groups is as the algebraic version of the homotopy fixed points of the $\mathbb{Z}_2$-action on $C \otimes C$. That is $\mathrm{Map}_{\mathbb{Z}_2}(pt,X \times X)$ is the set of equivariant maps of a point into $X \times X$, or in other words the fixed points of the $\mathbb{Z}_2$-action. Instead go to homotopy theory, and you replace “point” with “homotopy $\mathbb{Z}_2$-point” which is $S^{\infty}$. Then go to algebra and take homology and you get the Q groups.
Maybe a slightly abstract viewpoint but from certain points of view I think this is very important. For example, the quadratic Q-groups are then the homotopy orbits of the $\mathbb{Z}_2$-action.

1. It does not look like you defined C_{\bullet}^{op}

Comment by Mayer A. Landau — May 9, 2010 @ 10:32 pm

• Thanks! I corrected this. Usually it seems to be suppressed in textbook definitions of the cap product… but I wanted to stress that the left copy of C_\bullet is being viewed as a right R-module rather than as a left R-module, because that, after all, is the reason R needs to come equipped with an involution.

Comment by dmoskovich — May 11, 2010 @ 10:43 pm

2. On my computer, one cannot see the pics (they are replaced by a sign saying that they are hosted by Angelfire). Frustrating…

Comment by anon — May 5, 2012 @ 8:45 am

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