# Reference for intersection and linking in algebraic topology

I have a feeling that I have seen some kind of theory of linking and intersection that applies in spaces that are not manifolds. I've found two kinds of theories in the books I've checked:

1) intersection product of homology classes, defined in terms of Poincare duality,

2) linking numbers defined for disjoint subsets of $\mathbb{R}^n$ using the vector space structure of $\mathbb{R}^n$.

What I really want to do is to talk about intersection/linking of subcomplexes of a finite simplicial complex. Can anyone point me to a reference?

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You want a notion of linking that applies to arbitrary subcomplexes of arbitrary simplicial complexes? I suspect if you want this idea of "linking" to be at all a non-trivial reflection of what happens in manifolds, at least at this level of generality you're sunk. "Poincare Duality complexes" are more general than manifolds and have everything you need to get the machinery moving. But they're much more restrictive than arbitrary simplicial complexes. –  Ryan Budney Jul 29 '10 at 20:10
You can define the cup product in the singular cohomology of an arbitrary simplicial complex, and the information that it carries is what is left of intersection. Similarly, Massey products play the role of linking numbers. –  Charlie Frohman Jul 30 '10 at 17:41
This is my general understanding, too. But can you make the phrase "what is left of intersection" precise? (Or point me to a reference?) –  Jeff Strom Aug 2 '10 at 12:02
The interpretation of the cup product as an intersection depends on Poincare Duality, which only happens for manifolds, or for restricted kinds of homology for restricted kinds of complexes. Yet there is a way through to definining a product structure which has a lot of properties of the intersection pairing, like graded commutativity. There is a formulas for linking number in terms of intersection with a chain that is reminiscent of the formula for Massey products. Check out Griffiths and Morgan "Rational Homotopy theory and Differential Forms". –  Charlie Frohman Aug 2 '10 at 20:00

I'm not quite sure if this is what you're thinking of, but intersection homology has a good theory of intersection products for simplicial pseudomanifolds. See Goresky-MacPherson, "Intersection Homology Theory"

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I quote Andrew Ranicki's answer here.

The linking form appears in Example 12.44 of my recent book "Algebraic and geometric surgery" (Oxford University Press, 2002), and also in Chapter 3 of my earlier book "Exact sequences in the algebraic theory of surgery" (Princeton University Press, 1981) which is available from http://www.maths.ed.ac.uk/~aar/books/exact.pdf
I don't know if these are "textbook references". At any rate, the L-theory localization exact sequence is a good algebraic surgery setting for linking forms and their non-simply-connected analogues, although maybe too elaborate and non-geometric for some tastes.

Abstractly, surely the localization exact sequence is the correct context for linking, and is precisely what I think you are looking for. It's also a very beautiful construction, in my opinion, which is fun to learn and good to know. On the other hand, I don't think Andrew would strongly disagree to the assertion that it's hellishly difficult to calculate explicit localizations and L-groups, except in the very simplest cases. So if you want to be able to work concretely in this day and age, you need a bit more structure than a symmetric structure (or a quadratic structure) on a chain complex. This, as Greg pointed out, intersection homology gives you, in a less general context of simplicial pseudomanifolds.
On the other hand, I really wish there were better techniques for calculating Cohn localizations, and higher L-groups, explicitly- if you find anything, please let us know!

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Here is a down-to-earth cohomological interpretation of the usual linking number.

Let $M$ be an oriented manifold (possibly non-compact) of real dimension $p$ and let $X\subset M$ be a closed subset, the support of a codimension $q$ Borel-Moore cycle $c$ homologous to 0 in $M$. An example: take $X$ to be a pseudo-manifold in the sense of Goresky-MacPherson, Intersection theory 1 (informally speaking, a manifold with singularities of real codimension $>1$); then the fundamental class of $X$ is well-defined and we require that it should be homologous to 0 in $M$. To be more specific, we can take $M=\mathbf{C}^n$ and $X$ a closed complex analytic subvariety of complex codimension $q/2$.

Suppose $H_{q-1}(M)=0$. Then the group $H^{q-1}(M)\cong H_{p-q+1}^{BM}(M)$ is finite (we use the universal coefficients formula for this and $H^{BM}$ stands for the Borel-Moore homology), and $c$ defines (via the Poincar\'e-Lefschetz duality) a unique element of $$H_{p-q+1}^{BM}(M\setminus X)/\mbox{torsion}\cong H^{q-1}(M\setminus X)/\mbox{torsion}.$$

Here is another equivalent definition: suppose that $c$ is represented by a smooth singular chain $\tilde c$, and consider the function $H_{q-1}(M\setminus X)\to\mathbf{Z}$ defined as follows: take a cycle in $$H_{q-1}(M\setminus X)=\ker(H_{q-1}(M\setminus X)\to H_{q-1}(M)),$$ represent it by a smooth singular chain $z$, find a smooth singular chain $w$ in $M$ that is bounded by $z$ and transversal to $\tilde c$, and calculate the intersection index of $w$ and $\tilde c$. This function also defines a unique element of $H^{q-1}(M\setminus X)/\mbox{torsion}$, which coincides with the above.

Example: if $X$ is a line or a circle in $\mathbf{R}^3$, then the corresponding cohomology class of the complement takes value $\pm 1$ on any circle linked with $X$.

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This (in a singular homological context) is in the link which I posted (for Andrew's answer): mathkb.com/Uwe/Forum.aspx/research/286/linking-form It's surprisingly hard to find in the references- is LaLa (Lannes-Latour) still the only place in the literature where the details are properly written down? –  Daniel Moskovich Aug 3 '10 at 14:56
Daniel -- I don't know any reference for that at all and I agree it's a shame that no standard textbook seems to mention this. I don't have the book by Lannes and Latour at hand, but from the description in the thread you give their version is a bit different from the above (although it uses exactly the same idea): it is defined on the torsion subgroup so I would guess it takes values in $\mathbf{Q}/\mathbf{Z}$. (The definition is probably as follows: take torsion classes $a$ and $b$, find $c$ that bounds $na,n\in\mathbf{Z}$, take the intersection of $c$ and $b$, divide by $n$ and reduce.) –  algori Aug 4 '10 at 4:03