Reference for intersection and linking in algebraic topology - MathOverflow

Reference for intersection and linking in algebraic topology

2010-07-29T19:59:02Z

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?

Answer by Greg Friedman for Reference for intersection and linking in algebraic topology

2010-07-29T22:30:24Z

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"

Answer by Daniel Moskovich for Reference for intersection and linking in algebraic topology

2010-08-01T21:40:46Z

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!

Answer by algori for Reference for intersection and linking in algebraic topology

2010-08-02T06:57:42Z

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 $X$.