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\title{Topics in Algebraic Topology\\September 2, 2004}
\author{Gabriel C. Drummond-Cole}
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Today I want to informally discuss some examples before we get into definitions.

$X$ is a Whitney-stratified space (which is a Thom-stratified space in our situation).  Roughly, $X=\dorp_{\alpha\in A}\Sigma_\alpha,$ where $\Sigma_\alpha$ are smooth manifolds locally closed in $X.$  There are many many conditions you have to impose, but right now let's do it intuitively.  Say you have $X=\mathbb{R}^2.$  Then $\Sigma_2$ can be the plane minus a line, $\Sigma_1$ the punctured line, and $\Sigma_0$ a point. 

You want to be dealing with analytic spaces, because the smooth case is harder.  There is a nice theorem that given any analytic subspace of an analytic space can be stratified.

You can also have a stratified singular space; the locus $xy=0$ can have $\Sigma_0=(0,0)$ and $\Sigma_1$ the remainder.

Typically, you will have something like $X=\dorp^n \Sigma_n.$  There is a notion of dimension here, so that $\Sigma_n$ has dimension $n.$  There is the dimension ($n$) of $X$ and then the dimension of the strata.  Actually, I may regret this later.  For example, $X$ could be $\mathbb{R}^n,$ and inside of $X$ take a stratum $Y_0$ and close it to $Y=\bar{Y}_0.$  A requirement that you usually put is that $Y\backslash Y_0$ is a union of strata.  Now you define the dimension of $Y$ to be the dimension of the manifold $Y_0.$  This is usually denoted $n.$

Now $\tilde{\Sigma}$ is the set of strata of $Y$ not in $Y_0.$  In the case above, $\tilde{\Sigma}_2=\{\Sigma_0,\Sigma_1\}.$  $\Sigma$ is the union of the singular strata of lower dimension so that $Y\backslash \Sigma$ is smooth.  In the obvious case this is $Y_0$ and sometimes it's a smaller thing which is whatever the smooth part is.

I'm giving this rough background because I'm aiming for the definition of an intersection homology.  The space we're going to consider is the closure of a stratum.  This may sound restrictive, but it is good for algebraic varieties.  You have $Y=\dorp \Sigma_\alpha,$ wherein you have $\Sigma,$ a singular set, and $\tilde{\Sigma},$ the strata in the singular set.  So if $Y_0=\Sigma_2$ above then $\Sigma$ is the line.

This is not the day for complete definitions, but once you have a stratification you can stratify further, as much as needed.  So if I use only the point, the line, and so on, I won't get very far.  So I want to look at further refinements of a stratification.  I can take the plane as a single stratum, or I can discard the circle and then look at the circle as a separate stratum.  If I only have the plane then the only cycle is the empty cycle, so I want to be looking at every possible refinement.

Singular homology was good because you had no hypotheses on the type of space.  The basic idea was to construct a geometric homology; algebraic homology came much later.  What I'm going to describe now can only be made watertight with a theory of stratification.  When the time comes, we'll give definitions, but for now we'll just think that if something is interesting, it is in my class of stratified spaces.

For homology we need the notion of cycles.
\begin{defi}
This is preliminary, but essentially correct.
\\
A geometric cycle on $Y$ is a closed subset $\xi$ of $Y_0$ (not $Y$) which is a union of strata for some refinement of the original stratification of $Y.$
\end{defi}
We're going to consider the closure $|\xi|$ of $\xi,$ which is again a union of strata in $Y.$  Some are contained setwise in the original singular set $\Sigma.$  

We take these geometric cycles as cycles and will impose conditions to get our abelian group.  How does the closure of a cycle meet the boundary?
\begin{defi}
A perversity is a function $\tilde{\Sigma}\to\mathbb{Z}.$  To every stratum of the original and fixed stratification I associate a number.  
\end{defi}
This is not the original definition of perversity, and I'm not going to explain why right now.  Let's describe how such a function can be used to impose restrictions on how the closure of $\xi$ meets the boundary $\Sigma.$
\begin{rem}
Let $Z$ be closed in $Y$ or $X$ and is a union of strata for some refinement (we call such a $Z$ admissable).  Then $Z\cap \Sigma_\alpha$ is a union of strata of a refinement.  This means that this intersection has a dimension; it's the maximum of the dimensions of the strata in this union.  For example, the upper closed half-circle would be a good $Z$ in our $\mathbb{R}^2$ example.  This is very key because the definition of intersection homology passes through the definition of dimension.  
\end{rem} 
\begin{defi}
Let $p:\tilde{\Sigma}\to\mathbb{Z}$ be a perversity, $i\in\mathbb{Z},$ and let $Z$ be admissable.  Then $Z$ is $(p,i)$ admissable if $dim\ Z\le i$ and
for every $\Sigma_\alpha\subset \Sigma$ the dimension of $Z\cap\Sigma_\alpha\le i- codim_Y(\Sigma_\alpha)+p(\Sigma_\alpha).$
\end{defi}
Here the codimension term is the expected dimension.  In $\mathbb{R}^n$ if $\mathbb{R}^k$ and $\mathbb{R}^l$ intersect, we imagine that the dimension is $k+l-n,$ so if $Z=\mathbb{R}^k$ and $\Sigma_\alpha=\mathbb{R}^l,$ we see that $i$ takes the place of $k$ and $n-l$ is the codimension.  Then $p$ is a correction term which shows how far we're allowing things to deviate.  For $p$ very large we are imposing no condition at all (a large perversity is a large deviation from the norm).

So a perversity measures how badly a cycle can intersect the boundary.

Now let's define the chain complex that arises from all of this.
\begin{defi}
A geometric $i$-chain $\xi$ in $Y_0$ (with twisted coefficients to be discussed later) is a chain of dimension $i.$
\\
It is $p$-allowable if its closure $|\xi|$ in $Y$ is $(p,i)$-allowable.  We're taking closures of each cycle separately.
\end{defi}
For simplicity, we'll let $\xi$ be a single cycle.  Then $\boundary \xi$ in $Y_0$ is an $i-1$-chain, and we can consider $|\boundary \xi|,$ its closure.  There's a point to keep in mind that the boundary of the closure is not necessarily the closure of the boundary.  Let the boundary be a point and let the cycle be a ray from that point.
\begin{defi}
Let $C_.(Y)$ be the chain complex of usual chains.  We're interested in $I^pC_.(Y),$ the subset of those elements of $C_.(Y)\ \{\xi||\xi|$ is $(p,i)$-allowable and $|\boundary\xi|$ is $(p,i-1)$ allowable$\}$  Then $I^pH_.(Y)=H_.(I^pC_.(Y),\boundary).$
\end{defi}
So at first they had trouble convincing anyone that this would be important, because there are so many different groups you can get according to the perversity.  For example, for some spaces, with one perversity you get the homology and with another you get the cohomology.  If the perversity is extreme enough in one direction, you get an empty complex.  In the other direction, if $p$ is very large, you're imposing no restriction on the cycles and so you get the normal homology.  If $p=0$ you're discussing something quite strong.  It only looks at transverse intersections.

So for $p=0$ the condition is $dim(Z\cap\Sigma_\alpha)\le 1-codim(\Sigma_\alpha).$  Then for $\xi$ to be in the complex means that $codim(\Sigma_\alpha\cap |\xi|)\ge m-dim(\Sigma_\alpha),$ where $m$ is the dimension of $Y.$
\\This is $dim |\xi|-dim |\xi|\cap\Sigma_\alpha \ge m-\dim \Sigma_\alpha.$  This is as little as you would expect in a general case.  You're asking for the best possible thing.  I'm not saying this very well.  Having good behaviour when you intersect the boundary means intersecting it as little as possible.

Next time I'll repeat the definition and go on from there.
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