I am having trouble understanding the definition of a topologically stratified space as presented in Laurentiu Maxim's Intersection Homology & Perverse Sheaves: with Applications to Singularities (inserted below). In particular, the local normal triviality condition seems unintuitive and I have two main questions:
- What does this condition buy us? (Why are cones important to the theory?)
- How should a working mathematician think about this condition? (Is everything cones if you look at it the right way?)
In effort to address (1), I have come to understand that this is the definition used and introduced by McPherson and is equivalent to the notion wielded by Whitney. Additionally, it implies that the space admits nice triangulations (which I can easily believe are important for a homology theory). Later in the text, the author explores the intersection homology of open cones and then uses an inductive argument on cones to prove a version of Poincaré duality. So perhaps the definition is made to set up that proof. However, as this seems to be a vital object for the book, I feel like I am missing a justification for its usefulness.
Below, the notation $\mathring{c}L$ refers to the open cone of $\mathring{c}L=L\times[0,1)/L\times\{0\}$.
Definition 2.1.2 A topologically stratified space is defined by induction on dimension as follows:
(i) A $0$-dimensional topologically stratified space is a countable set of points with discrete topology.
(ii) For $n> 0$, an $n$-dimensional topologically stratified space is a Hausdorff topological space with a filtration $$X=X_n\supseteq X_{n-1} \supseteq \dots \supseteq X_1 \supseteq X_0 \supseteq X_{-1}=\emptyset $$
by closed subspaces $X_j$, so that the following local normal triviality condition is satisfied: if $x \in X_j − X_{j−1}$, there is a neighborhood $U_x$ of $x$ in $X$ and a compact $n−j −1$ dimensional topologically stratified space $L$ with a filtration $$L=L_{n-j-1}\supseteq \dots \supseteq L_1 \supseteq L_0 \supseteq \emptyset,$$ and a homeomorphism $$\phi: U_x\to \mathbb{R}^j\times \mathring{c}L$$ such that $\phi$ takes $U_x\cap X_{j+i+1}$ homeomorphically onto $\mathbb{R}^j\times\mathring{c}L_i$ when $n−j − 1 \geq i \geq 0$, and $\phi:U_x\cap X_j\to \mathbb{R}^j\times \{x\}$ is a homeomorphism.
