We know that the Thom class $\tau_W$ is defined on a disk bundle $W\rightarrow L$, where $L$ is a $p$-dimensional manifold and the rank of $W$ is $k$.
Let $[W]_0$ denote the fundamental class of the pair $(W,W\setminus L)$ or better, $(W,\partial W)$ as both are homotopy equivalent pairs.
Definition 1: $\tau_W$ is the Poincaré dual to the canonical zero section of $W$ w.r.t this class. That is, $\tau_W$ is the unique class satisfying $\tau_W\cap [W]_0=[L]$ where $L$ is the zero section of the disk bundle.
Definition 2: $\tau_W$ is the unique class that restricts to the positive generator of $H^k(D^k,\partial D^k)$
I tried arguing that if $W$ has a simplicial structure such that the fibre at some point is the $k^{th}$ front face and $L$ is the $p^{th}$ back face, then the cap product definition itself, straightaway gives the equivalence, if I start from Definition 2 and arrive at the former definition.
My question: Is there such a simplicial structure? I am thinking whether cell structures help.
OR
Is there a better way to prove the equivalence?
A way of seeing this is to see $W$ as a vector bundle, so that $H^\ast(W,W-L)$ is the same thing as $H^\ast_{c}(W)$, the compactly supported cohomology. Now taking the cap-product with $[W_0]$ ressembles integrating along the fibre, and saying that $\tau_W$ is the unique class that restricts to 1 in $H^k_c(\mathbb{R}^k)$ amounts to say that the integration of $\tau_W$ along the fibers is the constant 1 function on $[L]$. Then juggling with Poincaré duality gives the answer.
For a more precise treatment, I recommend reading the chapter on the Thom isomorphism in Bott and Tu's excellent Differential Forms in Algebraic Topology.