On geometric representation of $\mathbb{L}$-homology classes

In this chapter we give a geometric representation of $H_n(B;\mathbb{L})$ classes, where $\mathbb{L}$ is the $4$-periodic surgery spectrum, by establishing a relationship between the normal cobordism classes ${\mathcal{N}}_n^H(B,\mathrm{\partial })$ and the $n$-th $\mathbb{L}$-homology of $B$, representing the elements of $H_n(B;\mathbb{L})$ by normal degree one maps with a reference map to $B$. More precisely, we prove that for every $n\ge 6$ and every finite complex $B,$ there exists a map $\mathrm{\Gamma }:H_n(B;\mathbb{L})⟶{\mathcal{N}}_n^H(B,\mathrm{\partial })\mathrm{.}$