Concerning your first question, I think a good place to learn about it is Markus Banagl's book "Topological invariants of stratified spaces" (chapter 6) or have a look at Paul Siegel's paper "$KO$-homology at odd primes" Amer. J. Math. 105 (1983), 1067-1105 (in particular the appendix where Sullivan's approach is explained).

In Siegel's paper you have a bordism theory of some singular spaces called Witt spaces. Such spaces carry rational L-classes and this bordism theory when tensored by $\mathbb{Z}[1/2]$ is naturally isomorphic to $ko_*(-)\otimes \mathbb{Z}[1/2]$.

Thus for any Witt space in particular for smooth manifolds you have a fundamental class in Witt bordism, when tensored over the rational you get Goresky-McPherson L-classes (obtained thanks to intersection cohomology) and thus in the smooth case "ordinary" L-classes. Then use the natural isomorphism between this bordism theory and $ko$ at odd-primes to get your formula.

Concerning your second question you can have a look at theorem 11.1 "Multiplicativity of the L-theory fundamental class" of this paper: "The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture" by Markus Banagl, Gerd Laures and Jim McClure (available on Banagl's homepage).

Concerning your first question, I think a good place to learn about it is Markus Banagl's book "Topological invariants of stratified spaces" (chapter 6) or have a look at Paul Siegel's paper "$KO$-homology at odd primes" Amer. J. Math. 105 (1983), 1067-1105 (in particular the appendix where Sullivan's approach is explained).

In Siegel's paper you have a bordism theory $\Omega^{Witt}_*(-)$ build from cobordisms of singular spaces called Witt spaces. A Witt space $X$ carries rational homology L-classes $l_i\in H_{dim(X)-4i}(X,\mathbb{Q})$. This homology $L$-classes were defined by Goresky and McPherson in this singular context extending the homology $L$-classes of a manifold $M$ which are poincaré dual to Hirzebruch $L$-classes $L_i\in H^{4i}(M,\mathbb{Q})$: $$l_i(M)=L_i(M)\cap [M]$$

• In Witt bordism any Witt space has a fundamental class $[X]\in \Omega^{Witt}_{dim(X)}(X)$ represented by $id:X\rightarrow X$.

• Moreover we have a natural transformation $$\Phi:\Omega^{Witt}_k(X)\rightarrow \oplus_{i} H_{k-4i}(X,\mathbb{Q})$$such that for a Witt space we have $\Phi([X])=l_0+l_1+\ldots$, for a manifold we get that $\Phi([M])=L(M)\cap [M]$.

• And this bordism theory when tensored by $\mathbb{Z}[1/2]$ is naturally isomorphic to $ko_*(-)\otimes \mathbb{Z}[1/2]$. Siegel used Sullivan's construction of $ko_*(-)\otimes \mathbb{Z}[1/2]$ to get a natural transformation $$\mu: \Omega^{Witt}_*(-)\rightarrow ko_*(-)\otimes \mathbb{Z}[1/2].$$ such that $\mu([M])=\Delta_M.$

Concerning your second question you can have a look at theorem 11.1 "Multiplicativity of the L-theory fundamental class" of this paper: "The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture" by Markus Banagl, Gerd Laures and Jim McClure (available on Banagl's homepage).

Concerning your first question, I think a good place to learn about it is Markus Banagl's book "Topological invariants of stratified spaces" (chapter 6) or have a look at Paul Siegel's paper "$KO$-homology at odd primes" Amer. J. Math. 105 (1983), 1067-1105 (in particular the appendix where Sullivan's approach is explained).

In Siegel's paper you have a bordism theory of some singular spaces called Witt spaces. Such spaces carry rational L-classes and this bordism when tensored by $\mathbb{Z}[1/2]$ is naturally isomorphic to $ko_*(-)\otimes \mathbb{Z}[1/2]$.

Thus for any Witt space in particular for smooth manifolds you have a fundamental class in Witt bordism, when tensored over the rational you get Goresky-McPherson L-classes (obtained thanks to intersection cohomology) and thus in the smooth case "ordinary" L-classes. Then use the natural isomorphism between this bordism theory and $ko$ at odd-primes to get your formula.

Concerning your second question you can have a look at theorem 11.1 "Multiplicativity of the L-theory fundamental class" of this paper: "The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture" by Markus Banagl, Gerd Laures and Jim McClure (available on Banagl's homepage).

Concerning your first question, I think a good place to learn about it is Markus Banagl's book "Topological invariants of stratified spaces" (chapter 6) or have a look at Paul Siegel's paper "$KO$-homology at odd primes" Amer. J. Math. 105 (1983), 1067-1105 (in particular the appendix where Sullivan's approach is explained).

In Siegel's paper you have a bordism theory of some singular spaces called Witt spaces. Such spaces carry rational L-classes and this bordism theory when tensored by $\mathbb{Z}[1/2]$ is naturally isomorphic to $ko_*(-)\otimes \mathbb{Z}[1/2]$.

Thus for any Witt space in particular for smooth manifolds you have a fundamental class in Witt bordism, when tensored over the rational you get Goresky-McPherson L-classes (obtained thanks to intersection cohomology) and thus in the smooth case "ordinary" L-classes. Then use the natural isomorphism between this bordism theory and $ko$ at odd-primes to get your formula.

Concerning your second question you can have a look at theorem 11.1 "Multiplicativity of the L-theory fundamental class" of this paper: "The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture" by Markus Banagl, Gerd Laures and Jim McClure (available on Banagl's homepage).