**Recap: bordism group**

An *oriented singular $n$-manifold* in $X$is a map $f:M^n\to X$ where $M$ is a finite disjoint union of $n$-dimensional smooth manifolds.
The empty set is an admissible oriented singular $n$-manifold.

Two oriented singular manifolds $f_i:M_i^n\to X$ $i=1,2$ are *bordant* in $X$ if exists a map $F:W^{n+1}\to X$ ($W^{n+1}$ smooth oriented) such that $M_1\sqcup -M_2\subset \partial W$ (orientation is induced), $F|_{M_1\sqcup -M_2} \equiv f_1\sqcup f_2$.

This defines an equivalence relation and the equivalence classes $[M^n,f]$ of oriented singular $n$-manifolds up to bordism define the *oriented $n$-bordism group of $X$* denoted $\Omega_n(X)$.
The (abelian) group structure is given by the operation $[M_1,f_1]+ [M_2,f_2] = [M_1\sqcup M_2, f_1\sqcup f_2]$.
If we forget all the orientation requirement above, we get the (unoriented) n-bordism group of $X$ denoted as $\eta_n(X)$.
With $\Omega_n$ is intended $\Omega_n(pt)$ i.e. $X={*}.$

**Question 1:Thom's theorem**

Thom discovered an isomorphism of groups: $$\phi:\Omega_n \to \pi_{n+k}(MSO_k)$$ where $k>n+1$ and $MSO_k$ is the Thom space of the universal $SO_k$ bundle $ESO_k\to BSO_k$. Roughly $\phi([M])$ is constructed from an embedding $M\to \mathbb{R}^{n+k}$, taking the classifying map for the normal bundle $\tau_\nu:M\to BSO_k$ and extending it to a map $\mathbb{R}^{n+k}\to ESO_k$ that is constant in a neighbourhood of $\infty$. This yelds a map $\mathbb{S}^{n+k}\to MSO_k$.

Question 1: This works for $\Omega_n(pt)$, what kind of isomorphism do we have for $\Omega_n(X)$? How is it defined? References?

I hope there is a geometric definition similar to the one I sketched above (that clarifies the relation with the normal bundle of an embedding for example).

**Question 2: Thom-Pontryagin theorem on characteristic numbers**

We have the following theorem: $[M]= 0 \in \Omega_n$, i.e. $M$ bounds an oriented manifold iff all the Stiefe-Whitney numbers and all the Pontryagin numbers of $M$ vanish: $$\langle w_{i_1}(M)\cup \cdots \cup w_{i_{k'}}(M), [M]\rangle = 0\ \ \ \ \ \langle p_{i_1}(M)\cup \cdots \cup p_{i_k}(M), [M]\rangle = 0 $$

For $\eta_n(X)$ things are similar: $[M,f]= 0\in \eta_n(X)$ iff $$\langle w_{i_1}(M)\cup \cdots \cup w_{i_k}(M)\cup f^*(h), [M]\rangle = 0 \in \mathbb{Z}_2\ \ \ \forall h \in H^*(X,\mathbb{Z}_2) \text{ and SW classes}$$ i.e. all SW numbers of the map $f$ vanish.

Curiously, for $\Omega_n(X)$ the only statement I've found requires an additional assumption: if all torsion classes in $H_*(X,\mathbb{Z})$ have order exactly $2$, then $[M,f]= 0 \in \Omega_n(X)$ iff the SW and Pontryagin numbers of $f$ vanish. Also usually it's required $X$ to be (h-equivalent to) a finite CW complex.

Question 2a: how to prove the statement for $\eta(X)$ and $\Omega(X)$? What are some good references? Is there any

geometric interpretationof why the vanishing of the characteristic numbers entails the extendability of the map $f$?Question 2b: Can we get rid of the assumption on the torsion of $H_*(X)?$ Why is it needed?

Question 3: What important achievements regarding bordism theory has been reached after the 60s?

Note: the reference I have used here is *Conner P.E and E.E. Floyd: Differentiable Periodic Maps. Bull. Am. Math. Soc. 68, 76-86 (1962)*.