# Two definitions of framed manifolds/cobordism

One notion of framed cobordism is that $\Omega_{fr}^n(X)$ is a certain equivalence class of $n$-manifolds $M^n$ in $X$ with a trivialization of the normal bundle $N M^n\subseteq TX$. For compact manifolds, there is the correspondence $\Omega^n_{fr}(X)\simeq [X,S^{m-n}]$.

Another definition I have seen is free of the ambiend manifold $X$: a framing on $M^n$ is the trivialization of the stable tangent bundle $\tau_M\oplus\epsilon_M^1$ and a framed cobordism class $\Omega_{fr}^n$ is defined without $X$.

Are these two different mathematical objects, or is there some simple connection, such as $\Omega_{fr}^n\simeq \lim_{N\to\infty} \Omega_{fr}^n(S^N)$ in "some sense"? If there is a clear connection, why?

According to the definition I have seen $\Omega^n_{fr}(X)$ is the set of cobordism classes of continuous functions $f\colon M^n\to X$. The map $f_1\colon M\to X$ is said to be cobordant to $f_2\colon N\to X$ if there is a cobounding $(n+1)$-manifold $W$ so that the maps $f_1$ and $f_2$ can be extended over all of $W$ to a map to $X$. I'm spelling this out because your post makes it sound like these manifolds have to be embedded in $X$. Also I am suppressing framing information. Now the bordism groups $\Omega^n_{fr}$ which are defined without reference to an ambient manifold are actually just the bordism groups of a point $\Omega^n_{fr}=\Omega^n_{fr}(*)$. There is an obvious homomorphism $\Omega^n_{fr}(X)\to \Omega^n_{fr}(*)$.
• I know this definition, but had something different in mind -- I mean the notion of "framed cobordism" as defined in Milnors Topology from differentialbe viewpoint, i.e. a submanifolds of $X$ with a framing of the normal bundle. Is there a conection to $\Omega_{fr}^n(*)$ and this? Commented Jul 16, 2014 at 23:06
• In this source it is denoted by $\Omega^{fr}_{(m-n);X}$.. Commented Jul 16, 2014 at 23:10
• I mean, I still can't see the conection between the Pontryagin-Thom construction as explained in Milnor (ambient framed cobordisms in $X$) and "your" $\Omega_{fr}^n(*)$ and the notion of stable tangent bundle. Probably I should study more; could you recommend me an understandable source where all this is explained? Commented Jul 16, 2014 at 23:12