# Additional structure on complex cobordism?

So I want to understand does the following addition give stricter condition for equivalence of two stably-complex manifolds?

Assume the following relation: let two stably-complex $$n$$-manifolds $$M^n,N^n$$ be complex' cobordant if

1. They are a boundary of stably-complex $$n+1$$-manifold $$W^{n+1}$$ such that stably-complex structure is induced from $$W$$;
2. Any automorphism of $$M$$ that preserves stably-complex structure on M could be extended to $$W$$ (same condition for $$N$$). In more detail it means that for any such automorphism $$f$$ pullback of a bundle equivalent to tangent bundle $$f^*(TM\oplus \epsilon^k)\cong TM\oplus \epsilon^k$$ is isomorphic to itself as complex bundles. (@MichaelAlbanese thank you for clarification)

My question: is this relation something new as compared to standard complex cobordism definition (i.e. condition 1)?

• In $2$, what do you mean by an automorphism of $M$ that preserves the stably-complex structure on $M$? Do you mean a diffeomorphism $f : M \to M$ such that $f^*(TM\oplus\varepsilon^k) \cong TM\oplus\varepsilon^k$ as complex bundles? Dec 31, 2019 at 22:05
• @MichaelAlbanese Yes, I think so. Dec 31, 2019 at 23:04

This is not the same as complex cobordism. For example, $$S^6$$ with the stably-complex structure given by $$TS^6\oplus\varepsilon^2_{\mathbb{R}} \cong \varepsilon^4_{\mathbb{C}}$$ is complex nullcobordant as it is the boundary of the closed ball $$B^7$$ with the stably-complex structure $$TB^7\oplus\varepsilon^1_{\mathbb{R}} \cong \varepsilon^4_{\mathbb{C}}$$. For any diffeomorphism $$f : S^6\to S^6$$, $$f^*\varepsilon^4_{\mathbb{C}} \cong \varepsilon^4_{\mathbb{C}}$$ and hence is an automorphism in the sense you suggested. However, not every diffeomorphism of $$S^6$$ extends to a diffeomorphism $$B^7 \to B^7$$. This is related to the existence of exotic spheres in dimension $$7$$.