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I am currently trying to read Quillen's Cobordism paper - On the formal group laws of unoriented and complex cobordism theory Bull. Amer. Math. Soc, 75 (1969) pp. 1293–1298. Still on the first page though so the emphasis is on the trying :P.

A map of manifolds $f: Z \to X$ can be give an complex orientation in the following way: If $\dim (f) = \dim T_z Z - \dim T_{f(z)}X$ is even then an orientation for $f$ is an equivalence class of factorizations $Z \to E \to X$ where $E$ is a complex bundle over $X$ and the map $Z \to E$ is an embedding with a complex structure on the normal bundle.

I am assuming that the definition of the normal bundle is: $$ \upsilon_{i(z)}=T_{i(z)}E/i_*(T_zZ) $$ above each point of $Z$. I am taking everything to be real the previous equation. The dimension of this is: $\dim E - \dim Z$. Considering the whole bundle over $Z \subset E$ the dimension of the normal bundle is $ \upsilon_iZ =\dim E - \dim Z + \dim Z =\dim E$.

Obviously this is where I am mistaken. This would mean that the requirement that the function be even dimensional is entirely arbitrary.

I cannot think of anything nice to fix this.

Two factorizations are equivalent if there is an isotopy between them that preserves the complex structure. This means if $E$ and $E'$ are two factorisations and $E,E'$ are subbundles of $E''$ then there is an embedding endowed with a complex structure on the normal bundle $i'': Z \times I \to E \times I$ that restricts to $i$ and $i'$ with the same complex structure on either end.

Quillen says $i'':X \times I \to E \times I$ but that does not make sense. I haven't gotten to this bit yet.

For $\dim f$ odd consider $(f, \epsilon): Z \to X \times \mathbb{R}$ where $\epsilon(z)=0$. Then for generalised maps split it up.


This is a different way to look at it given to me by a professor in my department. I have added little bit of explanation to what he gave me.

The map $\pi$ is a submersion so we can pull back $TX$ to $TE$. This map has a kernel, lets call it $T_FE$, which is fiberwise, over $X$, isomorphic to $TF$. This gives a short exact sequence of bundles: $$ 0 \to T_FE \to TE \stackrel{T_\pi}{\to} \pi^*TX \to 0 $$ (This whole thing is a little like lifting $F \to E \to X$ to a SES of bundles in $TE$.) This SES splits $TE=\pi^*TX+T_FE$

With everything as in Quillen (just like above), now look at the normal bundle $$\begin{align} v_i & = TE - i_*TZ \\ &= \pi^*TX + T_FE - i_*TZ \end{align} $$ Which has the dimension work out in both the even and odd cases for $f$ and to me at least feels better.

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The paper to which Sven is referring is Elementary proofs of some results of cobordism theory using Steenrod operations in Advances in Mathematics 7, 29-56 (1971) – Vitaly Lorman Sep 5 '11 at 23:29
We want a $\text{GL}_n(\mathbb{C})$ structure on the normal bundle $v_i$ (i.e. a continuously varying complex structure on the fibers of $v_i$). Not a complex structure on the manifold $v_i$. I have answered my own question before so I am just going to add this as a comment. – Sven Sep 7 '11 at 18:49

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