I was reading the book "Differentiable Periodic Maps" by P.E. Conner (1979). I am stuck at the following problem given at the end of section 21:


Let $\xi\to V^n$ be a $k$-plane bundle over a closed n-manifold then $[\mathbb{R}P(\xi)]\_2=0$ if either $k=2$ or $n=1$.


I was able to solve the $k=2$ case just by using the expression of total Stiefel Whitney class for the projectivization of the bundle given at the starting of the section.
But I could not complete the second case using a similar approach.


Can anyone suggest an alternate approach?



Cobordism class of projectivization of a bundle
First, we can assume wlog that $\xi$ has an inner product.


Next, given a $2$-dimensional real vector space $U$ with inner product, let $QU$ be the space of self-adjoint endomorphisms of trace zero. Given a unit vector $u$, define $\phi(u)\in QU$ by $\phi(u)(v)=\sqrt{2}(\langle v,u\rangle u - \langle u,u\rangle v/2)$. This satisfies $\phi(-u)=\phi(u)$ and $\|\phi(u)\|=\|u\|^2$. It therefore induces a map from the projective space $PU$ to the unit circle $SQU$, which is a diffeomorphism.


Now suppose that $\xi$ is a $2$-dimensional real bundle over a base $V$. We can apply the above fibrewise to identify $P\xi$ with the circle bundle associated to $Q\xi$, which is the boundary of the corresponding disc bundle. Thus, $P\xi$ is nullbordant.


Now consider instead the case of a vector bundle $\xi$ (with inner product) of dimension $k$ over a closed $1$-manifold $V$. By dividing $V$ into path components, we reduce to the case where $V=S^1$. Here standard arguments identify the total space $E\xi$ with $(\mathbb{R}\times\mathbb{R}^k)/\sim$, where $(t,v)\sim(t+1,Av)$ for some fixed matrix $A\in O(k)$. The isomorphism type of $\xi$ depends only on the path-component of $A$ in $O(n)$. Because $-I$ acts as the identity on $\mathbb{R}P^{k-1}$, we also see that the diffeomorphism type of $P\xi$ depends only on the path component of $\pm A$ in $O(k)/\{\pm I\}$. If $k$ is odd then $O(k)/\{\pm I\}$ is connected so $P\xi\simeq S^1\times \mathbb{R}P^{k-1}$, which is the boundary of $B^2\times \mathbb{R}P^{k-1}$. The same applies if $k$ is even but $A$ lies in the base component.


The case where $A$ is not in the base component is less obvious. When $k=2$ we get the Klein bottle, which is diffeomorphic to the connected sum of two copies of $\mathbb{R}P^2$. This is nullbordant by the first part of this answer, or by general properties of connected sums. I think that there should be a direct geometric construction for $k>2$, but I don't see one just now.


If I remember rightly, there are various constructions like this in Stong's book on cobordism, but I don't have a copy.


