Title of Paper :Secondary Brown-Kervaire Quadratic forms and $\pi$-manifolds
Author(s) :Fuquan Fang and Jianzhong Pan
AMS Classification numbers
Addresses of Authors:
Fuquan Fang
Nankai Institute of Mathematics, Nankai University,
Tianjin 300071, P.R.C
email:ffang@sun.nankai.edu.cn
and
Jianzhong Pan
Institute of Math.,Academia Sinica ,Beijing 100080 ,China
email:pjz@math03.math.ac.cn
Text of Abstract (try for 20 lines or less) :
In this paper we assert that for each $\Phi$-oriented
$2n$-manifold (c.f : Definition 1.1) $M$ where $n\ge 4$ and
$n\ne 3(mod 4)$, there is a well-defined quadratic function
$\phi_M: H^{n-1}(M, \Z_4)\to \Q/\Z$, we call the secondary Brown-Kervaire quadratic forms, so that
\begin{itemize}
\item{ $\phi _{M}(x+y)=\phi _{M}(x)+\phi _{M}(y)+j(x\cup Sq^2y)[M]$},
\item{ the Witt class of $\phi _M$ is a homotopy invariant, if
the Wu class $ v_{n+2-2^i}(\nu _M)=0$ for all $i$.}
\end{itemize}
where $j: \Z_2 \to \Q/\Z$ is the inclusion homomorphism and
$\nu _M$ the stable normal bundle of $M$.
Among the applications we obtain a complete classification of
$(n-2)$-connected $2n$-dimensional $\pi$-manifolds up to
homeomorphism and homotopy equivalence, where $n\geq 4$ and
$n+2\neq 2^i$ for any $i$. In particular, we prove that the
homotopy type of such manifolds determine their homeomorphism
type.