Obstruction Theory in Algebra and Topology: A Homotopy Perspective

Abstract

Based on the Homotopy theorem of Madhav V. Nori on smooth vector bundles and his Homotopy question on algebraic vector bundles \cite{M2}, we develop the theories of topological and algebraic obstructions as follows.\bE \item Let $M$ be a \textit{smooth manifold} of \textit{dimension} $d$ and ${\SV}$ be a \textit{smooth vector bundle} of \textit{rank} $n$ over $M$. We define an \textit{obstruction set} $\pi_0\left({\mathcal LO}({\SV})\right)$, to be called \textbf{Nori homotopy set}, and an \textit{obstruction class} $\varepsilon({\SV}) \in\pi_0\left({\mathcal LO}({\SV})\right) $. Then, if $2n\geq d+3$, we prove that $$ \varepsilon(\SV)=neutral \Llra {\SV}\cong {\SV}_1\oplus {\BR} $$ \item Let $A$ be an \textit{essentially smooth ring} of \textit{dimension} $d$ over an infinite \textit{perfect field} $k$, with $1/2\in k$, and $P$ be a \textit{projective $A$-module} with $rank(P)=n$. We define a similar \textit{obstruction set} as above $\pi_0\left({\mathcal LO}(P)\right)$, and an\textit{obstruction class} $\varepsilon(P) \in \pi_0\left({\mathcal LO}(P)\right)$. Then, if $2n\geq d+3$, we prove that $$ \varepsilon(P)=neutral \Llra P\cong Q\oplus A $$ \item Further, for \textit{real smooth affine schemes} $X=\spec{A}$, we reconcile these two theories, as follows. Let $M$ be the \textit{manifold of real points} in $X$. Let $P$ be a projective $A$-module, with $rank(P)=n$. Let ${\SV}_{Top}(P)$ be the smooth vector bundle over $M$ induced by $P$. Then, there is a natural map $$ \pi_0\left({\mathcal LO}(P)\right) \lra \pi_0\left({\mathcal LO}({\SV}_{Top}^{\star}(P))\right) $$ where ${\SV}_{Top}^{\star}(P)$ denote the dual bundle. \eE