# 09-15 Notes

\theoremstyle{plain} \newtheorem{thm}{Theorem}%[section] \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{cor}{Corollary} \newtheorem*{nonumthm}{Theorem} \newtheorem*{nonumprop}{Proposition}

\theoremstyle{definition} \newtheorem{definition}[equation]{Definition} \newtheorem*{nonumdef}{Definition} \newtheorem*{nonumrem}{Remark} \newtheorem{example}[equation]{Example} \newtheorem*{nonumex}{Example}

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Here we have still fixed a manifold $X$, and principal $G$-bundle $\pi:P\to X$ with associated vector bundle $E\to X$.

We need to understand the local topology of the space $\B^p_k=\A^p_k/\G^p_{k+1}$. The best hope we could have is