\documentclass[12pt]{article}
\usepackage{amsthm,amssymb,amsmath}
\usepackage[all]{xy}


\begin{document}
\section{1}


\begin{equation}
\xymatrix{
                                                                               & \mathcal{U}_\alpha \ar[dr]^{I^\alpha}  &  \\
 \Sigma_g \ar[ur]^{i^\alpha} \ar[dr]_{i^\beta} &                                      & \mathcal{U}_\alpha \cup_\Sigma \mathcal{U}_\beta   \\
                                                                               &  \mathcal{U}_ \beta \ar[ur]^{I^ \beta}  &  }
\end{equation}
\begin{equation*}
\xymatrix{ & BSpin_n^\mathbb{C}  \ar[dd] \ar[dr]^-\alpha \\
 M \ar[ur]^-{ \tilde{f}} \ar[dr]_-f &  & B( SO_n \times U(1)) \ar[dl] \\
 & BSO(n) }
\end{equation*}

\newpage
--
\begin{equation}
\xymatrix{\mathbb{Z} \ar[d]|\approx \ar@{}[r]|\subset  & U(1) \ar[r] \ar[d] & U(1) \ar[d] \\
\mathbb{Z} \ar@{}[r]|\subset  & Spin_n^\mathbb{C} \ar[r] \ar[d] & SO_n\times U(1) \ar[d]\\
 &  SO_n \ar@{}[r] |\approx & SO_n } 
\end{equation}
\end{document}