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\title{Topology}
\author{Last Updated: January 1994}
\date{}

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The candidate should know the following definitions and theorems and
be able to apply them to draw conclusions about specific topological
spaces and continuous maps.

\subsection*{Point Set Theory}

\paragraph{Definitions:}
Hausdorff, regular, and normal spaces.  Connected, locally connected
and path connected spaces.  Compact, locally compact and para-compact
spaces.  Metric spaces, uniform continuity, and the compact-open
topology for function spaces.  Products of spaces and quotients of
spaces.  Embeddings and homeomorphisms.

\paragraph{Key Ideas and Theorems:}
Properties preserved by continuous maps, products, quotients and
subspaces.  Tychonoff's Theorem, Urysohn's Lemma and the Heine-Borel
Theorem.

\subsection*{Algebraic Topology:}

\begin{enumerate}
\renewcommand{\theenumi}{\alph{enumi}} 
\item Basic Homotopy.  Homotopy of maps, retracts, deforma-tion
  retracts, homotopy type.  The fundamental group and covering spaces.
  Compu-tation of the fundamental group (Van Kampen's Theorem and the
  edge-path group).
\item Homology Theory.  Definitions and basic properties of singular
  homology theory: the Eilenberg-Steenrod axioms.  Computation of
  homology groups using $CW$-complexes.  Basic cohomology theory.
\item The Algebra of Topology: Exact sequences, chain and cochain
  complexes, chain homotopy, introduction to categories and functors,
  the functors tensor, $\mathrm{Hom}$, $\mathrm{Tor}$ and
  $\mathrm{Ext}$.  The universal coefficient theorems.
\end{enumerate}

\paragraph{Theorems:}
The Mayer-Vietoris sequence, the Euler-Poincare formula, the Brouwer
fixed point theorem.

\subsection*{Differential Topology}

Definitions, basic properties, and examples of the following: Smooth
manifolds and smooth maps.  The tangent space and the differential of
a smooth map.  Embeddings, immersions diffeomorphisms.  Submanifolds
and product manifolds.  The inverse image of a regular value.
Partitions of unity.  Orientability.  Vector fields, integral curves
and flows, the bracket operation.  Definitions of Lie group and Lie
algebra, examples of matrix Lie Groups.  Differential Forms.  Exterior
differentiation and integration of differential forms.  Riemannian
metrics.

\paragraph{Theorems:}
(In each case the statement and simple applications but not proofs are
required.)  The inverse function theorem for manifolds, Sard's
Theorem, Stokes' Theorem.

\subsection*{References}

The student is not expected to read all the books on the list.  The
major references are indicated with an asterisk.  The student should
consult with a faculty member to determine which sources cover which
material.

\subsubsection*{Point Set Topology}

\begin{enumerate}
\item Munkres, \textit{Topology: A First Course}
\item Willard, \textit{General Topology}
\end{enumerate}

\subsubsection*{Basic Homotopy}

\begin{enumerate}
\item Massey, \textit{Algebraic Topology: An Introduction } (Chapters
  2, 4, 5)
\item Munkres, \textit{Topology: A First Course} (Chapter 8)
\end{enumerate}

\subsubsection*{Homology Theory and the Algebra of Topology}

\begin{enumerate}
\item Massey, \textit{Singular Homology Theory}
\item Rotman, \textit{An Introduction to Algebraic Topology}
\item Spanier, \textit{Algebraic Topology}
\end{enumerate}

\subsubsection*{Differential Topology}

\begin{enumerate}
\item Boothby, \textit{An Introduction to Differentiable Manifolds and
    Riemannian Geometry}
\item Bott and Tu, \textit{Differential Forms in Algebraic Topology}
\item Guillemin and Pollak, \textit{Differential Topology}
\item Spivak, \textit{Calculus on Manifolds} (This should be read
  first.)
\item Spivak, \textit{A Comprehensive Introduction to Differential
    Geometry}, Volume 1
\end{enumerate}

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