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\title{Topology}
\author{Last Updated: 2009}
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\subsection*{General Topology}
\noindent
Some basic ideas from general topology, while not tested explicitly, are regarded as prerequisites for qualification in topology and will be assumed during the qualifying exam. These ideas include: the definitions and basic facts about topological spaces, bases, neighborhoods, continuous functions and
homeomorphisms; connectedness, local connectedness
and path-connectedness; compactness, local compactness and compactness in Euclidean space; Hausdorff
spaces, normal spaces and metric spaces; the quotient topology and the product topology.
\subsection*{Algebraic Topology}
\begin{enumerate}
\renewcommand{\theenumi}{\alph{enumi}}
\item Elementary Homotopy and the Fundamental Group: Basic homotopy of maps, deformation
retracts, homotopy equivalences and homotopy type. The fundamental group and its main properties. Computation of the fundamental
group. The theory of covering spaces and its relation to the fundamental group.
\item Homology Theory: Construction and basic properties of singular homology theory including excision, the
homotopy property and exactness. The Eilenberg-Steenrod axioms. CW complexes and cellular homology theory.
Computation of homology groups. Applications of homology theory. Elementary cohomology theory.
\item Homological Algebra: Exact sequences, chain and cochain complexes, chain homotopy and the exact homology sequence of a short exact sequence of chain complexes. Introduction to categories
and functors.
%The functors $\mathrm{Tor}$ and $\mathrm{Ext}$ and the Universal Coefficient Theorems.
\end{enumerate}
\paragraph{Theorems:} There are two types of theorems: The first consists of theorems whose proofs are intricate and the student is expected to be able to state them and apply them, but is not expected to prove them. The second type consists of theorems which the student is expected to know how to prove. In the first type are: Van Kampen's Theorem, the existence of a universal cover, the excision property of singular homology theory, the isomorphism of singular and cellular homology.
In the second type are: Lifting theorems for covering spaces, calculation of the fundamental group of the circle,
the Mayer-Vietoris sequence, the Euler-Poincare formula,
the Brouwer
fixed point theorem.
Parts {\bf b} and {\bf c} are normally covered in detail in Math 114. Part {\bf a} is often covered in Math
74; it can also be learned by reading independently under the direction of the student's committee.
\subsection*{Differential Topology}
Smooth manifolds. Submanifolds, product manifolds, boundary of a manifold and examples of these.
The tangent space, cotangent space and the differential of
a smooth map. Embeddings, immersions, submersions and diffeomorphisms. Applications of partitions of unity. Vector bundles, vector fields and flows. Orientability. The
Lie derivative and the Lie bracket. Differential forms and operations on them. Integration on manifolds. Applications of the Stokes's Theorem to cohomology.
\paragraph{Theorems:} Theorems which the student is expected to state and apply: the inverse function
theorem, the existence of partitions of unity, the existence and uniqueness of flows of vector fields and
their properties, the general theorem of Stokes. Theorems which the student is expected to be
able to prove: the theorem on rank, the existence of a Riemann metric on a manifold, the theorem on
embedding of a closed manifold into $\mathbb R^n$.
The material on differential topology is generally covered in Math 124, which assumes as undergraduate
preparation a course on analysis on manifolds at the level of Spivak's ``Calculus on manifolds''. Students
who do not have this background should normally enroll during the first year in Math 73, which furnishes the
necessary prerequisites for Math 124.
\subsection*{References}
The student is not expected to read all the books on the list. The
major references are indicated with an asterisk. If in doubt, the student should
consult with a faculty member to determine which sources cover which
material.
\subsubsection*{General Topology}
\begin{enumerate}
\item Dugundji, \textit{Topology}
\item *Munkres, \textit{Topology}
\item Willard, \textit{General Topology}
\end{enumerate}
\subsubsection*{Algebraic Topology}
\begin{enumerate}
\item Fulton, \textit{Algebraic Topology: A First Course}
\item *Hatcher, \textit{Algebraic Topology}
\item *Massey, \textit{A Basic Course in 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 Guillemin and Pollak, \textit{Differential Topology}
\item *Lee \textit{Introduction to smooth manifolds.}
\item *Spivak, \textit{Calculus on Manifolds} (This should be read
first.)
\item Tu, \textit{An Introduction to Manifolds}
\end{enumerate}
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