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UF MTG 4303/5317\\
Introduction to Topology 2\\
Spring 2024
\begin{center}
{\large \bf Homework 6}\\
Due Friday, April 19, anytime, on Canvas
\end{center}

{\bf Reading.}
Munkres \S73--77.


{\bf Problems.}
\begin{enumerate}

\item \S74 \#1.
Find a presentation for the fundamental group of $\RP^2 \# T$.

\item \S75 \#1.
Calculate $H_1(\RP^2 \# T)$.
Assuming that the list of compact surfaces given in Theorem~75.5 is a complete list, to which of these surfaces is $\RP^2 \# T$ homeomorphic?

\item \S75 \#3.
Let $X$ be the quotient space obtained from an 8-sided polygonal region $P$ by pasting its edges together according to the labelling scheme $acadbcb^{-1}d$.

(a) Check that all vertices of $P$ are mapped to the same point of the quotient space $X$ by the pasting map.

(b) Calculate $H_1(X)$.

(c) Assuming $X$ is homeomorphic to one of the surfaces given in Theorem~75.5 (which it is), which surface is it?

\end{enumerate}



\vfill

{\bf Recommend Problems (not to turn in).}
\begin{itemize}

\item By cutting and pasting, show that the connected sum of a torus and a projective plane is homeomorphic to the connected sum of a Klein bottle and a projective plane.

\item \S73 \#2

\item \S74 \#6

\end{itemize}

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