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\parbox[b]{2in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}:
  \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \href{http://www.math.toronto.edu/~drorbn/classes/\#2526}{2025-26}:
  \newline \href{http://drorbn.net/26-1301}{MAT 1301 Algebraic Topology}:
}~{\LARGE\bf Homework Assignment 7}
\hfill\parbox[b]{1.5in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill Due March 27 at 11:59pm
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\noindent\includegraphics[width=\linewidth]{../25-1301-AlgebraicTopology/HockingYoungFig4-12pp177+RedLoop.png}

\noindent{\bf Problem 1.} On page 177 of their topology textbook, Hocking and Young display an embedded interval in $\bbR^3$ whose complement $X$ is not simply connected. I took the liberty of adding a little red circle to the picture, which represents a class $\gamma$ in $H_1(X)$. But by a theorem from class, $H_1(X)=0$, so $\gamma$ must be the boundary of some 2D object $\beta$ in $X$. Draw it!

If you need scratch paper, I've left multiple paper copies of the above picture in an envelope near my office door (Bahen 6178). Feel free to take some (yet leave some for others).

Not for credit, ponder the following: Everything we did in class was in-principle constructive: the prizm construction, barycentric subdivisions, the long exact sequence of a short exact sequence, etc. How exactly did these relatively benign constructions ``discover'' the relatively sophisticated surface that you must have discovered when you answered this problem?

\parpic[r]{\scalebox{0.6}{\parbox{2.8in}{
  \[ \includegraphics[width=2.8in]{../25-1301-AlgebraicTopology/BabyMobile.jpg}\] 
  \tiny\url{https://www.canadiantire.ca/en/pdp/tiny-love-tiny-princess-soothe-n-groove-mobile-2744075p.html}
}}}
\noindent{\bf Problem 2.} Search your memories and I'm sure you can go back to these times when you were lying in a crib looking up at a baby mobile, a lovely toy such as in the picture on the right. Little did you expect that twenty-something years later baby mobiles will come back to haunt you in an algebraic topology homework assignment.

If $(X_i,x_i)$ are connected based topological spaces for $i=1,\ldots,n$, we let $BM((X_i,x_i))$ be the topological space obtained by connecting each of the $X_i$'s by a string to some central point $y_0$. In formulas, let $Y$ be a star-shaped tree with centre $y_0$ and leafs $y_1,\ldots,y_n$, and let
\[ BM((X_i,x_i))\coloneqq \left(Y\sqcup X_1\sqcup\dots\sqcup X_n\right)/(\forall i\ x_i\sim y_i). \]

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Using the Mayer-Vietoris sequence and/or whatever else we studied, compute the homology of $BM((X_i,x_i))$ in terms of the homologies of the individual $X_i$'s.

\noindent{\bf Problem 3.} The suspension $\Sigma X$ of a topological space $X$ is $X$ multiplied by an interval, with the top and the bottom sides crashed into points $S$ and $N$ (that are not in $X$):
\[ \Sigma X\coloneqq \left(X\times[-1,1]\sqcup\{S,N\}\right) / (\forall x\ (x,1)\sim S,\,(x,-1)\sim N). \]
\begin{enumerate}
\item (0 points) Identify the colonial roots of the discomfort you felt regarding the choice of directions, signs, and poles used in this definition.
\item Using the same tools as in Problem 2, compute the homology of $\Sigma X$ in terms of the homology of $X$.
\end{enumerate}

\needspace{20mm}
\noindent{\bf Problem 4.}
\begin{enumerate}
\item Compute the homology groups of the torus $T^2=S^1\times S^1$.
\item (Hatcher's problem 28a on page 157). Use the Mayer-Vietoris sequence to compute the homology groups of the space obtained from a torus $T^2$ by attaching a M\"obius band via a homeomorphism from the boundary circle of the M\"obius band to the circle $S^1\times
    \{x_0\}$ in the torus.
\end{enumerate}

\noindent{\bf Problem 5.}
\begin{enumerate}
\item Formulate and prove a naturality property for the Mayer-Vietoris sequence. Your property must be at least strong enough to answer part 2 of this question.
\item Use part 1 of this question to prove that if $f\colon S^n\to S^n$ then $\deg(f)=\deg(\Sigma f)$ where $\Sigma$ is the suspension functor, mentioned previously both in class and in HW7. 
\end{enumerate}

\vskip 2mm\noindent{\bf Problem 6.} Suppose $n$ is even.
\begin{enumerate}
\item Show that for any continuous map $f\colon S^n\to S^n$ there is a point $x$ such that $f(x)=\pm x$.
\item Show that any continuous map $f\colon\bbR P^n\to\bbR P^n$ has a fixed point.
\end{enumerate}

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