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\def\ctln{\centerline}
\def\msk{\medskip}
\def\bsk{\bigskip}
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\def\hsk{\hskip.3in}
\def\ra{\rightarrow}
\def\ubr{\underbar}

\def\mt{{\mathcal T}}
\def\mb{{\mathcal B}}
\def\ms{{\mathcal S}}
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\def\bbz{{\mathbb Z}}
\def\bbq{{\mathbb Q}}
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\def\del{\partial}
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\ctln{\bf Math 872 Problem Set 9}

\msk

Starred (*) problems are due Thursday, April 23.


\begin{description}

\item{60.} Find the (singular) homology groups of the M\"obius band 
$M=[0,1]\times [0,1]/\{(0,t)\sim (1,1-t)\ :\ t\in [0,1]\}$, and compute
the inclusion-induced maps $i_*:H_k(\del M)\ra H_k(M)$.

\msk

\item{61.} Use the computations from problem \# 60 and the Mayer-Vietoris
sequence to compute the (reduced) homology groups of the Klein bottle, 
which can be viewed as two M\"obius bands with their boundary circles
glued together $K^2=M_1\cup_{\del M_1 = \del M_2} M_2$.

\msk

\item{(*)} 62. [Hatcher, p.158, \# 32] Show, using a Mayer-Vietoris sequence,
that for the suspension $\Sigma X = cX\cup_X cX$ of a space $X$ (the union of two cones
on $X$ glued together along the two copies of $X$) that $H_n(\Sigma X)\cong H_{n-1}(X)$
for every $n$.

\msk

\item{63.} Find an example of a pair of graphs $\Gamma_1,\Gamma_2$,
so that one is a finite sheeted covering of the other $p:\Gamma_1\ra\Gamma_2$,
but the induced map on homology $p_*:H_1(\Gamma_1)\ra H_1(\Gamma_2)$ is 
\underbar{not} injective. [Hint: it actually almost never \underbar{is}
injective...]

\msk

\item{64.} Show that the inclusion map $\iota$, thought of as a map of pairs 
$\iota:(D^n,\del D^n)\ra (D^n,D^n\setminus \{0\})$, is a homotopy
equivalence both of the spaces and of the subspaces, but is \u{not} a 
{\it homotopy equivalence of pairs}, that is, there is \underbar{not} a map 

\item{\spc} $f:(D^n,D^n\setminus \{0\})\ra (D^n,\del D^n)$ so that $f\circ \iota$ and $\iota\circ f$
are both homotopic, as maps of pairs, to the identity maps.

\msk

\item{(*)} 65. Show that $H_i(X\times S^n)\cong H_i(X)\oplus H_{i-n}(X)$ for every $i$;
here $H_i(X)=0$ if $i<0$.

\ssk

\item{\spc} [One approach: show that $H_i(X\times S^n)\cong H_i(X)\oplus H_i(X\times S^n,X\times D^n_+)$,
where $D^n_+$ is the upper hemisphere of $S^n$ 
(Problem \#57 will help), and that $H_i(X\times S^n,X\times D^n_+)\cong H_{i-1}(X\times S^{n-1},X\times D^{n-1}_+)$
by excision and the long exact sequence of the triple $(X\times D^{n-1}_+,X\times S^{n-1},X\times D^{n}_-)$.]

\ssk

\item{\spc} [Hatcher, p.158, \# 36 gives a different approach.]

\msk

\item{(*)} 66. Show by induction on $n$ that, for the {\it $n$-torus}, 
the Cartesian product ${\mathbb T}^n = (S^1)^n = S^1\times\cdots\times S^1$ of $n$ copies of the circle,
we have for 
every $k$ that $H_k({\mathbb T}^n)$
is isomorphic to $\bbz^{N(n,k)}$ for some $N(n,k)\in\bbz$, and find a formula for 
$N(n,k)$. 

\msk

\item{\spc} [Working out the homology of ${\mathbb T}^n$ for small values of $n$ and looking for a pattern 
will probably help. Note that this says \underbar{find}, not \underbar{prove}, although a familiar formula
for some familiar numbers probably constitutes a fine proof....]

\msk

\item{67.} Show that, for any collection of finitely generated abelian groups $G_1,\ldots ,G_n$,
there is a path-connected space $X$ with $\wtih_i(X)\cong G_i$ for all $i=1,\ldots,n$ and
$\wtih_i(X)=0$ for all other $i$.


\end{description}
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