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\centerline{\Head Mathematics 205B, Winter 2018, Assignment 1}
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This will be due on {\bf Monday, March 19, 2018, at 9:00 A.M, } which 
is the beginning of the final examination.  If you wish to use 
some version of \TeX ~in writing up your answers, please feel free 
to do so.  The plain \TeX ~file for this document is in the course directory.
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{\sl You must show the work behind or reasons for your answers. }
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\noindent
{\bf 1. }\quad 
$(a)$ 
Consider a hollow octahedron in 3-space 
with vertices $A$, $B$, $C$, $D$, $E$, $F$ 
and 2-simplex faces listed below (take the usual alphabetical ordering for 
the vertices):
$$ACD~,~ADE~,~AEF~,~ACF~,~BCD~,~BDE~,~BEF~,~BCF$$ 
(It is highly recommended that you sketch this figure or construct a cardboard 
model!).  It is known that 
this space is homeomorphic to $S^2$ and hence its 2-dimensional 
homology is isomorphic to $\bbZ$.  An explicit generator for the 
simplicial homology group is given by a linear combination of the 
2-simplices such that the coefficient of each generator is $\pm\,1$.  
Find the generator for which the coefficient of $ACD$ is $+1$.
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$(b)$ 
The boundary of a triangular prism can be described as a polyhedron 
with vertices $A$, $B$, $C$, $D$, $E$, $F$ and 2-simplex faces listed 
below (the conventions and suggestions of $(a)$ also apply here):
$$ABC~,~DEF~,~ADE~,~ABE~,~BEF~,~BCF~,~CDF~,~ACD$$ 
Once again the 2-dimensional 
homology is isomorphic to $\bbZ$.  An explicit generator for the 
simplicial homology group is given by a linear combination of the 
2-simplices such that the coefficient of each generator is $\pm\,1$.  
Find the generator for which the coefficient of $DEF$ is $+1$.
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{\bf 2. }\quad 
$(a)$ 
Let $\bbR^n_+\subset \bbR^n$ be the set of points whose first coordinates 
are nonnegative, and let $W$ be the open subset of points whose first 
coordinate is positive.  By results in the lectures, if $p\in W$ then 
$$H_q(\bbR^n_+,\bbR^n_+-\{p\})\cong H_q(W,W-\{p\})\cong 
H_q(\bbR^n,\bbR^n-\{p\})~\cong~\bbZ~ (q=n)~,\quad 0~({\rm otherwise})~.$$
Prove that if $p\in\{0\}\times \bbR^{n-1}\subset \bbR^n_+$, then  
$H_q(\bbR^n_+,\bbR^n_+-\{p\})\cong 0$ for all $q$.  [{\it Hint:}  Why is the 
pair $(\{1\}\times \bbR^{n-1},\{1\}\times \bbR^{n-1})$ a strong deformation 
retract of the pair in the preceding sentence, 
and why is $H_q(X,X)$ always zero?  How does the long 
exact homology sequence of a pair imply the latter?]
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$(b)$ 
Prove the following generalization of Proposition VII.1.4:  {\sl Let $U$ and $V$ be open 
subsets of a topological space $X$, and suppose that $y\in U$ and $z\in V$.  Let 
$f:U\to V$ be a homeomorphism such that $f(y)=z$.  
Then  the local homology groups $H_*(X,X-\{y\})$ and 
$H_*(X,X-\{z\})$ are isomorphic.}
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$(c)$
Consider the special case of $(b)$ where $X= \bbR^n_+$, 
and let $U,V,y,z,h$ be as given 
there.  Prove that the first coordinate of $y$ is zero if and only if the first coordinate of 
$z$ is zero; {\it i.e.\/}, $h$ takes boundary points of $\bbR^n_+$ to boundary points 
and nonboundary points to nonboundary points.
\bye

