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Defn:  $M$ is an n-dimensional manifold ({\Blue{with boundary}}) if 

1.)  For all $x \in M$, there exists a neighborhood $V_x$ such that $V_x$ is homeomorphic to an open set in $R^n$ 
{\Blue{or ${R_+}^n$}}

2.)  M is $T_2$ and ...

Give an example of a topological space which satisfies (1), but is not $T_2$.

Answer:  Friday's Lecture.



$M$ is a closed manifold if $M$ is a compact manifold without boundary.

Wild knot:

\v\v
\vfill
Alexander horned sphere:  see handout


To avoid such pathologies, we will work in the differentiable ($C^{\infty}$) or piecewise linear (PL) category. 
\eject

Examples of  n-manifolds:  

$D^n = B^n = \{ {\bf x} \in R^{n} ~|~ ||x|| \leq 1\}$

$S^n = \{ {\bf x} \in R^{n+1} ~|~ ||x|| = 1\} = \partial B^{n+1}$

$P^n = S^n/ ({\bf x} \sim -{\bf x})$

$T^n = S^1 \times S^1 \times ... \times S^1$

Forming new manifolds from old manifolds:


If $M$ is an m-manifold and $N$ is an n-manifold, then $M \times N$ is a 
(m+n)-manifold.

If $M$ is an m-manifold, then $\partial M$ is an (m-1)-manifold.

Suppose $M$ and $N$ are n-manifolds and \hfil \break
$f:$ a component of $\partial M \rightarrow$ a component of $\partial N$ is a 
homeomorphism, then
\vskip 5pt
\centerline{$M \cup_f N = M \cup N / (x \sim f(x))$}

In particular, $M \# N = (M - B^n) \cup_i (N - B^n)$ where $i: S^{n-1} \rightarrow S^{n-1}$.

$F_g = \# T^2 =  (S^2 - \cup_{i=1}^{2g} D^2) \cup (\cup_{i=1}^{g}{A^2})$
where $A^2 $ = annulus

$N_g = \# P^2  = (S^2 - \cup_{i=1}^{g} D^2) \cup (\cup_{i=1}^{g} V^2)$ where $V^2 $ = mobius band


\vskip 15pt
{\line Euler characteristic = $\chi(M) =$ vertices - edges + faces - ... }

\centerline{= $\Sigma_{i=0}^\infty (-1)^i \alpha_i(M)$ where $\alpha_i(M) =$ number of $i$ cells.}

\centerline{= $\Sigma_{i=0}^\infty (-1)^i \beta_i(M)$ where $\beta_i(M) = dim H_i(M)$}

$\chi(M_1 \cup_F M_2) = \chi(M_1) + \chi(M_2) - \chi(F)$



$\chi(S^{2n-1}) = 0$. \hfil  $\chi(S^{2n}) = 2$. \hfil $\chi(D^{n}) = 1$. 

If $S$ is a surface (compact connected 2-manifold) consisting of disjoint disks with bands attached, then 

\centerline{$\chi(S) = $
\# of disks - \# of bands.}

$\chi(T^2)  = 0$. \hfil $\chi(T^2 \# T^2) =  -2$,
\hfil  $\chi(F_g) = 2-2g$
\vskip -15pt
$\chi(P^2)  = 1$. \hfil $\chi(P^2 \# P^2) =  0$,
\hfil  $\chi(N_g) = 2-g$


\vfill



Casson: ``For three-dimensional topology, intuitive understanding is much more important than technical details."



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