Defn: M is an n-dimensional manifold (with boundary) if 1.) For all x ∈ M, there exists a neighborhood V_{x} such that V_{x} is homeomorphic to an open set in R^{n} 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:

Alexander horned sphere: see handout To avoid such pathologies, we will work in the differentiable (C^{∞}) or piecewise linear (PL) category. Examples of n-manifolds: D^{n} = B^{n} = {x ∈ R^{n} | ||x|| ≤ 1} S^{n} = {x ∈ R^{n+1} | ||x|| = 1} = ∂B^{n+1} P^{n} = S^{n}/(x ∼ −x) T^{n} = S^{1} × S^{1} × ... × S^{1} Forming new manifolds from old manifolds: If M is an m-manifold and N is an n-manifold, then M × N is a (m+n)- manifold. If M is an m-manifold, then ∂M is an (m-1)-manifold. Suppose M and N are n-manifolds and f : a component of ∂M → a component of ∂N is a homeomorphism, then

M ∪_{f} N = M ∪ N/(x ∼ f(x))

In particular, M#N = (M − B^{n}) ∪_{i} (N − B^{n}) where i : S^{n−1} → S^{n−1}. F_{g} = #T^{2} = (S^{2} − ∪_{i=1}^{2g}D^{2}) ∪ (∪_{i=1}^{g}A^{2}) where A^{2} = annulus N_{g} = #P^{2} = (S^{2} − ∪_{i=1}^{g}D^{2}) ∪ (∪_{i=1}^{g}V^{2}) where V^{2} = mobius band Euler characteristic = χ(M) = vertices - edges + faces - ...

= Σ_{i=0}^{∞}(−1)^{i}α_{i}(M) where α_{i}(M) = number of i cells.

= Σ_{i=0}^{∞}(−1)^{i}β_{i}(M) where β_{i}(M) = dimH_{i}(M)

χ(M_{1} ∪_{F} M_{2}) = χ(M_{1}) + χ(M_{2}) − χ(F) χ(S^{2n−1}) = 0. χ(S^{2n}) = 2. χ(D^{n}) = 1. If S is a surface (compact connected 2-manifold) consisting of disjoint disks with bands attached, then χ(S) = # of disks - # of bands.

χ(T^{2}) = 0.  χ(T^{2}#T^{2}) = −2,  χ(F_{g}) = 2 − 2g
χ(P^{2}) = 1.  χ(P^{2}#P^{2}) = 0,  χ(N_{g}) = 2 − g

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