/sci/ - Science & Math

The "number of holes" is defined as the genus [math]g = \operatorname{dim}H_2(M)[/math], namely the dimension (or the number of generators) of the second homology group. For a space [math]X[/math] that is homeomorphic to a cylinder [math]S^1\times I[/math] such as a straw, it can be homological-invariantly retracted to [math]S^1[/math]. Using [math]U \cong V \cong \mathbb{R}[/math] as subsets of [math]S^1[/math] without the north/south poles, respectively, we can compute the homology group with the Meyer-Vietoris sequence
[eqn]
\dots \rightarrow H_2(U\cap V) \rightarrow H_2(U)\oplus H_2(V) \rightarrow H_2(X) \rightarrow H_1(U\cap V) \rightarrow \dots
[/eqn]
which is equivalent to
[eqn]
0 \rightarrow H_2(X) \rightarrow 0
[/eqn]
which by exactness of the sequence means that [math]H_2(X)[/math] is [math]0[/math], so there are no holes in the straw.

ITT freshmen.