# Homology of Klein bottle

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology and the topological space/family is Klein bottle
Get more specific information about Klein bottle | Get more computations of homology

## Statement

The homology of the Klein bottle $K$ is given by:

$H_0(K) = \mathbb{Z}$

and:

$H_1(K) = \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$

All other homology groups are 0.

## Related invariants

These are all invariants that can be computed in terms of the homology groups.

Invariant General description Description of value for Klein bottle
Betti numbers The $k^{th}$ Betti number $b_k$ is the rank of the $k^{th}$ homology group. $b_0 = b_1 = 1$, $b_k = 0$ for $k > 1$
Poincare polynomial Generating polynomial for Betti numbers $1 + x$
Euler characteristic $\sum_{k=0}^\infty (-1)^k b_k$ 0 -- consistent with the fact that its double cover is the 2-torus, which, being a nontrivial compact connected Lie group, has Euler characteristic zero, and the fact that Euler characteristic of covering space is product of degree of covering and Euler characteristic of base