Embed the Klein bottle into the 3-manifold $S^{2} \times S^{1}$

The Klein bottle is the space $S^1 \times I/\sim$, where the equivalence relation identifies $(z,0) \sim (\bar z, 1)$, where $\bar z$ denotes complex conjugation (aka reflection across the x-axis).

Denote by $\text{rot}_\theta$ the rotation of 3-space which moves the z-axis down theta degrees toward the y-axis (this is a rotation whose fixed set is the x-axis). Write $g: S^1 \to S^2$ for the standard inclusion of the equator.

$f: S^1 \times I \to S^1 \times S^2$, given by $f(z,t) =(2\pi t, \text{rot}_{\pi t} g(z))$ descends to an embedding of the Klein bottle.

The Klein bottle can be given by $S^1 \times [0,2\pi] $ with the identificition

$$(\theta, 0) \sim (-\theta, 2\pi)$$

(where I used $\theta$ to parametrizes the circle). Now define

$$S^1 \times [0,2\pi] \to S^2 \times S^1, \ f(\theta, t) =\big( a(\theta, t), e^{it} \big),$$

where

$$a(\theta, t ) = \sin\theta (\cos\frac{t}{2} , \sin \frac{t}{2}, 0)+ \cos \theta (0,0,1).$$

Note that

$$f(-\theta, 2\pi) = (a(-\theta, 2\pi ), e^{i2\pi}) = (a(-\theta, 2\pi ), e^{i0}) $$

and

\begin{equation} \begin{split} a(-\theta, 2\pi ) &=\sin(-\theta) (\cos \pi , \sin \pi, 0)+ \cos (-\theta) (0,0,1)\\ &= \sin \theta (\cos 0 , \sin 0, 0)+ \cos \theta (0,0,1) \\ &= a(\theta, 0) \end{split} \end{equation}

Thus $f(\theta, 0) = f(-\theta, 2\pi)$ and so $f$ descends to a map

$$\tilde f: K \to S^2 \times S^1. $$

Note that $\tilde f$ is injective.