A Survey on Seifert Fiber Space Theorem

<table>The nonorientable 3-manifold <svg height="11.175" id="M335" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 11.7375 11.175" width="11.7375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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v-245q31 -12 73 -12q59 0 102 38t43 124zM290 131v390q0 62 -7.5 79.5t-30.5 17.5q-24 0 -31.5 -17.5t-7.5 -79.5v-390q0 -62 7.5 -79.5t31.5 -17.5q23 0 30.5 17.5t7.5 79.5z" id="x2119"></path></g>
</svg> and a fibered annulus in the Klein bottle component of <svg height="12.4375" id="M336" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 20.137501 12.4375" width="20.137501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M471 379q0 -56 -17 -126t-47 -124q-80 -141 -227 -141q-81 0 -119 46t-38 114q0 92 45 161.5t108 101.5q73 37 137 37q45 0 80 -33v27q0 83 -21 133q-38 90 -138 90q-68 0 -123 -31q-12 0 -26.5 18.5t-14.5 31.5q52 28 138 28q139 0 219 -134q44 -74 44 -199zM382 335
q0 22 -4.5 33t-18.5 20q-35 21 -79 21q-23 0 -57 -14q-36 -15 -74.5 -75t-38.5 -156q0 -58 23 -97t67 -39q17 0 38 7q50 15 97 93t47 207z" id="x1D715"></path></g><g transform="matrix(.017,-0,0,-.017,8.46,12.162)"><path d="M646 468q0 -91 -56.5 -143t-138.5 -62q-23 -3 -45 -1l-74 19v-157q0 -63 15.5 -77t83.5 -19v-28h-393v28q62 5 76 19.5t14 77.5v400q0 63 -14 77.5t-76 19.5v28h375q112 0 169 -41q64 -49 64 -141zM550 464q0 77 -42.5 115.5t-108.5 38.5q-44 0 -56 -12q-11 -8 -11 -47
v-245q31 -12 73 -12q59 0 102 38t43 124zM290 131v390q0 62 -7.5 79.5t-30.5 17.5q-24 0 -31.5 -17.5t-7.5 -79.5v-390q0 -62 7.5 -79.5t31.5 -17.5q23 0 30.5 17.5t7.5 79.5z" id="x2119"></path></g>
</svg>.</table>

International Scholarly Research Notices

A Survey on Seifert Fiber Space Theorem

Figure 2