Here are instructions for making various topological spaces using wool. See also the published article

Irving, Claire. Making the Real Projective Plane. Math. Gazette November 2005, 417--423.

Techniques

There are two main crocheting stitches which will be used in the pattern for Steiner's Roman Surface. The first type of stitch will be referred to as single and abbreviated to S. To create this stitch, insert the crochet hook into a stitch and pull the wool through this stitch so that there are two loops on the hook, then pull the wool through both loops. The second type of stitch is used to increase the diameter of the work, and will be referred to as enlarge (E). Proceed as for single stitches, but after hooking the wool through both loops, pull the wool through the remaining loop.

Knitting Abbreviations: K=Knit, P=Purl, K2tog=Knit two together.

The technique used to join up the lobes of the model of Boy's Surface is called grafting. To do this, thread the end of the wool on to a large, blunt sewing needle, and put the stitches to be grafted together on to a pair of parallel knitting needles with right sides facing you. Thread the needle knitwise through the first stitch on the front knitting needle and slip the stitch off. Then thread the needle purlwise through the front of the second stitch on the front needle, but do not slip the stitch off. Thread the needle purlwise through the first stitch on the other knitting needle and slip the stitch off, then thread the needle knitwise through the second stitch on the same knitting needle, without slipping the stitch off. Continue in the same way until all stitches are grafted together. The joining stitches should be pulled tight so that they look like the rest of the knitted stitches. To finish, sew in the loose end of the wool.

Boy's Surface

This is a pattern I developed myself after studying this immersion of the real projective plane in great detail.

Using size 10 (3.25mm) double pointed needles, cast on 90 stitches in a contrasting colour and knit two rounds. Then cast on the main colour and knit 10 rounds. Take the 30 stitches on one of the needles, knit them on to 3 double pointed needles, and work as follows to create a piece which will be referred to as a lobe.

Knit 1 round
K19, turn, P18, turn, K17, and so on until P2, then turn and K11.
Turn, P20, then turn and (K2tog K4) 5 times. Turn and (P2tog P3) 5 times. Thread the remaining 20 stitches on to some spare wool.

Take another set of 30 stitches and knit them in the same way. Finally, knit the last set of 30 stitches in the same way.

Pass 10 stitches from the inner edge of one of the three lobes through the wall of the shape by hooking stitches through individually using a crochet hook, spreading them evenly in a small circle, as shown in the diagram below. 

                        

Put the stitches on to three double pointed needles, then knit 1 round with the knit side inside the circle. Knit 9 stitches, turn and P8, and so on until P2, then turn and K6, turn and P10. Graft these 10 stitches to the 10 stitches on the outer edge of the next lobe. Treat the other two lobes in the same way.

Now pick up the 90 stitches in the main colour at the casting on edge using three double pointed needles. Knit 10 rows, then work round as follows.

(K8 K2tog) 9 times
Knit 1 round
(K7 K2tog) 9 times
Knit 1 round
(K6 K2tog) 9 times
Knit 1 round
(K5 K2tog) 9 times
Knit 1 round
(K4 K2tog) 9 times
Knit 1 round
(K3 K2tog) 9 times
Knit 1 round
(K2 K2tog) 9 times
Knit 1 round
(K1 K2tog) 9 times
(K2tog) 9 times.

Cut the contrasting wool out of the shape and sew in the loose ends around the lobes. Stuff the shape firmly, then thread the wool through the remaining 9 stitches and pull tight.

Steiner's Roman Surface

This is another of the patterns I developed myself.
The hardest part of making this object is starting off. Using a size 11 (3mm) crochet hook, begin by making a chain of 48 stitches and forming it into a loop. Then twist it into shape as shown below, so there are three loops of 12 stitches, with 4 stitches linking each loop.

                                               
Tie pieces of wool, in a contrasting colour, around each twist (using a half-hitch), then tie each loop to every other loop at the points marked in the above diagram, leaving 4 stitches between every pair of knots. Then using single stitch, work around the shape, crossing over at every knot. Using a second colour (so that you can see where you are up to) join the wool on to the chain at the point marked b in the above diagram and work in the same way, using single stitch. Finally, join a third colour on to the chain at the point c and work around the shape using single stitch. At this point, it should be clear that you have three intersecting discs. Remove the pieces of contrasting wool, then work as follows.
Second round: Using S1 E1, work around the discs using each colour in turn.
Third round: Work in single using each colour in turn.
Fourth round: Work S1 E1 in each colour.
Fifth round: Work in single in each colour.
Sixth round: Work E1 S5 in each colour.
Seventh round: Work in single in each colour.
Eighth round: Work E1 S6 in each colour.
Ninth round: Work in single in each colour.
If you wish to make a larger shape, continue crocheting in a similar manner.
Now choose a base for your Roman surface. Fill in all triangles which share a side with this base, and the triangle vertically above the base, in the following way. Using single stitch, work into two stitches, then miss one out. Continue in this manner all around the triangle. On the second, third and fourth rounds, use single stitch all round the triangle, but omit the stitches at each of the corners of the triangle. On the fifth and subsequent rounds, work into every other stitch until the hole is completely filled in. Fasten off and sew in the ends of the wool.

Links
Other patterns for making topological shapes can be found elsewhere on the Web.

Knitting and crocheting patterns to make a sphere-with-crosscap.

Various topological spaces including Moebius bands and hyperbolic planes.

Miles Reid has written an article discussing mathematical knitting, which includes patterns for making a sphere, torus, Klein bottle and Moebius band.