MATERIAL FOR A MASTERCLASS ON THE POINCARE–HOPF

THEOREM

Timeline

10 minutes Introduction and History (slides 1-4)

5 minutes Discussion of regular polyhedra1

5 minutes Talk about contour maps (slides 5-8)

10 minutes Exercise session: contour maps.

10 minutes Talk about flow diagrams (slides 9-15)

10 minutes Exercise session: flow diagrams.

5 minutes Talk about indices of stationary points (slides 15-17)

15 minutes Exercise session: indices of stationary points.

15 minutes BREAK (resume by 1hr 30)

10 minutes Talk on surfaces (slides 18-21)

15 minutes Exercises on surfaces.

10 minutes Talk on flows on surfaces (slides 22-25)

15 minutes Exercise session: flows on surfaces.

10 minutes Finish: Partial proof of the Poincare–Hopf theorem.

1Sheet used as ‘problems while arriving’.

1

Introductory problems: Regular polyhedra

Complete the following table:

Picture

Name Tetrahedron Octahedron Icosahedron

Faces (F) 4 20

Edges (E) 12 30

Vertices (V) 4 6 20

V-E+F 2

Can you find a relationship between the numbers of edges and vertices for a given solid?

Can you find a relationship between the numbers of edges and faces for a given solid?

Contour Maps

Can you construct contour maps with the following certain properties?

• A contour map with precisely two points such that if a ball was placed thereit would not roll.

• A contour map with three such points.

• A contour map with infinitely many such points.

Points at which an object placed there will not move are called stationary points.

More on stationary points.

• Can I make a contour plot with a stationary point such that if I move alittle away from this point I always roll further away?

• ... or always roll back?

• ... or roll away in some directions and back in others?

Flow diagrams

Can you construct flow diagrams for the contour plots you made in the lastexercise? Here is an example of how to do this, remember the essential point is tokeep flow lines and contours at right angles to each other at all points.

Example: Flow for a saddle point. The contour map is shown below (darkerregions indicate going down, lighter ones going up).

Figure 1. Contours Figure 2. Flow diagram

To draw the flow diagram, choose a point on the contour plot and draw a curvemoving downward, just as a rolling ball would, always at right angles to the contourlines. Each flow curve also runs ‘back’ from a point, uphill (as if we were playingthe film of the rolling ball in reverse). NB different flow lines never intersect.

The Vortex

Figure 3. A flow diagram of a vortex

Above is a picture of a special type of flow. It has a single stationary point atthe centre, and every point lies on a closed path (every point eventually returns towhere it started under the flow).

(1) Is there a contour map which gives this flow?(2) Can you draw the flow diagram for a stream flowing left to right, in which a

small vortex has formed somewhere. How many stationary points are therein your flow? Be careful that your stream isn’t suddenly jumping directionanywhere!

Indices of stationary points

We’ve seen how to find the index of a stationary point by walking around thestationary point with a little needle which always points in the direction of the flowand counting the number of times it spins round. Can we find the index of thesaddle point, source and sink flows?

Figure 4. Can you find the indices of these flows?

Figure 5. Can you find the index of the dipole stationary point?

Euler numbers

Lets start by investigating the first rule (Euler number is V − E + F ) in somecases where there are no faces. Here are three planar graphs - shapes drawn byconnecting points in a plane without edges crossing. We can still use the ruleV −E + F to work out the Euler number, just remember that F = 0 for all cases.

Figure 6. Three planar graphs

Can you find a connection between the Euler number and the ‘number of holes’(the number of enclosed regions) for each graph? Is there a pattern? If you needmore inspiration feel free to draw more graphs of your own and calculate their Eulernumbers. Working out the Euler numbers of the following sequence of graphs maymake things clearer.

Figure 7. A sequence of planar graphs

Now we’ll look at Euler numbers for a range of different kinds of shape - some ofthese you’ve seen already, some are new. All these shapes have Euler numbers youcan find using rules 1 and 2.

Note that in the final figure there are no faces on the ‘caps’ of the cuboid. Thenext five questions are all about how rule 3 can be used.

(1) I can make a circle by gluing the end points of two pieces of string, canyou use rule three to work our the Euler number of the circle from thisdescription?

(2) Explain how removing a small disc from a shape changes the Euler number.

(3) Can you compute the Euler number of a cylinder using only rule 3? (Canyou do it using the other rules?)

(4) Gluing two cylinders together we can make a ‘doughnut’ (or torus), usingrule 3 can you work out the Euler number of a torus?

(5) What is the Euler number of a ball (or ‘filled in sphere’), what about asolid torus?

Here some slightly more involved questions for those who might be interested.

(1) For any whole number n > 0 can you find a shape with Euler number n?What about −n?

(2) Can you think of a shape which must be given an ‘infinite’ Euler numberby our rules? Or a shape which our rules say nothing about?

(3) Can you find two shapes with the same Euler number which cannot bestretched or deformed into each other?

Flows on surfaces

Our final exercise is to study flows on surfaces. We’ve looked at flows in theplane and their singular points, but flows can lie on more complicated surfaces:imagine a weather pattern over the whole earth for example, this is a flow on asphere.

Figure 8. Some flows on a sphere

Figure 9. Another flow on a sphere

For these flows can you find the sum of the indices of all stationary points of theflow? Can you do this for a flow of your own? Below is an image of a torus, canyou construct a flow on this shape with no stationary points? Can you construct aflow with more than one stationary point? What is the sum of the indices of thesepoints? (Harder) Can you construct a flow on a 2-holed torus? What is its Eulernumber?

Figure 10. A torus

Figure 11. A two holed torus