Do Homeomorphisms , Diffeomorphisms Preserve Intersections?

[Bacle]

Hi, Everyone:

Say f:M-->N is a non-identity homeomorphism. Does f preserve intersections, both
number and sign-wise? Maybe a more precise statement (for a 2n-manifold), given
the intersection form Q on H_n . Is it the case that Q(a,b)=Q(f(a),f(b))?

Thanks.

 

[lavinia]

Hi, Everyone:

Say f:M-->N is a non-identity homeomorphism. Does f preserve intersections, both
number and sign-wise? Maybe a more precise statement (for a 2n-manifold), given
the intersection form Q on H_n . Is it the case that Q(a,b)=Q(f(a),f(b))?

Thanks.

If f is a diffeomorphism then the number of points in the transverse intersection of two half dimensional manifolds is preserved as is transversality. If one takes the orientation on M to be the induced orientation under f then the oriented intersection number is preserved I think. If not, then the intersection number could reverse sign.

E.G.Map a torus into itself ( the square [0,1] x [-1/2,1/2] with opposite edges identified) by negating the y coordinate. This map is orientation reversing. The intersection number of the two xy-axis circles is reversed unless one also reverses the orientation of the torus.

For homeomorphisms I am not sure how to generalize the idea of intersection number. Maybe by approximation through diffeomorphisms. One might try approximating the homeomorphism with a homotopy where for each time t except time,1, the maps are diffeomorphisms and at time,1, the map is the original homeomorphism. But I do not know if there is such a procedure. Just guessing.

 

[Greg Bernhardt]

Hello,

Yes, a non-identity homeomorphism f:M-->N will preserve intersections, both number and sign-wise. This means that for any two submanifolds a and b of M, the intersection form Q(a,b) will be equal to the intersection form Q(f(a),f(b)) in N. This is because homeomorphisms preserve the topological structure of spaces, and intersections are a topological property.

To understand this better, let's consider the intersection of two submanifolds a and b in M. This intersection can be represented by the intersection form Q(a,b), which is a bilinear form that measures the number and sign of intersections between a and b. Now, when we apply the homeomorphism f to both submanifolds, we are essentially mapping a and b onto their images f(a) and f(b) in N. Since f preserves the topological structure, the intersection between f(a) and f(b) will be the same as the intersection between a and b in M. Therefore, the intersection form Q(f(a),f(b)) will be equal to Q(a,b).

In general, homeomorphisms preserve all topological properties, including intersections. This is why we can say that f preserves intersections, both number and sign-wise.

I hope this helps clarify things. Let me know if you have any further questions. Thanks.