What Maps Induce Isomorphisms on Lower but Not Higher Homology Groups?

[Bacle]

Hi, All:

I am curious to find examples of maps f:X-->X ; X an n-dimensional manifold

that induce isomorphisms on , say, the first k<n homology groups, but not

so on the remaining n-k groups. I can see if we had maps g:X-->Y, we could start

with Y=X, let f be an automorphism, and then cap some boundaries of X, i.e., all j-

boundaries for j>k , but not so for maps f:X-->X . Any Ideas?

Thanks.

others, so that the induced maps on H_k(X) are not isomorphsims

 

[Tinyboss]

Hint: "skeleton".

 

[Bacle]

Tinyboss:

I guess you're suggesting some obstruction theory issues; spin structure, etc?

Unfortunately, I haven't been able to find much in this area from a geometric

perspective; most of the info nowadays seems to be done in terms of abstract

obstruction theory, spin structures, etc. Still, I ordered Steenrod's book on the

geometry of bundles from the library recently. Is this what you were referring to?

 

[lavinia]

Hi, All:

I am curious to find examples of maps f:X-->X ; X an n-dimensional manifold

that induce isomorphisms on , say, the first k<n homology groups, but not

so on the remaining n-k groups. I can see if we had maps g:X-->Y, we could start

with Y=X, let f be an automorphism, and then cap some boundaries of X, i.e., all j-

boundaries for j>k , but not so for maps f:X-->X . Any Ideas?

Thanks.

others, so that the induced maps on H_k(X) are not isomorphsims

Take any map of degree higher than one from a sphere to itself. Using Cartesian products of spheres with other manifolds I think you should be able to get all of the examples except iso up to dimension n-1 and not iso in the top dimension. Cartesian product of spheres with tori should do it.

 

[blue_raver22]

, but the induced maps on H_j(X) for j>k are isomorphisms. One example is the map f:S^2 --> S^2 given by f(x,y,z) = (-x,-y,z). This map induces an isomorphism on H_1(S^2) but not on H_2(S^2). Another example is the map g:T^2 --> T^2 (a torus) given by g(x,y) = (2x,2y). This map induces an isomorphism on H_1(T^2) but not on H_2(T^2). In general, any map that induces an isomorphism on H_k(X) but not on H_j(X) for j>k can be constructed by composing a map that induces an isomorphism on H_k(X) with a boundary map that removes some higher homology groups.