Two arXiv posts caught my attention the other day, both by Siddhartha Gadgil. The first paper [1] shows that an embedded surface in a 3-manifold is incompressible if and only if it is isotopic to an (embedded) minimal area surface, i.e. minimal over its homotopy class, for every Riemannian metric for the 3-manifold. This seems like a very natural result, and the paper is just six pages.

The second paper [2] is on degree one maps between 3-manifolds. A degree one map between closed, orientable *n*-manifolds is a map from one into the other that induces an isomorphism of their *n*th homology groups (each of which are isomorphic to *Z*). In dimension two, its not too hard to construct a degree one map from a given surface to any lower genus surface. There are no degree one maps from a surface to a higher genus surface, though I don’t recall a proof that’s nice enough to explain in one sentence. But one can think of a degree one map as indicating that the image manifold is somehow smaller or simpler than the domain manifold.

In dimension three, things are much more complicated. One can always construct a degree one map from a closed 3-manifold into the 3-sphere. Also, a homotopy equivalence map is degree one, so all those lens spaces with the same fundamental groups have degree one maps between them. Determining in general when there is a degree one map from *M* to *N* (both 3-manifolds) was (as far as I know) a very difficult open problem. However, Gadgil gives is surprisingly simple solution: He shows that there is a degree one map from *M* to *N* if and only if *M* can be produced from *N* by Dehn surgery on a link in which each component is an unknot. The paper is, again, quite short at 11 pages, and is based on interpretting Dehn surgery in terms of 4-manifolds. (In fact, he states a version of the main theorem in terms of 4-manifolds, but I won’t try to repeat it here.)

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A degree one map is surjective on fundamental groups (otherwise the map would factor through a covering space), and so the rank of forbids a degree one map from a surface of genus to one of genus .

Comment by Richard Kent — September 24, 2008 @ 8:44 am |

>A degree one map is surjective on fundamental groups

Alternatively, if you have a map from lower genus to higher, there is an element A in the kernel of the induced map on first cohomology (say over Q) and so the fundamental class=A cup B (for some B, by Poincare duality) goes to zero (by naturality of cup product).

Comment by Max — September 24, 2008 @ 11:39 pm |

A much more sophisticated property is the homotopy theoretic result of Hopf, which is th following propetry; if f and are homotopic, then d(f)=d(g)(where d is the degree map), the Hopf result is the converse of this property, i.e, if d(f)=d(g) then f and g are homotopic. Thus, the degree is a complete algebraic invariant for studying homotopy classes of maps from S^n to S^n. Moreover, for a given map f:S^n–>S^n, n>=0, there is associated a map

g:S^n+1–>S^n+1 called the suspension of f (denoted Sigma(f)). Intuitively, the idea is that the restriction to the equator S^n ic S^n+1 should be f and each slice in S^n+1 parallel to the aquator chould be mapped into the corresponding slice in the manner prescribed by f. So, we can prove that if

f:S^n–>S^n, n>=1 is a map, then d(Sigma(f))=d(f).

Comment by Hicham YAMOUL — April 21, 2010 @ 8:47 am |

The homotopy interpretation of degree one normal maps

Let (X,Y) be a pair of finite complexes which satisfy Poincaré duality in dimension n, and (M^n,dM) an n-dimensional with boundary. a degree one normal map f:(M^n,dM)—>(X,Y) is a continuous map f so that f_*([M,dM])=[X,Y] together with a stable vector bundle $psi$ on X and a bundle isomorphism g:$eta_M$

—>f^!(psi). Now let X be an oriented Poincaré duality complex of dimension n with oriented class [X]. Suppose that f:M^n—->X^n, g:eta_M—>f^!(psi) is a degree one normal map then, since eta_M is only stably well defined, we may as well assume that we are free to modify psi by adding arbitrary copies of the trivial bundle epsilon, and that the dimension of the fibers of psi, d, is greater than n. Since the Thom space of epsilon (+)psi=SigmaT(psi) it follows that the degree one normal data gives a map S^n+p–PT–>T(eta_M)–T(g)–>t(psi) which is well defined up to suspension, and consequently gives a well defined element in the stable homotopy group lim_s–infinity pi_n+s+d(SigmaT(psi))=pi^s_n(T(psi))

Comment by Hicham YAMOUL — May 5, 2010 @ 9:26 pm |