# Tag Info

14

There does not, even if you don’t require the fiber and base to be manifolds (or even connected, just that $F$ is not a single point). See Borel, Armand; Serre, Jean-Pierre, Impossibilité de fibrer un espace euclidien par des fibres compactes, C. R. Acad. Sci. Paris 230 (1950), 2258–2260.

13

I guess the most conceptual proof is the one using Morse theory: Take a Morse function on the (closed, orientable) surface S. If it has no saddle points, then (using the gradient flow) $S\cong S^2$. Assume by induction that a surface with k-1 saddle points is $S^2$ with finitely many handles added. For the inductive step, consider a Morse function with k ...

11

On the other hand, if you only mean "foliation" as in your title, and not "fibration", then there is Vogt's foliation of R^3 by circles! (But it is not C^1, only differentiable). Vogt, Elmar, "A foliation of R3 and other punctured 3-manifolds by circles", Publications Mathématiques de l'IHÉS, Tome 69 (1989), p. 215-232 http://www.numdam.org/item/...

9

This is more an extended comment than an answer to the question. The first thing to note is that there are different strenghts of the classification theorem for surfaces. Of course, there are the differentiable, triangulated and topological setting. But even if we choose such a setting, there are two statements one has to prove (at least in one approach): ...

9

The proof of Zeeman described in this note is by a substantial margin the easiest and most conceptual proof I know. To simplify the exposition I restrict to orientable surfaces in the note, but it is trivial to also do the non-orientable case (and see the edit below for one description of how to arrange this to avoid using the fact that three cross caps is ...

9

Yes, the Chern–Weil homomorphism lifts to differential cohomology, which guarantees that periods are integral. See the original paper by Cheeger and Simons, or the paper by Hopkins and Singer. The (modernized) construction of such a refinement relies on the computation of the de Rham complex of the stack B_∇(G) of principal G-bundles with connection and ...

5

Your first question can be answered by using the splitting principle. In order to answer the second question, one would need to know what $c_2(\Sigma^-)$ is, but I do not. If $V \to X$ is a complex vector bundle of rank two, then $c_1(S^3V) = 6c_1(V)$ and $c_2(S^3V) = 11c_1(V)^2 + 10c_2(V)$. Proof: By the splitting principle, there is a map $p : Y \to X$...

4

Using a little bit of real algebraic geometry, there is a conceptual proof at least in the critical case $\chi=-1$, i.e. the case you're talking about explicitly. Indeed, let $S$ be a compact connected smooth surface without boundary with $\chi(S)=-1$. Choose a conformal structure on $S$. Since $\chi$ is odd, $S$ is nonorientable. Let $\tilde S$ be its ...

4

This answers the first (simple) half of the question, asking just about a smooth map. In fact, you've already given an answer to it, in some sense. Apply the map $f: re^{i \theta} \to \sigma_1re^{i(\sigma_2/\sigma_1) \theta}$ to a unit disk that doesn't contain $(0,0)$, say radius $1$ disk $D$, centred at $(2,0)$. Then, the image $f(D)$ is contained in the ...

3

If $t$ is a regular value, then it is a property of Morse functions that there is some small open neighborhood $U$ of $t$ in $\mathbb{R}$ such that $u$ is also a regular value for all $u\in U$. In particular, we can take $U=(t-\delta,t+\delta)$ for some $\delta>0$. But then $\pi^{-1}(U)\cap S\cong(\pi^{-1}(t)\cap S)\times U$, ie. the surface is a product ...

3

The cited (early) work by Cerf proves that, given a submanifold Y in a manifold X, the obvious map Diff(X)->Emb(Y,X) is a locally trivial fibration. I guess that Budney and Gabai mean the following. By Palais, all embeddings D^3->S^4 are isotopic. Hence, for i=0, 1, the complement C_i of a small open tubular neighborhood U_i of Delta_i in S^4 is ...

2

Ian's argument of mean curvature is wonderfully simple. Here is another one. Rotate your surface to put it in generic position with respect to the heigth function z; then, the preimage of z is a Morse function f on RP^2, which has no critical point of index 1 (saddle point) since the surface is locally convex. Hence, every critical point of f has index 0 or ...

2

There is perhaps some confusion over the terminology. Wall (chapter 13A) uses the term multisignature to denote a collection of invariants of certain Hermitian forms over group rings, giving rise to a function from $L_{2k}(\pi) \to \mathbb{Z}^n$. In that chapter, he interprets the multisignature in terms of equivariant signatures. Such signatures occur in ...

1

Such a homotopy exists and in fact you can assume that it is an isotopy. This is a "standard fact" in the theory of mapping class groups. See Proposition 2.2 of the "Primer" by Farb and Margalit.

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