On 250 page of Scorpan's book Wild world of 4-manifolds. there is a construction of an exotic $\mathbb{R}^4$. It starts from taking manifold $M = \mathbb{C}P^2 \# 9 \overline{\mathbb{C}P}^2$ and considering an element $\alpha \in H_2(M;\mathbb{Z})$ such that it spans $[+1]$ in associated to intersection form $Q_M = -E_8 \oplus [-1] \oplus [+1]$ (i.e. $Q_M(\alpha, \alpha) = 1$). Then it is claimed that by Casson handles we can represent $\alpha$ by a topologically-embedded sphere $\Sigma$ in $M$ and via construction of such Casson handle we can find its neighborhood which is homeomorphic to $+1$-disk bundle over $S^2$. Unfortunately there is no reference for that result and I cannot see how given in book exposition on Casson handles may imply that fact. At the first glance I thought that it may be conclusion from Casson's Embedding Theorem stated below but I don't see how the assumptions might apply to described case.

I'll be grateful for reference to theorem which implies that and explanation how does it implies that.

Moreover does this approach via Casson handles generalize? Namely is it true that for every closed, simply-connected, oriented $4$-manifold for every $\alpha \in H_2(M; \mathbb{Z})$ we can find topologically embedded sphere $\Sigma$ representing $\alpha$ such that its neighborhood is homeomorphic to $Q_M(\alpha, \alpha)$-disk bundle over $S^2$?

(Casson's Embedding Theorem) Let $M$ be a simply-connected $4$-manifold, with non-empty boundary. Let $f_1, \ldots , f_n$ be immersions $f_i:D^2 \to M$ such that $f_k|_{\partial D^2}$ are disjoint embeddings into $\partial M$.

Assume that, when $i \neq j$, we have intersection numbers $f_i \cdot f_j = 0$. Further, assume therea are classes $\alpha_1,\ldots, \alpha_n \in H_2(M; \mathbb{Z})$ such that $\alpha_k \cdot f_k = 1$ but $\alpha_i \cdot f_j = 0$ if $i \neq j$, and so that all self-intersections $\alpha_k \cdot \alpha_k$ are even.

Then there must exist disjoint open sets $C_1, \ldots, C_n$ such that:

  • we have proper homotopy equivalences $(C_k, C_k \cap \partial M) \sim (D^2 \times \mathbb{R}^2, \partial D^2 \times \mathbb{R}^2)$,
  • $C_k \cap \partial M$ are open tubular neigborhoods of the circles $f_k[\partial D^2]$ in $\partial M$,
  • $f_k$ are homotopic, relative to its boundary $S^1$, to a map into $C_k$.

1 Answer 1


At least for locally-flat topological embeddings (which are necessary for the spheres to be attaching spheres of some handle decomposition), your generalization is false. Lee and Wilczynski [LW] actually compute topological genus functions (the function sending a $\Bbb Z$-homology class to the minimum genus of an embedded surface representing it) for some for simply-connected $4$-manifolds.

No, Freedman's theorem (that a Casson handle is homeomorphic to a standard $2$-handle) does not follow immediately from Casson's embedding theorem. The only somewhat readable proof is probably in Freedman and Quinn's book [FQ] (in the book it is really that pinched neighborhoods of infinite towers of capped-gropes are homeomorphic to $2$-handles, but you can find those precisely when you can find Casson towers for simply-connected manifolds). If you are feeling very brave, you can try Freedman's original paper [F]. A very useful auxiliary source to [FQ] or [F] is Ancel's paper [A].

[LW], R. Lee and D. Wilczynski, Representing Homology Classes by Locally Flat Surfaces of Minimum Genus , Amer. J. Math 19(1997), 1119-1137.

[FQ], M. Freedman and F. Quinn, Topology of 4-manifolds ,PMS 39 , (1990)

[F], M. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17 (1982) 357-453.

[A], F. Ancel, Approximating cell-like maps of $S^4$ by homeomorphisms, Contemp. Math. 35 (1984) Four Manifold Theory edited by C. Gordon and R. Kirby


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