View source for Casson handle

A certain kind of smooth [[Manifold|manifold]] pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c1101101.png" /> that, by a foundational theorem of M. Freedman [[#References|[a3]]], is homeomorphic to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c1101102.png" />-dimensional open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c1101103.png" />-handle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c1101104.png" /> (cf. [[Handle theory|Handle theory]]). Casson handles are the key to understanding topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c1101105.png" />-manifolds [[#References|[a4]]]. The fundamental theorems of high-dimensional manifold topology (cf. [[Topology of manifolds|Topology of manifolds]]), namely the surgery and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c1101106.png" />-cobordism theorems, fail for smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c1101107.png" />-manifolds because they depend on finding an embedded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c1101108.png" />-dimensional disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c1101109.png" /> in a given manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011010.png" />, with specified boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011011.png" />. In dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011012.png" />, such an embedding is easily constructed by [[General position|general position]], but in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011013.png" />, immersed surfaces cannot be made embedded by perturbation (cf. [[Immersion of a manifold|Immersion of a manifold]]; [[Immersion|Immersion]]). By work of A. Casson [[#References|[a2]]], it is often possible to embed a Casson handle in the required <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011014.png" />-manifold, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011015.png" /> mapping to the required circle, so Freedman's theorem provides a homeomorphically embedded disc. This leads to proofs of the above fundamental theorems for topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011017.png" />-manifolds (that is, manifolds without specified smooth structures), provided that the fundamental groups involved are not too  "large" . In particular, Freedman obtained a complete classification of closed simply-connected topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011019.png" />-manifolds in terms of the intersection pairing (cf. also [[Intersection theory|Intersection theory]]).

A Casson handle is constructed as a union of kinky handles. A kinky handle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011020.png" /> can be defined as the smooth, oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011021.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011022.png" /> arising as a closed regular neighbourhood of a generically immersed (but not embedded) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011023.png" />-disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011024.png" /> in an oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011025.png" />-manifold, together with the boundary circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011026.png" />. For each pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011027.png" /> of non-negative integers, not both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011028.png" />, there is a unique oriented diffeomorphism type of kinky handle, corresponding to a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011029.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011030.png" /> positive and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011031.png" /> negative self-intersections. The attaching circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011032.png" /> has a canonical framing of its normal bundle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011033.png" />, obtained by restricting any normal framing of an embedded, compact, oriented surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011034.png" />. Equivalently, the framing is obtained from the normal framing of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011035.png" /> by adding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011036.png" /> right twists (relative to the boundary orientation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011037.png" />). There is also a canonical (up to diffeomorphism) embedded collection of normally framed circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011038.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011039.png" />, with the property that attaching <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011040.png" />-handles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011041.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011042.png" /> along these circles (identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011043.png" /> with a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011044.png" /> so that the framings correspond) transforms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011045.png" /> into a standard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011046.png" />-handle.

An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011048.png" />-stage Casson tower <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011049.png" /> is defined inductively, as follows: A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011050.png" />-stage tower is a kinky handle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011051.png" /> with canonical circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011052.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011053.png" /> an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011054.png" />-stage tower <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011055.png" /> is obtained from an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011056.png" />-stage tower <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011057.png" /> by attaching a kinky handle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011058.png" /> to each of the canonical circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011059.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011060.png" />, identifying tubular neighbourhoods (cf. [[Tubular neighbourhood|Tubular neighbourhood]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011062.png" /> so as to match the canonical framings. The canonical framed circles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011063.png" /> are those of the newly attached kinky handles. If one continues this construction to form an infinite sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011064.png" />, the interior of the resulting manifold

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011065.png" /></td> </tr></table>

together with a tubular neighbourhood of the attaching circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011066.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011067.png" />, is a Casson handle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011068.png" />. According to Freedman, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011069.png" /> is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011070.png" /> with the canonical framing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011071.png" /> corresponding to the product framing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011072.png" />. A newer, more powerful version of the theory [[#References|[a4]]] relies on generalized Casson handles that have occasionally been called Freedman handles. These have most layers of kinky handles replaced by manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011073.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011074.png" /> a compact, oriented surface with boundary a circle.

Although all Casson handles are homeomorphic, gauge theory shows that the [[Differential topology|differential topology]] is much more complex. There are uncountably many diffeomorphism types of Casson handles [[#References|[a6]]]. Casson handles are indexed by based trees without any finite branches, with signs attached to the edges. Each vertex represents a kinky handle (with the base point representing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011075.png" />) and each edge represents a self-intersection. It is not presently known whether different signed trees can correspond to diffeomorphic Casson handles. While it is also not known if there is any Casson handle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011076.png" /> that admits a smoothly embedded disc bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011077.png" />, such a disc cannot exist if the corresponding tree has an infinite branch (from the base point) for which all signs are the same [[#References|[a1]]], [[#References|[a7]]]. For any non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011078.png" />, not both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011079.png" />, there is a Casson handle with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011080.png" /> having exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011081.png" /> positive and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011082.png" /> negative self-intersections, such that any generically immersed smooth disc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011083.png" /> bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011084.png" /> also has at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011085.png" /> positive and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011086.png" /> negative intersections [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Ž. Bižaca,   "An explicit family of exotic Casson handles"  ''Proc. Amer. Math. Soc.'' , '''123'''  (1995)  pp. 1297–1302</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Casson,   "Three lectures on new infinite constructions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011087.png" />-dimensional manifolds" , ''A la Recherche de la Topologie Perdue'' , ''Progress in Mathematics'' , '''62''' , Birkhäuser  (1986)  pp. 201–244  (notes prepared by L. Guillou)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Freedman,   "The topology of four-dimensional manifolds"  ''J. Diff. Geom.'' , '''17'''  (1982)  pp. 357–453</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Freedman,   F. Quinn,   "Topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110110/c11011088.png" />-manifolds" , ''Princeton Math. Ser.'' , '''39''' , Princeton Univ. Press  (1990)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Gompf,   "Infinite families of Casson handles and topological disks"  ''Topology'' , '''23'''  (1984)  pp. 395–400</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Gompf,   "Periodic ends and knot concordance"  ''Topology Appl.'' , '''32'''  (1989)  pp. 141–148</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Rudolph,   "Quasipositivity as an obstruction to sliceness"  ''Bull. Amer. Math. Soc.'' , '''29'''  (1993)  pp. 51–59</TD></TR></table>