The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4.
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Four-dimensional manifolds are the most unusual: they are not geometrizable (as in lower dimensions), and surgery works topologically, but not differentiably. Since topologically, 4-manifolds are classified by surgery, the differentiable classification question is phrased in terms of "differentiable structures": "which (topological) 4-manifolds admit a differentiable structure, and on those that do, how many differentiable structures are there?" Four-manifolds often admit many unusual differentiable structures, most strikingly the uncountably infinitely many exotic differentiable structures on R4. Similarly, differentiable 4-manifolds is the only remaining open case of the generalized Poincaré conjecture.
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In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonomic function is an element of a holonomic module of smooth functions. Holonomic functions can also be described as differentiably finite functions, also known as D-finite functions. When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called holonomic.
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In dimension 5 and above (and 4 dimensions topologically), manifolds are classified by surgery theory. The Whitney trick requires 2+1 dimensions (2 space, 1 time), hence the two Whitney disks of surgery theory require 2+2+1=5 dimensions. The reason for dimension 5 is that the Whitney trick works in the middle dimension in dimension 5 and more: two Whitney disks generically don't intersect in dimension 5 and above, by general position (2+2 < 5). In dimension 4, one can resolve intersections of two Whitney disks via Casson handles, which works topologically but not differentiably; see Geometric topology: Dimension for details on dimension.
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Posner was born on August 10, 1933, in Brooklyn, and graduated from Stuyvesant High School in 1950; at Stuyvesant, one of his close friends was mathematician Paul Cohen. He took only two years to complete his undergraduate studies in physics at the University of Chicago, graduating in 1952, and he then switched to mathematics for a master's degree in 1953 and a PhD in 1957.. While a graduate student, he also visited Bell Labs, and later claimed that he had been assigned to the desk there that had formerly been Harry Nyquist's. His doctoral thesis, supervised by Irving Kaplansky, was on the subject of ring theory and entitled Differentiably Simple Rings; at only 26 pages long, it held the record for the shortest doctoral thesis at the university. After finishing his studies, he became a mathematics instructor at the University of Wisconsin and then an assistant professor of mathematics at Harvey Mudd College.
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Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low- dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as exotic differentiable structures on R4. Thus the topological classification of 4-manifolds is in principle easy, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture; see Gluck twists. The distinction is because surgery theory works in dimension 5 and above (in fact, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above is controlled algebraically by surgery theory. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work, and other phenomena occur.
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