By John Hempel

It sort of feels ordinary that no-one has reviewed this booklet earlier, yet absolutely this is often a result of popularity of the ebook and the truth that nearly all of these searching for it don't have any want for a assessment; in spite of the fact that, a minority might discover a evaluate of what this ebook is and is not valuable of their determination to shop for or not.

What this ebook is not: 1) An advent to topology, or perhaps to low-dimensional topology. an individual who has heard of 3-manifolds and gotten excited could do greater to get a style of the topic somewhere else first, e.g. in Rolfsen's _Knots and Links_. 2) A study monograph designed to convey the reader in control on present study on 3-manifolds. This publication is set 30 years previous and does not even point out the Geometrization Conjecture of Thurston. three) A booklet at the function of knot conception in 3-manifolds. Knots play an immense position within the conception, not just theoretically, yet as a wealthy resource of examples to sharpen the instinct and attempt conjectures (through Dehn surgical procedures on knots and links). This position isn't mentioned during this book.

What this ebook is: 1) A primer for topologists looking to turn into experts in 3-manifolds. the elemental theorems concerning major decomposition, loop and sphere theorems, Haken hierarchy, and Waldhausen's theorems on Haken manifolds are defined intimately. those may be thought of many of the highlights even though a lot proper fabric is inevitably additionally defined. As might be befitting a primer, the JSJ decomposition and attribute submanifold idea isn't integrated. Jaco's e-book enhances Hempel via masking this fabric. 2) A reference for these already acquainted with the cloth. The writing sort is especially concise and to the purpose. This makes it basic to appear up a theorem to refresh one's reminiscence on a sticky element in an explanation. As an creation to the cloth, a few passages should be terse, yet unavoidably after a few attempt, they are often "decoded" thoroughly, not like a few texts which may be extra verbose yet can by no means be completely deciphered. i believe there can be a lot extra images; there aren't very many, to claim the least. but when the reader attracts his/her personal images, this isn't an excessive amount of of a problem.

Some ultimate comments: This ebook serves its twin position as a primer and reference admirably, however the reader may well wander away within the info and lose the wooded area for the bushes. regrettably, the one technique to rectify this seems to learn a number of papers at the topic to get a great think of a number of the threads that inspire present examine. yet with Hempel's _3-manifolds_ in hand, this job is way more straightforward and stress-free.

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A function u : Ω → [−∞, ∞] is upper semi-continuous if for every s ∈ (−∞, +∞] the set u−1 [−∞, s) is open. Equivalently, u is upper semi-continuous if lim sup u(ζ) ≤ u(z). 8. Subharmonic functions 15 Note that every upper semi-continuous function u is Lebesgue measurable. In fact, for any measurable subset U ⊂ C f dA := inf U U f dA ; f ∈ C0∞ (U ) and f ≥ u . If X is a Hausdorff space and f : X → R, the upper regularization of f is the function f ∗ (x) := lim sup f (y). y→x f∗ is upper semi-continuous, f ∗ ≥ f , and that if g ≥ f is upper It follows that semi-continuous then g ≥ f ∗ .

Let f : ∂D → R be a continuous function. 6) u(z) := |ζ|=1 f (ζ) dζ 1 − |z|2 √ , 2 |ζ − z| 2π −1ζ |z| < 1 and u(z) = f (z) for |z| = 1 is a continuous solution of the Dirichlet Problem Δu = 0 u=f in D on ∂D 14 1. Complex Analysis for the unit disk. 6). We shall return to the Dirichlet Problem in due course. 6. Regularity of harmonic functions. If we interpret the Laplacian classically, we must require that harmonic functions be a priori C 2 . However, even if we interpret the definition in the sense of distributions, harmonic functions are still smooth—a fact sometimes known as Weyl’s Lemma, which we now prove.

To see that uε is decreasing, we use a “double smoothing trick”: let ϕ ∈ C0∞ (D) be non-negative. Then u ∗ ϕδ is smooth and subharmonic (as we just argued), and hence uε ∗ ϕδ = u ∗ ϕδ ∗ ψ ε is a subharmonic family decreasing with ε. Letting δ → 0 shows that uε is decreasing. 3 that u is subharmonic. Conversely, suppose u is subharmonic. Define uε = u ∗ ψ ε with ψ as above. 4. 8 that Δuε ≥ 0, and therefore if h ≥ 0 has compact support then uΔh = lim uε Δh = lim Δuε h ≥ 0. Thus Δu ≥ 0 in the sense of distributions, as claimed.