By Nikolai Saveliev

Growth in low-dimensional topology has been very quick over the past 20 years, resulting in the ideas of many tricky difficulties. one of many outcomes of this "acceleration of historical past" is that many effects have simply seemed in specialist journals and monographs. those are hardly ever available to scholars who've accomplished just a simple direction in algebraic topology, or maybe to a couple researchers whose instant strong point isn't topology. one of the highlights of this era are Casson’s effects at the Rohlin invariant of homotopy 3-spheres, in addition to his l-invariant. The objective of this booklet is to supply a much-needed bridge to those smooth issues. The booklet covers a few classical subject matters, akin to Heegaard splittings, Dehn surgical procedure, and invariants of knots and hyperlinks. It proceeds throughout the Kirby calculus and Rohlin’s theorem to Casson’s invariant and its functions, and provides a quick comic strip of hyperlinks with the most recent advancements in low-dimensional topology and gauge concept. The publication may be available to graduate scholars in arithmetic and theoretical physics acquainted with a few undemanding algebraic topology, together with the basic workforce, easy homology conception, and Poncar? duality on manifolds.

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For the spectral counting function, this follows from the following computation: ∞ ∞ Nν (x) = #{j : lj−1 ≤ x/k} = 1= k=1 j: k·l−1 ≤x j k=1 ∞ NL k=1 x . k Observe that this is a ﬁnite sum, since NL (y) = 0 for y < l1−1 . The second expression is derived similarly: ∞ ∞ Nν (x) = 1= j=1 k≤lj x [lj x]. j=1 For the spectral zeta function, we have successively ∞ ζν (s) = k,j=1 k −s ljs = ∞ ∞ ljs j=1 k −s = ζL (s)ζ(s). k=1 This completes the proof of the theorem. 37) for Nν (x), we can derive Weyl’s asymptotic law for fractal strings.

RN and gaps scaled by g1 , . . 4). 3. The gaps of L have lengths g1 L, . . , gK L. By abuse of language, we will usually refer to the quantities g1 , . . , gK as the gaps of the self-similar string L. 4. 1. Then the geometric zeta function of this string has a meromorphic continuation to the whole complex plane, given by ζL (s) = K s k=1 gk , N s j=1 rj Ls 1− for s ∈ C. 10) Here, L = ζL (1) is the total length of L, which is also the length of I, the initial interval from which L is constructed.

We note that by construction, the boundary ∂Ω of the Cantor string is equal to the ternary Cantor set. In general, the volume of the tubular neighborhood of the boundary of L is given by (see [LapPo2, Eq. 2), p. 48]) lj = 2ε · NL 2ε + V (ε) = j: lj ≥2ε j: lj <2ε 1 2ε lj . 3 When the two endpoints of an interval of length lj are covered by intervals of radius ε, then these discs overlap if lj < 2ε, covering a length lj , or they do not overlap if lj ≥ 2ε, in which case they cover a length of 2ε.