By Edward H. Spanier

Meant to be used either as a textual content and a reference, this booklet is an exposition of the basic rules of algebraic topology. the 1st 3rd of the e-book covers the elemental crew, its definition and its software within the research of overlaying areas. the focal point then turns to homology thought, together with cohomology, cup items, cohomology operations, and topological manifolds. the rest 3rd of the publication is dedicated to Homotropy concept, overlaying easy proof approximately homotropy teams, purposes to obstruction idea, and computations of homotropy teams of spheres. within the later elements, the most emphasis is at the software to geometry of the algebraic instruments built previous.

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**Example text**

The original proof in [Hul] was much more complicated and was under the assumption that R has no zero divisors. Though in this work we do not need that proof, the idea and the method used in that proof might be useful for further studies, especially of the maps E and E- 1 • Also, that proof involves some interesting formal calculus and is related to formal groups. So here we give the heuristic idea of that proof and leave the rigorization to the reader as an exercise. L. 00 3=30 F(z) = J (E:-1(a)) .

2) w=oo I = W=Zi lim '¢i(W) - z· W ..... ziW "# 0, i = 1, .. ,n. 7.. (1 .. 3 3) Proof. We prove only the case k = n. The other cases are the same. 1, any sphere is complex analytically isomorphic to the sphere C. Note that the complex analytic isomorphisms of C are the projective transformations. 4 ) be a sphere with tubes of type (1, n) and F: S-+C a complex analytic isomorphism. We have a sphere with tubes (C; F(po), ... , F(Pn); (F(Uo),

Let n E Z+ and k an integer satisfying 1 ::; k ::; n. Then any sphere with tubes of type (1, n) is conformally equivalent to a sphere with tubes of type (1, n) and of the form (C; 00, Zl,···, Zn; (B';:" 'l/Jo), (B::, 'l/Jl), ... 3. THE MODULI SPACES OF SPHERES WITH TUBES 23 where Zk = 0, Zi E ex for i "# k satisfying Zi "# Zj for i "# j, ri E lR+, i = 0, ... , n, and '¢o, ... , '¢n are analytic functions on B~, B;~, ... , B;:, respectively, such that '¢o( 00) = 0, d'¢o( w) . = wlim w'¢o(w) = .....