By Massey

This ebook is meant to function a textbook for a direction in algebraic topology firstly graduate point. the most themes coated are the category of compact 2-manifolds, the elemental team, protecting areas, singular homology conception, and singular cohomology idea. those issues are constructed systematically, fending off all unecessary definitions, terminology, and technical equipment. anywhere attainable, the geometric motivation at the back of a number of the recommendations is emphasised. The textual content comprises fabric from the 1st 5 chapters of the author's prior e-book, ALGEBRAIC TOPOLOGY: AN creation (GTM 56), including just about all of the now out-of- print SINGULAR HOMOLOGY conception (GTM 70). the fabric from the sooner books has been conscientiously revised, corrected, and taken modern.

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I. Gelfand. A certain category of g-modules. Funkt. Analis i ego Prilozh, 10(2):1–8, 1976. English Translation: Funct. Anal. Appl. 12:4 (1978), 79-80. [Car05] Roger William Carter. Lie algebras of finite and affine type. Number 96 in Cambridge studies in advanced mathematics. Cambridge University Press, 2005. [FH91] William Fulton and Joe Harris. Representation theory: a first course, volume 129. Springer, 1991. [Hum72] James E Humphreys. Introduction to Lie algebras and representation theory, volume 9.

Histoire Math, 5:249–284, 1999. B. Vinberg. Discrete linear groups generated by reflections. Izvestija AN USSR, 35:1072–1112, 1971. English Translation: Math. USSRIzvestija 5 (1971), 1083-1119.

9) 0 = U0 ⊆ U1 ⊆ . . Um ⊆ Ws−1 ⊆ V. 10) To know that these are filtrations of the right type, observe that by the second isomorphism theorem, Vt−1 /U ∼ = (Vt−1 + Ws−1 )/Ws−1 Ws−1 /U ∼ = (Vt−1 + Ws−1 )/Vt−1 . 11) That these composition factors have the right property follows. 5) share leading terms, L(µ) has the same multiplicity in each. 6). 10). Since these filtrations differ only by the first two factors, we check that the number of times L(µ) appears among the first two factors is the same.