By William S. Massey

"This booklet is meant to function a textbook for a path in algebraic topology in the beginning graduate point. the most issues coated are the category of compact 2-manifolds, the elemental team, overlaying areas, singular homology thought, and singular cohomology thought. those subject matters are constructed systematically, warding off all pointless definitions, terminology, and technical equipment. anyplace attainable, the geometric motivation at the back of a few of the options is emphasised. The textual content involves fabric from the 1st 5 chapters of the author's prior ebook, ALGEBRAIC TOPOLOGY: AN advent (GTM 56), including just about all of the now out-of-print SINGULAR HOMOLOGY conception (GTM 70). the cloth from the sooner books has been conscientiously revised, corrected, and taken as much as date."

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N ∈ F [X ]n×n be j the corresponding Jacobian matrix. Show that if detV is the trivial representation, then we have det J(f1 , . . , fn ) ∈ S[V ]G . ,n ) ∈ F [X ]n×n denote the correi sponding Hessian matrix. Show that if (detV )2 is the trivial representation, then we have det H(f ) ∈ S[V ]G . 13) Exercise: Jacobian criterion. Let F be a field such that char(F ) = 0 and let F [X ] := F [X1 , . . , Xn ]. Let n pn,k := i=1 Xik ∈ F [X ], for k ∈ N, be the power sums, and let en,1 , . . , en,n ∈ F [X ] be the elementary symmetric polynomials, where deg(en,i ) = i.

Let F be a field such that char(F ) = 2. An element f ∈ F [X ] := F [X1 , . . , Xn ] is called alternating, if f π = sgn(π) · f for all π ∈ Sn . 16). b) Show that F [X ]An = (1 · F [X ]Sn ) (∆n · F [X ]Sn ) as F [X ]Sn -modules. Conclude that F [X ]An is not a polynomial ring. 18) Exercise: Reflection representations of Sn . Let n ∈ N and let W be the natural permutation QSn -module, having permutation Q-basis {b1 , . . , bn } ⊆ W . n a) Show that W := i=1 bi Q ≤ W is a trivial QSn -submodule, and that V := W/W is an absolutely irreducible faithful reflection representation of Sn .

Let I ⊆ R1×3 be the regular icosahedron, one of the platonic solids. The faces of I consist of regular triangles, where at each vertex 5 triangles meet. Let f, e, v ∈ N be the number of faces, edges, and vertices of I, respectively. Hence by Euler’s Polyhedron Theorem we have f − e + v = 2. Since we have e = 3f 2 and v = 3f 5 , we conclude f = 20 and e = 30 as well as v = 12. Let G := {π ∈ O3 (R); Iπ = I} ≤ O3 (R) be the symmetry group of I, where we assume I ⊆ R1×3 to be centred at the origin, and where O3 (R) is the isometry group of the Euclidean space R3 .

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