Topology for physicists
"This is a very interesting book on an important topic both for physics and for mathematics. (...) It starts at the beginning, but is not really for beginners; the physics background develops rapidly, through seven short chapters, and the final eight chapters provide a lightning review of the mathematical topics encountered (...) Part II is the main part of the text, containing a selection of fascinating topics, beautifully presented, to many of which the author has been a significant contributor. The chapters on functional integration, on elliptic operators, their determinants and related index theorems, on calculating instanton contributions and on anomalies are particularly attractive. (...)""Bulletin London Mathematical Society"
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The Degree of a Map
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action of G adjoint algebra g antisymmetric Betti number boundary called cell decomposition characteristic classes charts Ck(X closed cochain coefficients cohomology class coincides compute consider construction coordinate system covering curve cycle defined definition denote dimension direct sum element equation exact example fc-cycle fc-form fiber fibration follows function G acts gauge field given GL(n gluing Hk(X homeomorphic homology groups homotopically equivalent homotopy class homotopy lifting property identity integral invariant isomorphic jacobian Lie algebra Lie group G linear loop map Sk matrix modulo n-dimensional neighborhood nk(E nonzero null-homotopic obtained one-form one-to-one correspondence open set open subset orbit oriented path principal fibration quotient regarded Riemannian metric satisfies scalar product Section sequence simply connected singular smooth manifold smooth map SO(n sphere spheroid subgroup subspace surface tangent space tangent vector tensor theorem topological space transformations trivial vector field vector space zero