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001 978-1-4612-4576-6
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007 cr nn 008mamaa
008 110927s1988 xxu| s |||| 0|eng d
020 _a9781461245766
_9978-1-4612-4576-6
024 7 _a10.1007/978-1-4612-4576-6
_2doi
050 4 _aQA612-612.8
072 7 _aPBPD
_2bicssc
072 7 _aMAT038000
_2bisacsh
072 7 _aPBPD
_2thema
082 0 4 _a514.2
_223
100 1 _aRotman, Joseph J.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
245 1 3 _aAn Introduction to Algebraic Topology
_h[electronic resource] /
_cby Joseph J. Rotman.
250 _a1st ed. 1988.
264 1 _aNew York, NY :
_bSpringer New York :
_bImprint: Springer,
_c1988.
300 _aXIV, 438 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aGraduate Texts in Mathematics,
_x0072-5285 ;
_v119
505 0 _a0 Introduction -- Notation -- Brouwer Fixed Point Theorem -- Categories and Functors -- 1.Some Basic Topological Notions -- Homotopy -- Convexity, Contractibility, and Cones -- Paths and Path Connectedness -- 2 Simplexes -- Affine Spaces -- Affine Maps -- 3 The Fundamental Group -- The Fundamental Groupoid -- The Functor ?1 -- ?1(S1) -- 4 Singular Homology -- Holes and Green's Theorem -- Free Abelian Groups -- The Singular Complex and Homology Functors -- Dimension Axiom and Compact Supports -- The Homotopy Axiom -- The Hurewicz Theorem -- 5 Long Exact Sequences -- The Category Comp -- Exact Homology Sequences -- Reduced Homology -- 6 Excision and Applications -- Excision and Mayer-Vietoris -- Homology of Spheres and Some Applications -- Barycentric Subdivision and the Proof of Excision -- More Applications to Euclidean Space -- 7 Simplicial Complexes -- Definitions -- Simplicial Approximation -- Abstract Simplicial Complexes -- Simplicial Homology -- Comparison with Singular Homology -- Calculations -- Fundamental Groups of Polyhedra -- The Seifert-van Kampen Theorem -- 8 CW Complexes -- Hausdorff Quotient Spaces -- Attaching Cells -- Homology and Attaching Cells -- CW Complexes -- Cellular Homology -- 9 Natural Transformations -- Definitions and Examples -- Eilenberg-Steenrod Axioms -- Chain Equivalences -- Acyclic Models -- Lefschetz Fixed Point Theorem -- Tensor Products -- Universal Coefficients -- Eilenberg-Zilber Theorem and the Künneth Formula -- 10 Covering Spaces -- Basic Properties -- Covering Transformations -- Existence -- Orbit Spaces -- 11 Homotopy Groups -- Function Spaces -- Group Objects and Cogroup Objects -- Loop Space and Suspension -- Homotopy Groups -- Exact Sequences -- Fibrations -- A Glimpse Ahead -- 12 Cohomology -- Differential Forms -- Cohomology Groups -- Universal Coefficients Theorems for Cohomology -- Cohomology Rings -- Computations and Applications -- Notation.
520 _aThere is a canard that every textbook of algebraic topology either ends with the definition of the Klein bottle or is a personal communication to J. H. C. Whitehead. Of course, this is false, as a glance at the books of Hilton and Wylie, Maunder, Munkres, and Schubert reveals. Still, the canard does reflect some truth. Too often one finds too much generality and too little attention to details. There are two types of obstacle for the student learning algebraic topology. The first is the formidable array of new techniques (e. g. , most students know very little homological algebra); the second obstacle is that the basic defini­ tions have been so abstracted that their geometric or analytic origins have been obscured. I have tried to overcome these barriers. In the first instance, new definitions are introduced only when needed (e. g. , homology with coeffi­ cients and cohomology are deferred until after the Eilenberg-Steenrod axioms have been verified for the three homology theories we treat-singular, sim­ plicial, and cellular). Moreover, many exercises are given to help the reader assimilate material. In the second instance, important definitions are often accompanied by an informal discussion describing their origins (e. g. , winding numbers are discussed before computing 1tl (Sl), Green's theorem occurs before defining homology, and differential forms appear before introducing cohomology). We assume that the reader has had a first course in point-set topology, but we do discuss quotient spaces, path connectedness, and function spaces.
650 0 _aAlgebraic topology.
650 1 4 _aAlgebraic Topology.
_0https://scigraph.springernature.com/ontologies/product-market-codes/M28019
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9781461289302
776 0 8 _iPrinted edition:
_z9780387966786
776 0 8 _iPrinted edition:
_z9781461245773
830 0 _aGraduate Texts in Mathematics,
_x0072-5285 ;
_v119
856 4 0 _uhttps://doi.org/10.1007/978-1-4612-4576-6
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
912 _aZDB-2-BAE
999 _c9419
_d9419