Description:

This  book  grew  out  of  courses which I taught  at  Cornell University  and

the University  of  Warwick during  1969  and  1970.  I wrote it because  of  a

strong belief  that  there should be readily available a semi-historical and geo-

metrically motivated exposition  of  J.  H. C. Whitehead's beautiful theory  of

simple-homotopy types; that the best way to understand this theory  is  to

know how  and  why it was built. This belief  is  buttressed by the fact that the

major uses of,  and  advances in, the theory in recent  times-for  example, the

s-cobordism theorem (discussed in  §25),  the use  of  the theory in surgery, its

extension  to  non-compact complexes (discussed  at  the end  of  §6)  and the

proof  of  topological invariance (given in the  Appendix)-have  come from

just such  an  understanding.

A second reason for writing the  book  is  pedagogical. This  is  an excellent

subject for a topology student  to  "grow  up"  on. The interplay between

geometry  and  algebra in topology, each enriching the other,  is  beautifully

illustrated in simple-homotopy theory. The subject  is  accessible (as in the

courses mentioned  at  the outset)  to  students who have  had  a good one-

semester course in algebraic topology. I have tried to write proofs which meet

the needs  of  such students. (When a  proof  was omitted  and  left as an exercise,

it was done with the welfare  of  the student in mind.  He  should do such

exercises zealously.)

There is some new material  here1-for  example, the completely geometric

definition  of  the Whitehead group  of  a complex in  §6,  the observations  on  the

counting  of  simple-homotopy types in  §24,  and  the direct  proof  of  the

equivalence  of  Milnor's definition  of  torsion with the classical definition,

given in  §16.  But my debt to previous works on the subject  is  very great.

I refer to [Kervaire-Maumary-deRham], [Milnor  1]  and  above all  [J.  H. C.

Whitehead  1,2,3,4].  The  reader should  turn  to  these sources for more material,

alternate viewpoints, etc.

I  am  indebted  to  Doug  Anderson  and  Paul Olum for many enlightening

discussions,  and  to  Roger Livesay  and  Stagg Newman for their eagle-eyed

reading  of  the original manuscript. Also I would like to express my apprecia-

tion  to  Arletta Havlik, Esther Monroe, Catherine Stevens  and  Dolores

Pendell for their competence  and  patience in typing the manuscript.

My research in simple-homotopy theory was partly supported by grants

from the  National  Science Foundation  and  the Science Research Council  of

Great Britain. I  and  my wife  and  my children are grateful  to  them.

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A Course in Simple Homotopy Theory PDF

124 Pages·1973·9.7244 MB·other
by  Marshall M. Cohen
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About A Course in Simple Homotopy Theory

<p>This&nbsp; book&nbsp; grew&nbsp; out&nbsp; of&nbsp; courses which I taught&nbsp; at&nbsp; Cornell University&nbsp; and</p><p>the University&nbsp; of&nbsp; Warwick during&nbsp; 1969&nbsp; and&nbsp; 1970.&nbsp; I wrote it because&nbsp; of&nbsp; a</p><p>strong belief&nbsp; that&nbsp; there should be readily available a semi-historical and geo-</p><p>metrically motivated exposition&nbsp; of&nbsp; J.&nbsp; H. C. Whitehead's beautiful theory&nbsp; of</p><p>simple-homotopy types; that the best way to understand this theory&nbsp; is&nbsp; to</p><p>know how&nbsp; and&nbsp; why it was built. This belief&nbsp; is&nbsp; buttressed by the fact that the</p><p>major uses of,&nbsp; and&nbsp; advances in, the theory in recent&nbsp; times-for&nbsp; example, the</p><p>s-cobordism theorem (discussed in&nbsp; §25),&nbsp; the use&nbsp; of&nbsp; the theory in surgery, its</p><p>extension&nbsp; to&nbsp; non-compact complexes (discussed&nbsp; at&nbsp; the end&nbsp; of&nbsp; §6)&nbsp; and the</p><p>proof&nbsp; of&nbsp; topological invariance (given in the&nbsp; Appendix)-have&nbsp; come from</p><p>just such&nbsp; an&nbsp; understanding.</p><p>A second reason for writing the&nbsp; book&nbsp; is&nbsp; pedagogical. This&nbsp; is&nbsp; an excellent</p><p>subject for a topology student&nbsp; to&nbsp; "grow&nbsp; up"&nbsp; on. The interplay between</p><p>geometry&nbsp; and&nbsp; algebra in topology, each enriching the other,&nbsp; is&nbsp; beautifully</p><p>illustrated in simple-homotopy theory. The subject&nbsp; is&nbsp; accessible (as in the</p><p>courses mentioned&nbsp; at&nbsp; the outset)&nbsp; to&nbsp; students who have&nbsp; had&nbsp; a good one-</p><p>semester course in algebraic topology. I have tried to write proofs which meet</p><p>the needs&nbsp; of&nbsp; such students. (When a&nbsp; proof&nbsp; was omitted&nbsp; and&nbsp; left as an exercise,</p><p>it was done with the welfare&nbsp; of&nbsp; the student in mind.&nbsp; He&nbsp; should do such</p><p>exercises zealously.)</p><p>There is some new material&nbsp; here1-for&nbsp; example, the completely geometric</p><p>definition&nbsp; of&nbsp; the Whitehead group&nbsp; of&nbsp; a complex in&nbsp; §6,&nbsp; the observations&nbsp; on&nbsp; the</p><p>counting&nbsp; of&nbsp; simple-homotopy types in&nbsp; §24,&nbsp; and&nbsp; the direct&nbsp; proof&nbsp; of&nbsp; the</p><p>equivalence&nbsp; of&nbsp; Milnor's definition&nbsp; of&nbsp; torsion with the classical definition,</p><p>given in&nbsp; §16.&nbsp; But my debt to previous works on the subject&nbsp; is&nbsp; very great.</p><p>I refer to [Kervaire-Maumary-deRham], [Milnor&nbsp; 1]&nbsp; and&nbsp; above all&nbsp; [J.&nbsp; H. C.</p><p>Whitehead&nbsp; 1,2,3,4].&nbsp; The&nbsp; reader should&nbsp; turn&nbsp; to&nbsp; these sources for more material,</p><p>alternate viewpoints, etc.</p><p>I&nbsp; am&nbsp; indebted&nbsp; to&nbsp; Doug&nbsp; Anderson&nbsp; and&nbsp; Paul Olum for many enlightening</p><p>discussions,&nbsp; and&nbsp; to&nbsp; Roger Livesay&nbsp; and&nbsp; Stagg Newman for their eagle-eyed</p><p>reading&nbsp; of&nbsp; the original manuscript. Also I would like to express my apprecia-</p><p>tion&nbsp; to&nbsp; Arletta Havlik, Esther Monroe, Catherine Stevens&nbsp; and&nbsp; Dolores</p><p>Pendell for their competence&nbsp; and&nbsp; patience in typing the manuscript.</p><p>My research in simple-homotopy theory was partly supported by grants</p><p>from the&nbsp; National&nbsp; Science Foundation&nbsp; and&nbsp; the Science Research Council&nbsp; of</p><p>Great Britain. I&nbsp; and&nbsp; my wife&nbsp; and&nbsp; my children are grateful&nbsp; to&nbsp; them.</p>

Detailed Information

Author:Marshall M. Cohen
Publication Year:1973
ISBN:387900551
Pages:124
Language:other
File Size:9.7244
Format:PDF
Price:FREE
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