243:
379:
154:
343:
318:
278:
181:
298:
104:
17:
186:
488:
107:, which was conceived as a first approximation to homotopy type. Thus Spanier–Whitehead duality fits into
353:
130:
511:
506:
108:
323:
346:
303:
467:
410:
388:
256:
159:
124:
8:
250:
392:
441:
419:
374:
283:
93:
484:
424:
357:
63:
44:
455:
414:
396:
476:
463:
406:
67:
40:
380:
Proceedings of the
National Academy of Sciences of the United States of America
99:
The basic point is that sphere complements determine the homology, but not the
36:
459:
500:
437:
370:
100:
89:
85:
428:
401:
446:
246:
28:
81:
71:
238:{\displaystyle \Sigma ^{-n}\Sigma '(\mathbb {R} ^{n}\setminus X)}
55:
326:
306:
286:
259:
253:
with the smash product as a monoidal structure. Here
189:
162:
133:
80:, but this can now cause possible confusion with the
377:(1953), "A first approximation to homotopy theory",
50:may be considered as dual to its complement in the
483:, European Mathematical Society Publishing House,
352:Taking homology and cohomology with respect to an
337:
312:
292:
272:
237:
175:
148:
103:, in general. What is determined, however, is the
498:
436:
369:
475:
418:
400:
216:
136:
96:, who developed it in papers from 1955.
444:(1955), "Duality in homotopy theory.",
14:
499:
43:, based on a geometrical idea that a
74:. The theory is also referred to as
62:is large enough. Its origins lie in
24:
328:
307:
204:
191:
25:
523:
226:
149:{\displaystyle \mathbb {R} ^{n}}
33:Spanier–Whitehead duality
232:
211:
13:
1:
363:
114:
70:, concerning complements in
7:
249:in the category of pointed
18:S-duality (homotopy theory)
10:
528:
354:Eilenberg–MacLane spectrum
345:are reduced and unreduced
460:10.1112/s002557930000070x
338:{\displaystyle \Sigma '}
313:{\displaystyle \Sigma }
339:
314:
294:
274:
239:
177:
150:
109:stable homotopy theory
402:10.1073/pnas.39.7.655
340:
315:
295:
275:
273:{\displaystyle X^{+}}
240:
178:
176:{\displaystyle X^{+}}
151:
324:
304:
284:
257:
187:
160:
131:
125:neighborhood retract
105:stable homotopy type
442:Whitehead, J. H. C.
393:1953PNAS...39..655S
375:Whitehead, J. H. C.
481:Algebraic topology
335:
310:
290:
270:
235:
173:
146:
94:J. H. C. Whitehead
88:. It is named for
490:978-3-03719-048-7
438:Spanier, Edwin H.
371:Spanier, Edwin H.
358:Alexander duality
293:{\displaystyle X}
64:Alexander duality
45:topological space
16:(Redirected from
519:
512:Duality theories
493:
477:tom Dieck, Tammo
470:
431:
422:
404:
344:
342:
341:
336:
334:
319:
317:
316:
311:
299:
297:
296:
291:
280:is the union of
279:
277:
276:
271:
269:
268:
244:
242:
241:
236:
225:
224:
219:
210:
202:
201:
182:
180:
179:
174:
172:
171:
155:
153:
152:
147:
145:
144:
139:
21:
527:
526:
522:
521:
520:
518:
517:
516:
507:Homotopy theory
497:
496:
491:
366:
327:
325:
322:
321:
305:
302:
301:
285:
282:
281:
264:
260:
258:
255:
254:
220:
215:
214:
203:
194:
190:
188:
185:
184:
167:
163:
161:
158:
157:
140:
135:
134:
132:
129:
128:
117:
68:homology theory
41:homotopy theory
23:
22:
15:
12:
11:
5:
525:
515:
514:
509:
495:
494:
489:
472:
471:
433:
432:
387:(7): 655–660,
365:
362:
349:respectively.
333:
330:
309:
289:
267:
263:
234:
231:
228:
223:
218:
213:
209:
206:
200:
197:
193:
170:
166:
143:
138:
116:
113:
37:duality theory
9:
6:
4:
3:
2:
524:
513:
510:
508:
505:
504:
502:
492:
486:
482:
478:
474:
473:
469:
465:
461:
457:
453:
449:
448:
443:
439:
435:
434:
430:
426:
421:
416:
412:
408:
403:
398:
394:
390:
386:
382:
381:
376:
372:
368:
367:
361:
359:
355:
350:
348:
331:
300:and a point,
287:
265:
261:
252:
248:
229:
221:
207:
198:
195:
168:
164:
141:
126:
123:be a compact
122:
112:
110:
106:
102:
101:homotopy type
97:
95:
91:
90:Edwin Spanier
87:
86:string theory
83:
79:
78:
73:
69:
65:
61:
57:
53:
49:
46:
42:
38:
34:
30:
19:
480:
451:
445:
384:
378:
351:
247:dual objects
120:
118:
98:
76:
75:
59:
51:
47:
32:
26:
447:Mathematika
347:suspensions
66:theory, in
29:mathematics
501:Categories
364:References
360:formally.
454:: 56–80,
356:recovers
329:Σ
308:Σ
227:∖
205:Σ
196:−
192:Σ
115:Statement
82:S-duality
77:S-duality
72:manifolds
479:(2008),
429:16589320
332:′
208:′
58:, where
468:0074823
420:1063840
411:0056290
389:Bibcode
251:spectra
156:. Then
487:
466:
427:
417:
409:
56:sphere
35:is a
485:ISBN
425:PMID
320:and
245:are
183:and
119:Let
92:and
456:doi
415:PMC
397:doi
127:in
84:of
39:in
27:In
503::
464:MR
462:,
450:,
440:;
423:,
413:,
407:MR
405:,
395:,
385:39
383:,
373:;
111:.
31:,
458::
452:2
399::
391::
288:X
266:+
262:X
233:)
230:X
222:n
217:R
212:(
199:n
169:+
165:X
142:n
137:R
121:X
60:n
54:-
52:n
48:X
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.