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The derivatives of homotopy theory
Author:
Brenda Johnson
Journal:
Trans. Amer. Math. Soc. 347 (1995), 1295-1321
MSC:
Primary 55P65
MathSciNet review:
1297532
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Abstract: We construct a functor of spaces,  , and show that its multilinearization is equivalent to the  th layer of the Taylor tower of the identity functor of spaces. This allows us to identify the derivatives of the identity functor and determine their homotopy type.
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A.
K. Bousfield and D.
M. Kan, Homotopy limits, completions and localizations,
Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin, 1972. MR 0365573
(51 #1825)
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M.
G. Barratt and J.
H. C. Whitehead, The first nonvanishing group of an
(𝑛+1)-ad, Proc. London Math. Soc. (3) 6
(1956), 417–439. MR 0085509
(19,52c)
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Graham
Ellis and Richard
Steiner, Higher-dimensional crossed modules and the homotopy groups
of (𝑛+1)-ads, J. Pure Appl. Algebra 46
(1987), no. 2-3, 117–136. MR 897011
(88j:55010), http://dx.doi.org/10.1016/0022-4049(87)90089-2
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Thomas
G. Goodwillie, Calculus. I. The first derivative of pseudoisotopy
theory, 𝐾-Theory 4 (1990), no. 1,
1–27. MR
1076523 (92m:57027), http://dx.doi.org/10.1007/BF00534191
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Thomas
G. Goodwillie, Calculus. II. Analytic functors,
𝐾-Theory 5 (1991/92), no. 4, 295–332.
MR
1162445 (93i:55015), http://dx.doi.org/10.1007/BF00535644
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-, Calculus III: The Taylor series of a homotopy functor,
 -Theory (to appear).
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Marshall
Hall Jr., A basis for free Lie rings and higher
commutators in free groups, Proc. Amer. Math.
Soc. 1 (1950),
575–581. MR 0038336
(12,388a), http://dx.doi.org/10.1090/S0002-9939-1950-0038336-7
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P.
J. Hilton, On the homotopy groups of the union of spheres, J.
London Math. Soc. 30 (1955), 154–172. MR 0068218
(16,847d)
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I.
M. James, Reduced product spaces, Ann. of Math. (2)
62 (1955), 170–197. MR 0073181
(17,396b)
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B. Johnson, The derivatives of homotopy theory, Thesis, Brown Univ., 1991.
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J. Milnor, On the construction
 , Algebraic Topology: A Student's Guide by J. F. Adams, London Math. Soc. Lecture Note Series 4, Cambridge Univ. Press, Cambridge, 1972, pp. 119-135.
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J. Rognes, The rank filtration in algebraic
 -theory, Thesis, Princeton Univ., 1990.
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Friedhelm
Waldhausen, Algebraic 𝐾-theory of topological spaces.
II, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus,
Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin,
1979, pp. 356–394. MR 561230
(81i:18014b)
- [Wh]
George
W. Whitehead, Elements of homotopy theory, Graduate Texts in
Mathematics, vol. 61, Springer-Verlag, New York, 1978. MR 516508
(80b:55001)
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E. Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math. 177 (1937), 152-160.
- [B-K]
- A. K. Bousfield and D. M. Kan, Homotopy limits, completions, and localizations, Lecture Notes in Math., vol. 304, Springer-Verlag, Berlin, 1972. MR 0365573 (51:1825)
- [B-W]
- M. G. Barratt and J. H. C. Whitehead, The first non-vanishing group of an
 -ad, Proc. London Math. Soc. 6 (3) (1956), 417-439. MR 0085509 (19:52c)
- [E-S]
- G. Ellis and R. Steiner, Higher dimensional crossed modules and the homotopy groups of
 -ads, J. Pure Appl. Algebra 46 (1987), 117-136. MR 897011 (88j:55010)
- [G1]
- T. G. Goodwillie, Calculus I: The first derivative of pseudoisotopy theory,
 -Theory 4 (1990), 1-27. MR 1076523 (92m:57027)
- [G2]
- -, Calculus II: Analytic functors,
 -Theory 5 (1992), 295-332. MR 1162445 (93i:55015)
- [G3]
- -, Calculus III: The Taylor series of a homotopy functor, -Theory (to appear).
- [Ha]
- M. Hall, Jr., A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc. 1 (1950), 575-581. MR 0038336 (12:388a)
- [Hi]
- P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955), 154-172. MR 0068218 (16:847d)
- [Ja]
- I. M. James, Reduced product spaces, Ann. of Math. 62 (1955), 170-197. MR 0073181 (17:396b)
- [J]
- B. Johnson, The derivatives of homotopy theory, Thesis, Brown Univ., 1991.
- [M]
- J. Milnor, On the construction , Algebraic Topology: A Student's Guide by J. F. Adams, London Math. Soc. Lecture Note Series 4, Cambridge Univ. Press, Cambridge, 1972, pp. 119-135.
- [R]
- J. Rognes, The rank filtration in algebraic -theory, Thesis, Princeton Univ., 1990.
- [Wa]
- F. Waldhausen, Algebraic
 -theory of topological spaces II, Algebraic Topology-Aarhus 1978, edited by J. F. Dupont and I. H. Madsen, Lecture Notes in Math., vol. 763, Springer-Verlag, Berlin, 1979, pp. 356-394. MR 561230 (81i:18014b)
- [Wh]
- G. W. Whitehead, Elements of homotopy theory, Springer-Verlag, New York, 1978. MR 516508 (80b:55001)
- [Wi]
- E. Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math. 177 (1937), 152-160.
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DOI:
http://dx.doi.org/10.1090/S0002-9947-1995-1297532-6
PII:
S 0002-9947(1995)1297532-6
Article copyright:
© Copyright 1995 American Mathematical Society
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