Induced mappings of homology decompositions

Martin Arkowitz

Banach Center Publications (1998)

  • Volume: 45, Issue: 1, page 225-233
  • ISSN: 0137-6934

Abstract

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We give conditions for a map of spaces to induce maps of the homology decompositions of the spaces which are compatible with the homology sections and dual Postnikov invariants. Several applications of this result are obtained. We show how the homotopy type of the (n+1)st homology section depends on the homotopy type of the nth homology section and the (n+1)st homology group. We prove that all homology sections of a co-H-space are co-H-spaces, all n-equivalences of the homology decomposition are co-H-maps and, under certain restrictions, all dual Postnikov invariants are co-H-maps. We give a new proof of a result of Berstein and Hilton which gives conditions for a co-H-space to be a suspension.

How to cite

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Arkowitz, Martin. "Induced mappings of homology decompositions." Banach Center Publications 45.1 (1998): 225-233. <http://eudml.org/doc/208905>.

@article{Arkowitz1998,
abstract = {We give conditions for a map of spaces to induce maps of the homology decompositions of the spaces which are compatible with the homology sections and dual Postnikov invariants. Several applications of this result are obtained. We show how the homotopy type of the (n+1)st homology section depends on the homotopy type of the nth homology section and the (n+1)st homology group. We prove that all homology sections of a co-H-space are co-H-spaces, all n-equivalences of the homology decomposition are co-H-maps and, under certain restrictions, all dual Postnikov invariants are co-H-maps. We give a new proof of a result of Berstein and Hilton which gives conditions for a co-H-space to be a suspension.},
author = {Arkowitz, Martin},
journal = {Banach Center Publications},
keywords = {dual Postnikov invariants; homology sections; co-H-space},
language = {eng},
number = {1},
pages = {225-233},
title = {Induced mappings of homology decompositions},
url = {http://eudml.org/doc/208905},
volume = {45},
year = {1998},
}

TY - JOUR
AU - Arkowitz, Martin
TI - Induced mappings of homology decompositions
JO - Banach Center Publications
PY - 1998
VL - 45
IS - 1
SP - 225
EP - 233
AB - We give conditions for a map of spaces to induce maps of the homology decompositions of the spaces which are compatible with the homology sections and dual Postnikov invariants. Several applications of this result are obtained. We show how the homotopy type of the (n+1)st homology section depends on the homotopy type of the nth homology section and the (n+1)st homology group. We prove that all homology sections of a co-H-space are co-H-spaces, all n-equivalences of the homology decomposition are co-H-maps and, under certain restrictions, all dual Postnikov invariants are co-H-maps. We give a new proof of a result of Berstein and Hilton which gives conditions for a co-H-space to be a suspension.
LA - eng
KW - dual Postnikov invariants; homology sections; co-H-space
UR - http://eudml.org/doc/208905
ER -

References

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  1. [Ar1] M. Arkowitz, The group of self-homotopy equivalences - A survey, Groups of Self-Homotopy Equivalences and Related Topics, Lecture Notes in Math. 1425, Springer-Verlag 1990, 170-203. 
  2. [Ar2] M. Arkowitz, Co-H-spaces, Handbook of Algebraic Topology, Elsevier Science, North Holland, 1995, 1143-1173. 
  3. [A-G] M. Arkowitz and M. Golasiński, Co-H-structures on Moore spaces of type (G,2), Can. J. of Math. 46 (1994), 673-686. Zbl0829.55006
  4. [A-M] M. Arkowitz and K. Maruyama, z Self homotopy equivalences which induce the identity on homology, cohomology or homotopy groups, Topology Appl. (to appear). 
  5. [B-H1] I. Berstein and P. Hilton, Category and generalized Hopf invariants, Ill. J. of Math. 4 (1960), 437-451. Zbl0113.38301
  6. [B-H2] I. Berstein and P. Hilton, On suspensions and comultiplications, Topology 2 (1963), 73-82. Zbl0115.40403
  7. [B-C] E. Brown and A. Copeland, An homology analogue of Postnikov systems, Mich. Math. J. 6 (1959), 313-330. Zbl0093.37203
  8. [Cu1] C. Curjel, On the homology decomposition of polyhedra, Ill. J. of Math. 7 (1963), 121-136. Zbl0115.17002
  9. [Cu2] C. Curjel, A note on spaces of category ≤ 2, Math. Zeit. 80 (1963), 293-299. Zbl0105.17102
  10. [G-K] M. Golasiński and J. Klein, On maps into a co-H-space, (preprint). 
  11. [Hi1] P. Hilton, Homotopy and Duality, Gordon and Breach, 1965. 
  12. [Hi2] P. Hilton, On excision and principal fibrations, Comm. Math. Helv. 35 (1961), 77-84. Zbl0107.16804
  13. [Sp] E. Spanier, Algebraic Topology, McGraw-Hill, 1966. 
  14. [Wh] G. Whitehead, Elements of Homotopy Theory, Graduate Texts in Math. 61, Springer-Verlag (1978). 

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