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Homology groups of types in stable theories and the Hurewicz correspondence [article]

John Goodrick, Byunghan Kim, Alexei Kolesnikov
2016 arXiv   pre-print
We give an explicit description of the homomorphism group H_n(p) of a strong type p in any stable theory under the assumption that for every non-forking extension q of p the groups H_i(q) are trivial for  ...  We call this the "Hurewicz correspondence" in analogy with the Hurewicz Theorem in algebraic topology.  ...  In the previous paper [5] , we introduced a notion of homology groups for a complete strong type in any stable, or even rosy, first-order theory.  ... 
arXiv:1412.3864v2 fatcat:xx6axk5xordu3e4zyr2c5zo2gm

Homology groups of types in stable theories and the Hurewicz correspondence

John Goodrick, Byunghan Kim, Alexei Kolesnikov
2017 Annals of Pure and Applied Logic  
We give an explicit description of the homology group H n (p) of a strong type p in any stable theory under the assumption that for every non-forking extension q of p the groups H i (q) are trivial for  ...  We call this the "Hurewicz correspondence" by analogy with the Hurewicz Theorem in algebraic topology. els. North Holland, 1990.  ...  In the previous paper [5] , we introduced a notion of homology groups for a complete strong type in any stable, or even rosy, first-order theory.  ... 
doi:10.1016/j.apal.2017.03.007 fatcat:pipqtxckhfdfxkykvjjqe4lcr4

Page 2020 of Mathematical Reviews Vol. 50, Issue 6 [page]

1975 Mathematical Reviews  
be a Moore space of type (Z,, 1), and let 4,(M,) | be the stable homotopy group {S*M,, M,}.  ...  homology groups and their limits [the reviewer, Topics in topology (Proc.  ... 

When certain natural maps are equivalences

Richard Holzsager
1972 Pacific Journal of Mathematics  
and homology groups.  ...  Under the assumption (made throughout) that the spaces have the homotopy type of connected CW-complexes, these are actually questions about relationships among the homotopy groups, stable homotopy groups  ...  Since both homology and stable homotopy are generalized homology theories, the same holds for M(Q, 2k + 2) = SM(Q, 2k + 1).  ... 
doi:10.2140/pjm.1972.42.69 fatcat:pcbpk6fk2jhpvp7lhtoyprpjvy

Page 3023 of Mathematical Reviews Vol. , Issue 85g [page]

1985 Mathematical Reviews  
Author’s summary: “We study the relation between a generalized homology theory and its coefficient group E,. For any element a € m.  ...  The author’s methods are to study the map j, together with the interplay of two natural types of homology operations in H.(T],>0QDn,qX).  ... 

Page 348 of Mathematical Reviews Vol. , Issue 98A [page]

1998 Mathematical Reviews  
Thompson (1-CUNYH-MS; New York, NY) 98a:55007 S5N20 19DS0 19DS55 55P42 55Q45 55845 Arlettaz, Dominique (CH-LAUS; Lausanne) The exponent of the homotopy groups of Moore spectra and the stable Hurewicz homomorphism  ...  For any abelian group A there is, up to homotopy, only one simply connected CW-complex X whose re- duced homology groups are isomorphic to A in degree n and vanish in any other degree.  ... 

Page 3956 of Mathematical Reviews Vol. , Issue 94g [page]

1994 Mathematical Reviews  
The authors also show in a clever way that this result can be stated in terms of the Brin-Thickstun proper homotopy groups and the homology groups of locally finite cycles.  ...  Math. 132 (1989), no. 3, 195-214; MR 90k:55014] proved a proper Hurewicz theorem involving the conical proper homotopy groups of Brin and Thickstun and the homology of a chain complex which is generated  ... 

The Hurewicz homomorphism and numerical forms

J. Hubbuck
1998 Banach Center Publications  
The data he chose to use in the classification of small complexes, stressed the role of the classical Hurewicz homomorphism from homotopy to homology groups.  ...  If X is a complex of finite type and X Y ∨ Z where Y is 2q − 1 connected and H 2q (Y, Z) is infinite, then there exists a stable map f : S 2q → Y inducing an injection in homology in degree 2q by inclusion  ... 
doi:10.4064/-45-1-235-240 fatcat:ozatl7fhzzbpre3xsamsnyirre

Page 743 of Mathematical Reviews Vol. 47, Issue 3 [page]

1974 Mathematical Reviews  
(iii) the homology suspensions H,(Q"X)—> H,,,(X) are isomorphisms for all r, and (iv) the maps from homotopy groups to stable homotopy groups are all isomorphisms?  ...  The main part of the paper consists of the proof of the following theorem: For n odd, the spaces CP,**+" and CP,,"*" are of the same stable homotopy type if and only if k—m =0mod M,,,.  ... 

Page 6413 of Mathematical Reviews Vol. , Issue 98J [page]

1998 Mathematical Reviews  
Zabrodsky’s lemma has been applied in connection with the Sullivan conjecture, p-compact groups, and unstable localization theory.  ...  6413 other extreme, taking ¥, to be the family of all subgroups of £,, for each n gives B,, equal to a point and so the group completion is Z= K(Z,0). The interest in the paper lies in the examples.  ... 

Page 879 of Mathematical Reviews Vol. , Issue 87b [page]

1987 Mathematical Reviews  
The author proves a Hurewicz theorem (absolute and relative case) and a Whitehead theorem for pointed (metric) continua, involving strong shape groups 7*, Steenrod homology groups H, and strong shape morphisms  ...  In particular, if MA is the Moore spectrum for the abelian group A (i.e. the suspension spectrum of the corresponding Moore space) we identify its func- tional dual even when A is not finitely generated  ... 

Page 4430 of Mathematical Reviews Vol. , Issue 92h [page]

1992 Mathematical Reviews  
For n > 3 the Hurewicz theorem gives an isomorphism of 2, X-modules p,X =C,X = H,(X", X"-') and so the homology groups of p(X) are H,(X) for n > 4, the image of the Hurewicz map for X for n = 3 and 2,,  ...  In this case the nonabelian part of ¢X, which lies in degrees 1, 2 and 3 and has the structure of a quadratic module, corresponds to the topological part X> of a homotopy system of order 4.  ... 

Page 168 of Mathematical Reviews Vol. 50, Issue 1 [page]

1975 Mathematical Reviews  
Suppose that Y¥>pXeS is a “chain functor with unit” corresponding to a homology theory with unit, and this Il, ¥—>II,pX is the Hurewicz homomorphism.  ...  The author generalizes Bott’s method for calculating the Hopf algebra structure of H,(QG) to the calculation of K,(QG), K, being the corresponding homology theory of K-theory.  ... 

Page 201 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 89, Issue 1 [page]

1958 American Mathematical Society. Transactions of the American Mathematical Society  
He outlined a new approach using inverse and direct systems of groups, and in many cases the limits of these were isomorphic to the corresponding “C” and “D” groups; but in some cases the limits gave the  ...  Moreover, a “local” theorem of Hurewicz type is proved in 4.35. Received by the editors March 4, 1957. 201  ... 

The equivariant Hurewicz map

L. Gaunce Lewis
1992 Transactions of the American Mathematical Society  
If Tty(Y) is the equivariant homotopy group of Y in dimension V and Hft(Y) is the equivariant ordinary homology group of Y with Burnside ring coefficients in dimension V , then there is an equivariant  ...  Let G be a compact Lie group, Y be a based G-space, and V be a G-representation.  ...  Finally, I would like to thank both the Alexander von Humboldt Foundation and Sonderforschungsbereich 170 in Göttingen for their support and hospitality during the completion of this paper.  ... 
doi:10.1090/s0002-9947-1992-1049614-9 fatcat:6yycwew2effklhxom7e7kqostm
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