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

2016
*
arXiv
*
pre-print

We give an explicit description

arXiv:1412.3864v2
fatcat:xx6axk5xordu3e4zyr2c5zo2gm
*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*. ...##
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Homology groups of types in stable theories and the Hurewicz correspondence

2017
*
Annals of Pure and Applied Logic
*

We give an explicit description

doi:10.1016/j.apal.2017.03.007
fatcat:pipqtxckhfdfxkykvjjqe4lcr4
*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*. ...##
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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

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). ...

##
###
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

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 ...

##
###
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

1992
*
Transactions of the American Mathematical Society
*

If Tty(Y) is

doi:10.1090/s0002-9947-1992-1049614-9
fatcat:6yycwew2effklhxom7e7kqostm
*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. ...
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