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Réponses 1-32 (sur 32)

845.19002
Cenkl, Bohumil; Vigue-Poirrier, Micheline
Hochschild and cyclic homology of an almost commutative cochain algebra associated to a nilmanifold. (English)
[CA] Broto, Carles (ed.) et al., Algebraic topology: new trends in localization and periodicity. Barcelona conference on algebraic topology (BCAT), Sant Feliu de Guixols, Spain, June 1-7, 1994. Basel: Birkhaeuser, Prog. Math. 136, 69-93 (1996). [ISBN 3-7643-5333-3/hbk]
A nilmanifold $N$ can be identified with an Eilenberg-MacLane space $K (G,1)$, where $G$ is a finitely generated torsion-free nilpotent group. The first author and {\it R. Porter} [Lect. Notes Math. 1509, 79-94 (1992; Zbl 755.55010)] associated with $G$ a finite differential graded cochain algebra $M(G)$, whose cohomology identifies with the cohomology of the nilmanifold. This cochain algebra is provided by the authors with a filtration and, when associative, enjoys a structure of an almost commutative differential algebra. With such data, a filtered complex is defined which computes the Hochschild homology of $M(G) \otimes \bbfQ$. The negative cyclic homology of $M(G)$ is also investigated, and explicit computations are done for the case of the Heisenberg group.
[ F.Patras (Nice) ]
Citations: Zbl.755.55010
Mots clés: nilmanifold; almost commutative differential algebra; Hochschild homology; negative cyclic homology; Heisenberg group

Classification:: *19D55 K-theory and homology; 57T15 Homology and cohomology of homogeneous spaces of Lie groups; 18G35 Chain complexes (homological algebra); 55N25 Homology with local coefficients, equivariant cohomology

960.42305
Solotar, Andrea; Vigue-Poirrier, Micheline
Dihedral homology of commutative algebras. (English)
[J] J. Pure Appl. Algebra 109, No.1, 97-106 (1996). [ISSN 0022-4049]

Classification:: *14B05 Singularities (algebraic geometry); 14F40 De Rham cohomology; 16E40 Homology and cohomology theories for assoc. rings; 18G60 Other (co)homologies

830.13012
Vigue-Poirrier, Micheline
Homologie et $K$-theorie des algebres commutatives: Caracterisation des intersections completes. (Homology and $K$-theory of commutative algebras: Characterization of complete intersections). (French)
[J] J. Algebra 173, No.3, 679-695 (1995). [ISSN 0021-8693]
Let $A = k[X\sb 1, \ldots, X\sb n]/I$, where $k$ is a field of characteristic zero and the $X\sb i$ are indeterminates. Let $HH\sb n (A) = \bigoplus\sb{0 \le p \le n} HH\sp p\sb n(A)$ and $HC\sb n (A) = \bigoplus\sb{0 \le p \le n} HC\sp p\sb n (A)$ be respectively the Hodge decompositions of Hochschild homology and cyclic homology. The ring $A$ is called a complete intersection if $I$ is generated by a regular sequence of $R$, and is called a local complete intersection if $I\sb{\germ p}$ is generated by a regular sequence $R\sb{\germ p}$ for every prime ideal ${\germ p}$ of $R$. Then it is proved (theorem 1) that $A$ is a local complete intersection if there exists an integer $N$ such that either $HH\sp p\sb n (A) = 0$ for $n > N$ and $0 < p < n/2$ or $HC\sp p\sb n (A) = 0$ for $n > N$ and $0 < p < n/2$. The converse (with $N = 0)$ of this result is well known and has been obtained by a number of people [including the present author: $K$-Theory 4, No. 5, 399-410 (1991; Zbl. 731.19004)]. If in addition the $X\sb i$ are of positive degree and $I$ is homogeneous then $I$ is a complete intersection (theorem $1'$). These results are applied to $K$-theory, one variant (corollary 3) of the result being that if $A$ is an artinian $k$-algebra such that for all maximal ideals $m$ of $k$ the residue field of $A$ at $m$ is isomorphic to $k$ $(k$ a number field) then $A$ is a local complete intersection if and only if $K\sp i\sb n (A) = 0$ for $n \ge 1$ and $i < (n + 1)/2$ (where for Quillen $K$-theory $K\sb *(A)$ and the rational numbers $\bbfQ$, $K\sb n (A) \bigotimes \bbfQ = \bigoplus\sb{i \ge 0} K\sp i\sb n (A)$ is the decomposition induced by the Adams operations). This proves a conjecture of Beilinson and Soule in the case of artinian algebras over a number field.
[ L.G.Roberts (Kingston/Ontario) ]
Citations: Zbl.731.19004
Mots clés: Quillen $K$-theory; Hochschild homology; cyclic homology; local complete intersection

Classification:: *13D15 K-theory (commutative rings); 13C40 Linkage, complete intersections and determinantal ideals; 19D55 K-theory and homology; 16E40 Homology and cohomology theories for assoc. rings; 13D03 (Co)homology of commutative rings and algebras

845.57029
Cenkl, Bohumil; Vigue-Poirrier, Micheline
The cyclic homology of $P(G)$. (English)
[CA] Bures, J. (ed.) et al., The proceedings of the Winter school Geometry and topology, Srni, Czechoslovakia, January 1992. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 32, 195-199 (1994).
Let $G$ be a finitely generated torsion free nilpotent group and $(P(G),d\sp*)$ the associated cochain algebra over a commutative ring $K$ [the first author with {\it R. Porter}, Lect. Notes Math. 1318, 73-86 (1988; Zbl 659.57021)] of the group $G$ (endowed with a norm $\Vert \cdot \Vert$). Put $H H\sb * (P(G), D\sp*)$ and $\text {HC}\sb *(P(G),d\sp*)$ for the Hochschild and cyclic homology of $(P(G), d\sp*)$.\par The authors show that (under some assumption on $G$) the norm $\Vert \cdot \Vert$ on $P(G)$ determines filtrations on the Hochschild and Connes complexes such that there are spectral sequences converging to $HH\sb *(P(G), d\sp*)$ and $HC\sb * (P(G), d\sp*)$ respectively.
[ M.Golasinski (Torun) ]
Citations: Zbl.659.57021
Mots clés: cyclic cohomology; differential graded algebra; Hochschild and Connes complex; nilpotent group; spectral sequence

Classification:: *57T99 Homology and homotopy of topological groups; 18G60 Other (co)homologies

802.55011
Vigue-Poirrier, Micheline
Homologie cyclique des espaces formels. (Cyclic homology of formal spaces). (French)
[J] J. Pure Appl. Algebra 91, No.1-3, 347-354 (1994).
A 1-connected topological space, $X$, is called $\bbfQ$-formal iff there exists a differential graded algebra $(A,d\sb A)$ and quasi-isomorphisms $C\sp*(X,Q)\leftarrow (A, d\sb A) \to H\sp* (X;Q)$. The author proves that the Connes operator $S$ is zero on the reduced cyclic homology of $X$. This implies that the cohomology of the free loop space on $X$ can be written as a direct sum $H\sp* (BS\sp 1)\oplus T\sp*$ where $T\sp*$ is a trivial $H\sp* (BS\sp 1)$-module. Explicit calculations have been done by {\it A. E. Tralle} [Czech. Math. J. 43, No. 4, 615-634 (1993; see the review above)].
[ J.C.Thomas (Villeneuve d'Ascq) ]
Citations: Zbl.802.55010
Mots clés: formal space; Connes operator; cyclic homology; free loop space

Classification:: *55P62 Rational homotopy theory; 55N91 Equivariant homology and cohomology

788.13007
Vigue-Poirrier, Micheline
Criteres de nullite pour l'homologie des algebres graduees. (Nullity criteria for homology of graded algebras). (French)
[J] C. R. Acad. Sci., Paris, Ser. I 317, No.7, 647-649 (1993).
Let $A$ be a finitely generated graded commutative $k$-algebra over a field $k$ of characteristic 0. If $A$ is not a polynomial algebra then the Hochschild homology $\text {HH}\sb n(A) \ne 0$ for each $n \in N$ (theorem 2) and there is an integer $r$ such that for the reduced cyclic homology of $A$ it is $\overline {\text {HC}\sb i} (A) \ne 0$ for each $i<r$ and $\overline {\text {HC}\sb{r+2n}} (A) \ne 0$ for all $n \in N$ (theorem 1).
[ L.Bican (Praha) ]
Mots clés: graded algebra; Hochschild homology; cyclic homology

Classification:: *13D03 (Co)homology of commutative rings and algebras; 18G60 Other (co)homologies; 13A02 Graded rings

767.17017
Kassel, Christian; Vigue-Poirrier, Micheline
Homologie des quotients primitifs de l'algebre enveloppante de ${\germ sl}(2)$. (Homology of primitive quotients of the universal enveloping algebra of ${\germ sl}(2)$). (French)
[J] Math. Ann. 294, No.3, 483-502 (1992).
The authors investigate the primitive quotients $A\sb \lambda=U({\germ {sl}}\sb 2(\bbfC))/(g\sb \lambda)$ [where $g\sb \lambda=2ef+2fe+h\sp 2-(\lambda\sp 2-1)$, $\lambda\in\bbfC$, and $\{e,f,g\}$ is the usual base of ${\germ {sl}}\sb 2(\bbfC)]$ of the universal enveloping algebra of ${\germ {sl}}\sb 2(\bbfC)$. {\it T. Hodges} [``Morita equivalence of primitive factors of $U({\germ {sl}}(2))$'', Preprint] has calculated the Grothendieck group of $A\sb \lambda$ for $\lambda\ne 0$: $K\sb 0(A\sb \lambda)\cong\bbfZ\sp 2$. In this article the author determines the higher cyclic homology groups of $A\sb \lambda$. The Chern character $\text{ch}\sb{2n}: K\sb 0(A\sb \lambda)\to HC\sb n(A\sb \lambda)$ is also determined.
[ J.Kubarski (Lodz) ]
Mots clés: Hochschild homology; universal enveloping algebra; higher cyclic homology groups; Chern character

Classification:: *17B55 Homological methods in theory of Lie algebras; 17B35 Universal enveloping algebras (Lie algebras); 19D55 K-theory and homology; 16E40 Homology and cohomology theories for assoc. rings

755.13006
Avramov, Luchezar L.; Vigue-Poirrier, Micheline
Hochschild homology criteria for smoothness. (English)
[J] Int. Math. Res. Not. 1992, No.1, 17-25 (1992).
The purpose of this paper is to give a converse for the Hochschild-Kostant-Rosenberg theorem: If a $k$-algebra $R$ is finitely generated and smooth, then the $R$-module $\Omega\sp 1\sb{k/k}$ is finitely generated and we have a natural isomorphism $\bigwedge\sp*\sb R\Omega\sp 1\sb{k/k}\to\text{Tor}\sb *\sp{R\otimes R}(R,R)$. The authors prove in fact that if $\text{Tor}\sb p\sp{R\otimes R}(R,R)=0$ for some positive even integer $p$ and for some positive odd integer then $R$ is smooth. They use arguments from differential graded homological algebra and intuition from the ``dictionaries'' between rational homotopy and local rings.
[ Y.Felix (Louvain-La-Neuve) ]
Mots clés: Hochschild homology; rational homotopy

Classification:: *13D03 (Co)homology of commutative rings and algebras; 55P62 Rational homotopy theory

746.55007
Felix, Y.; Thomas, J.C.; Vigue-Poirrier, M.
Free loop spaces of finite complexes have infinite category. (English)
[J] Proc. Am. Math. Soc. 111, No.3, 869-875 (1991).
If $X$ is a 1-connected space of finite type and there is some $r>0$ for which $H\sb r(X)\neq\{0\}$, then the authors show that the Lusternik-Schnirelmann category of the free loop space, $\hbox{map}(S\sp 1,X)$, is infinite. This bit of information may be established through the use of minimal models in the case of characteristic zero. To obtain a characteristic free argument requires more refined methods. These methods have been developed over the last decade by the authors with S. Halperin and J. M. Lemaire. For characteristic zero there is an algebraic notion of category that is related to $L$-$S$ category. This is further related to the notion of the depth of a Hopf algebra in the characteristic free setting --- if $k$ is a field and $A$ a Hopf algebra, then $\hbox{depth} A=\inf\{n\vert\hbox{ Ext}\sp n\sb A(k,A)\neq 0\}$.\par The lever for the proof of the main result of the paper is the fact that if $X$ is 1-connected and $H\sb r(X;k)$ is finite-dimensional for all $r$, then $\hbox{depth} H\sb *(\Omega X;k)\le\hbox{cat}(X)$.\par A further refinement of methods is required to compute the depth of $H\sb *(\Omega X;k)$ --- Hopf algebras over a field may satisfy certain conditions of growth, in particular, they may be elliptic in which case $\hbox{depth} A<\infty$ and $\dim\sb kA\sb n\le Cn\sp r$ for some constants $C$ and $r\ge 0$, and all $n$. Otherwise, $A$ is said to be hyperbolic. The difficult case is an elliptic Hopf algebra which is analyzed in some detail in the paper. Considering the difficulty in obtaining any information about the homology of free loop spaces, this is an impressive achievement.
[ J.McCleary (Poughkeepsie) ]
Mots clés: Lusternik-Schnirelmann category; free loop space; elliptic Hopf algebra

Classification:: *55P50 Category and cocategory; 55P62 Rational homotopy theory; 55P35 Loop spaces

732.16008
Vigue-Poirrier, Micheline
Cyclic homology of algebraic hypersurfaces. (English)
[J] J. Pure Appl. Algebra 72, No.1, 95-108 (1991).
The author gives a formula for the cyclic homology of commutative algebras of the form $H=k[x\sb 1,...,x\sb r]/(P)$, with P a weighted homogeneous polynomial and k a field of characteristic zero. Her formula is based on the interpretation of the Hochschild and cyclic homology of H in terms of the Sullivan minimal model of (H,0) [cf. {\it D. Burghelea} and the author, Lect. Notes Math. 1318, 51-72 (1988; Zbl. 666.13007)]. For $r\le 3$, the formulas are explicit. For instance $$ HC\sb{2n+1}(k[x\sb 1,x\sb 2]/(P))\cong k[x\sb 1,x\sb 2]/(\partial P/\partial x\sb 1,\partial P/\partial x\sb 2).$$
[ Y.Felix (Louvain-La-Neuve) ]
Citations: Zbl.666.13007
Mots clés: Hochschild homology; algebraic hypersurfaces; cyclic homology; commutative algebras; weighted homogeneous polynomial

Classification:: *16E40 Homology and cohomology theories for assoc. rings; 13D03 (Co)homology of commutative rings and algebras; 55P62 Rational homotopy theory

731.19004
Vigue-Poirrier, Micheline
Decompositions de l'homologie cyclique des algebres differentielles graduees commutatives. (Decompositions of cyclic homology of commutative differential graded algebras). (French)
[J] K-Theory 4, No.5, 399-410 (1991).
Let (A,d) be a differential graded commutative algebra over a field K of characteristic 0. Let HH(A,d) resp. HC(A,d) be the Hochschild resp. cyclic homology of (A,d). The author proves that there exist natural decompositions $$ HH\sb q(A,d)=HH\sb q\sp{(0)}(A,d)\oplus...\oplus HH\sb q\sp{(q)}(A,d), $$ $$ HC\sb q(A,d)=HC\sb q\sp{(0)}(A,d)\oplus...\oplus HC\sb q\sp{(q)}(A,d) $$ with $HH\sb q\sp{(0)}(A,d)=H\sb 0(A,d)$ and $HH\sb q\sp{(q)}(A,d)=HH\sb q\sp{(q)}(H\sb 0(A,d))$, similarly for HC. \par A map f: (A,d)$\to (B,d)$ inducing an isomorphism $H\sb*(A,d)\to H\sb*(B,d)$ induces isomorphisms on the groups $HH\sb q\sp{(p)}$ and $HC\sb q\sp{(p)}.$ \par The decomposition above coincides with the decomposition described by {\it D. Burghelea} and {\it M. Vigue-Poirrier} [Lect. Notes Math. 1318, 51-72 (1988; Zbl. 666.13007)], which was in terms of minimal models. \par In certain cases, the groups $HH\sb q\sp{(p)}$ and $HC\sb q\sp{(p)}$ vanish, if $p>p(q)$. Namely if A is of finite type and generated by r elements, then one may pose $p(q)=[(q+r)/2],$ if $A=K[x\sb 1,...,x\sb r]/I,$ where I is generated by a regular sequence, one can take $p(q)=[(n+1)/2]$.
[ H.M.Unsoeld (Berlin) ]
Citations: Zbl.666.13007
Mots clés: Hochschild homology; differential graded commutative algebra; cyclic homology; decompositions

Classification:: *19D55 K-theory and homology; 16E40 Homology and cohomology theories for assoc. rings; 55P35 Loop spaces

712.55011
Halperin, Stephen; Vigue-Poirrier, Micheline
The homology of a free loop space. (English)
[J] Pac. J. Math. 147, No.2, 311-324 (1991).
See the preview in Zbl. 666.55011.
Citations: Zbl.666.55011
Mots clés: free loop space; Hochschild homology; infinitely many distinct closed geodesics

Classification:: *55R20 Spectral sequences and homology of fiber spaces; 55P62 Rational homotopy theory; 53C22 Geodesics

728.19003
Vigue-Poirrier, Micheline
Homologie de Hochschild et homologie cyclique des algebres differentielles graduees. (Hochschild homology and cyclic homology of differential graded algebras). (French)
[CA] Theorie de l'homotopie, Colloq. CNRS-NSF-SMF, Luminy/Fr. 1988, Asterisque 191, 255-267 (1990).
[For the entire collection see Zbl. 721.00021.] \par Let (A,d) be a associative differential graded algebra over a commutative unitary ring k. Then there is a differential graded tensor algebra (T(V),d) and a morphism $\rho$ : (T(V),d)$\to (A,d)$ inducing an isomorphism in homology. The main result of the paper is a rather concrete description how the Hochschild homology $HH(A,d)=Tor\sp{A\otimes A\sp{op}}(A,A)$ and the cyclic homology HC(A,d) can be computed in terms of the model (T(V),d,$\rho$). \par There are explicit formulae for differentials $\delta$ resp. D such that $HH(A,d)\cong H(A\oplus A\otimes \bar V,\delta)$ resp. $HC(A,d)\cong H(k[u]\otimes (A\oplus A\otimes \bar V),D),\vert u\vert =2.$ \par The importance of these results for topology stems from the following facts: If X is an arcwise connected pointed topological space, then {\it D. Burghelea} and {\it Z. Fiedorowicz} [Topology 25, 303-317 (1986; Zbl. 639.55003)] and {\it T. G. Goodwillie} [Topology 24, 187-215 (1985; Zbl. 569.16021)] prove that the homology resp. equivariant homology of the free loop space $X\sp{S\sp 1}$ can be computed as $H\sb*(X\sp{S\sp 1};k)\cong HH(C\sb*(\Omega X);k)$ resp. $H\sb*\sp{S\sp 1}(X\sp{S\sp 1};k)\cong HC(C\sb*(\Omega X);k).$ \par On the other hand, {\it J. F. Adams} and {\it P. J. Hilton} [Comment. Math. Helv. 30, 305-330 (1956; Zbl. 71, 164)] prove, that for a 1-connected CW-complex X the algebra $C\sb*(\Omega X;k)$ has a model of the form (T(V),d), where the cells of X correspond to a basis of V.
[ H.M.Unsoeld (Berlin) ]
Citations: Zbl.721.00021; Zbl.639.55003; Zbl.569.16021; Zbl.071.164
Mots clés: associative differential graded algebra; differential graded tensor algebra; Hochschild homology; cyclic homology; model; equivariant homology; free loop space; CW-complex

Classification:: *19D55 K-theory and homology; 16E40 Homology and cohomology theories for assoc. rings; 55P35 Loop spaces

666.55011
Halperin, Stephen; Vigue-Poirrier, Micheline
The homology of a free loop space. (English)
[J] Pac. J. Math. 147, No.2, 311-324 (1991).
Denote by $X\sp{S\sp 1}$ the space of all continuous maps from the circle into a simply connected finite CW complex X. Theorem: Let ${\bbfK}$ be a field and suppose that either char ${\bbfK}>\dim X$ or that X is ${\bbfK}$-formal. Then the Betti numbers $b\sb q=\dim H\sb q(X\sp{S\sp 1};{\bbfK})$ are uniformly bounded above if and only if the ${\bbfK}$-algebra H*(X;${\bbfK})$ is generated by a single cohomology class. Corollary: If, in addition, X is a smooth closed manifold and ${\bbfK}$ is as in the theorem, and if H*(X;${\bbfK})$ is not generated by a single class then X has infinitely many distinct closed geodesics in any Riemannian metric.
[ S.Halperin ]
Citations: Zbl.666.55011
Mots clés: free loop space; Hochschild homology; infinitely many distinct closed geodesics

Classification:: *55R20 Spectral sequences and homology of fiber spaces; 55P62 Rational homotopy theory; 53C22 Geodesics

675.13008
Vigue-Poirrier, Micheline
Cyclic homology and Quillen homology of a commutative algebra. (English)
[CA] Algebraic topology, rational homotopy, Proc. Conf., Louvain-la-Neuve/Belg. 1986, Lect. Notes Math. 1318, 238-245 (1988).
[For the entire collection see Zbl. 652.00011.] \par Let $k\to A$ be a commutative differential graded algebra over a field of characteristic zero. In a joint work with {\it D. Burghelea} [ibid. 51-72 (1988; Zbl. 666.13007)], the author had shown that the cyclic homology HC(A) decomposes canonically into a direct sum $HC\sb*(k)\oplus (\oplus\sb{p\ge 1}HC\sp{(p)}(A) )$, with $HC\sp{(1)}(A)=A/k$. The purpose of the present paper is to identify the next direct summand, by establishing that $HC\sb 2\sp{(2)}=0$, $HC\sb 1\sp{(2)}(A)=\Omega\sp 1\sb{A/k}/dA$, and $HC\sb n\sp{(2)}(A)=T\sb{n-1}(A/k)$ for $n\ge 2$, where $T\sb i(A/k)$ denotes the i-th homology group of the k-algebra A, in the sense of Andre and Quillen.
[ L.L.Avramov ]
Citations: Zbl.652.00011; Zbl.666.13007
Mots clés: differential graded algebra; cyclic homology

Classification:: *13D03 (Co)homology of commutative rings and algebras; 16W50 Graded associative rings and modules

666.13007
Burghelea, Dan; Vigue Poirrier, Micheline
(Poirrier, Micheline Vigue )
Cyclic homology of commutative algebras. I. (English)
[CA] Algebraic topology, rational homotopy, Proc. Conf., Louvain-la-Neuve/Belg. 1986, Lect. Notes Math. 1318, 51-72 (1988).
[For the entire collection see Zbl. 652.00011.] \par The purpose of the paper (and subsequent ones with the same title) is to investigate the Hochschild and cyclic homology for commutative differential graded algebras over a field of characteristic zero. \par The definitions of the Hochschild and cyclic homology for chain differential graded algebras are stated and a formula for the Hochschild and cyclic homology of a free commutative differential graded algebra is given in section two. Using this formula, the Hochschild and cyclic homology of complete intersections are calculated in section four. In the case of a hypersurface, the homologies are precisely calculated. In section three it is shown that the Hochschild and cyclic homology have natural decomposition.
[ Y.Aoyama ]
Citations: Zbl.652.00011
Mots clés: Hochschild homology; cyclic homology; differential graded algebras; complete intersections

Classification:: *13D03 (Co)homology of commutative rings and algebras; 16W50 Graded associative rings and modules; 14M10 Complete intersections

618.55006
Vigue-Poirrier, Micheline
Sur l'algebre de cohomologie cyclique d'un espace 1-connex. Applications a la geometrie des varietes. (On the algebra of cyclic cohomology of a 1-connected space. Applications to the geometry of manifolds). (French)
[J] Ill. J. Math. 32, 40-52 (1988).
Let X be a 1-connected pointed space and k a characteristic zero field. It has been proved by {\it T. G. Goodwillie} [Topology 24, 187-215 (1985; Zbl. 569.16021)] and {\it D. Burghelea} and {\it Z. Fiedorowicz} [ibid. 25, 303-317 (1986)] that cyclic cohomology $HC\sp *(X,k)$ is isomorphic to the cohomology of the space $X\sp{S\sp 1} \times \sb{S\sp 1} ES\sp 1.$ Furthermore, $HC\sp *(X,k)$ is a graded module over the polynomial ring $H\sp *(BS\sp 1)=k[u]$ with $\deg u=2.$ \par In this paper, the author shows that, for any 1-connected space X, we have a decomposition, as k[u]-graded modules: $HC\sp *(X)=k[u]\oplus V\sp *$ where $V\sp *$ is a torsion module. Applications of this formula to Waldhausen algebraic K-theory are given. We have also results about rational homotopy of the group of diffeomorphisms of compact manifolds (precisely for $CP\sp n$, $HP\sp n$, $U(n)/U(k)\times U(n-k)).$
Citations: Zbl.569.16021; Zbl.618.55006
Mots clés: free loop space; minimal model; cyclic cohomology; Waldhausen algebraic K-theory; rational homotopy of the group of diffeomorphisms of compact manifolds

Classification:: *55P62 Rational homotopy theory; 18F25 Algebraic K-theory, etc.; 57R50 Diffeomorphisms; 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds; 55N91 Equivariant homology and cohomology; 55N15 K-theory (algebraic topology); 55P35 Loop spaces

591.55004
Vigue-Poirrier, Micheline
Cohomologie de l'espace des sections d'un fibre et cohomologie de Gelfand-Fuchs d'une variete. (Cohomology of the space of sections of a fibration and Gel'fand-Fuchs cohomology of a manifold). (French)
[CA] Algebra, algebraic topology and their interactions, Proc. Conf., Stockholm 1983, Lect. Notes Math. 1183, 371-396 (1986).
[For the entire collection see Zbl. 577.00005.] \par Soit $F\hookrightarrow E\to\sp{\pi}X$ un fibre nilpotent, ou les espaces sont connexes par arcs, nilpotents et ont le type d'homotopie de C.W. complexes de type fini. On suppose que $H\sp+(X,{\bbfQ})\ne 0$, qu'il existe $n\ge 1$ tel que X a le type d'homotopie d'un complexe simplicial de dimension n, et F a le type d'homotopie de $\bigvee\sp{r}\sb{i=1}S\sp{k\sb i+1}$ ou $r\ge 2$ et $\inf (k\sb i)\ge n.$ \par Soit $\Gamma$ l'espace des sections continues du fibre. L'auteur demontre qu'il existe $N\in {\bbfN}$ et une constante reelle $C>1$ tels que si $p\ge N$, on a $\sum\sp{p}\sb{i=0}\dim H\sp i(\Gamma,{\bbfQ})\ge C\sp p$ dans les deux cas suivants: ou bien, le fibre est trivial (i.e. $\Gamma =F\sp X)$, ou bien X a le type d'homotopie d'un bouquet $S\sp d\vee Y$ (ou Y est un complexe simplicial de dimension $\le n$ et $d\ge 1)$. Il en deduit que la suite des dimensions des groupes de la cohomologie de Gelfand-Fuchs d'une variete M, $C\sp{\infty}$, compacte, connexe, nilpotente de dimension $\ge 2$ telle que $H\sp+(M,{\bbfR})\ne 0$, et dont toutes les classes de Pontryagin sont nulles est a croissance exponentielle.
[ J.C.Thomas ]
Citations: Zbl.577.00005
Mots clés: nilpotent fibration; homology of the space of sections; Gelfand-Fuchs cohomology; minimal models; rational homotopy; theory

Classification:: *55P62 Rational homotopy theory; 55R05 Fiber spaces; 57R32 Classifying spaces for foliations; 54C35 Function spaces (general topology); 58H10 Cohomology of classifying spaces for pseudogroup structures

597.55008
Vigue-Poirrier, Micheline
Sur l'homotopie rationnelle des espaces fonctionnels. (French)
[J] Manuscr. Math. 56, 177-191 (1985).
The author studies the rational homotopy type of the (pointed) function spaces $(Y,y\sb 0)\sp{(X,x\sb 0)}$ and $Y\sp X$ where: X is a nilpotent (pointed) space such that there exists $k\ge 1$ with $H\sp p(X; {\bbfQ})=0$, $p>k$ and $H\sp k(X; {\bbfQ})\ne 0$, Y is an (m-1)-connected (pointed) space with $m\ge k+2$. She shows that the rational homotopy Lie algebra of $(Y,y\sb 0)\sp{(X,x\sb 0)}$ is isomorphic (as Lie algebra) to $H\sp+(X; {\bbfQ})\otimes (\pi\sb*(\Omega Y)\otimes {\bbfQ})$. She also finds sufficient conditions for the exponential growth of the sequence of Betti numbers. \par The context is the theory of minimal model [{\it D. Sullivan}, Publ. Math., Inst. Hautes Etud. Sci. 47, 269-331 (1977; Zbl. 374.57002)]. More precisely, the main tool is the Haefliger's model for $Y\sp X$ [{\it A. Haefliger}, Lect. Notes Math. 484, 121-152 (1975; Zbl. 316.57009; Trans. Am. Math. Soc. 273, 609-620 (1982; Zbl. 508.55019)] soon used by {\it K. Shibata} for the study of Gel'fand-Fuchs cohomology [Jap. J. Math., New Ser. 7, 379-415 (1981; Zbl. 525.57025)].
[ J.C.Thomas ]
Citations: Zbl.374.57002; Zbl.316.57009; Zbl.508.55019; Zbl.525.57025
Mots clés: nilpotent space; space of sections; rational homotopy type of; function spaces; rational homotopy Lie algebra; growth of the sequence of Betti numbers; minimal model

Classification:: *55P62 Rational homotopy theory; 55P45 H-spaces and duals; 53C22 Geodesics

595.55009
Vigue-Poirrier, Micheline; Burghelea, Dan
A model for cyclic homology and algebraic K-theory of 1-connected topological spaces. (English)
[J] J. Differ. Geom. 22, 243-253 (1985).
Let X be a 1-connected space of finite type, with Sullivan minimal model ($\bigwedge Z,d)$. Define a degree -1 derivation $\beta$ in $\bigwedge Z\otimes \bigwedge \bar Z$ $(\bar Z=\Sigma\sp{-1}Z)$ by $\beta(z)=\bar z$ and $\beta(\bar z)=0$. Construct degree $+1$ derivations $\delta$ in $\bigwedge Z\otimes \bigwedge \bar Z$ and D in $\bigwedge \alpha \otimes \bigwedge Z\otimes \bigwedge \bar Z$ $(\vert \alpha \vert =2)$ as follows: $\delta (z)=dz$, $\delta(\bar z)=-\beta(dz)$; $D(\alpha)=0$ and $D(u)=\delta(u)+ \alpha \cdot \beta (u)$ for $u\in \wedge Z\otimes \bigwedge \bar Z$. Main result: the algebraic fibration $$ (\bigwedge \alpha,0)\to\sp{incl}(\bigwedge \alpha \otimes \bigwedge Z\otimes \bigwedge \bar Z,D)\to\sp{proj}(\bigwedge Z\otimes \bigwedge \bar Z,\delta) $$ represents a model for the fibration defining equivariant cohomology of the free loop space: $X\sp{S\sp 1}\to ES\sp 1\times\sb{S\sp 1} X\sp{S\sp 1}\to BS\sp 1$. The usefulness of this is twofold: firstly, the Gysin sequence of this fibration was identified by {\it T. G. Goodwillie} [Topology 24, 187-215 (1985; Zbl. 569.16021)] and the second author and {\it Z. Fiedorowicz} [ibid. 25, 303-317 (1986)] with the Connes sequence relating Hochschild and cyclic cohomology of X; second, one has: $\tilde K\sb{*+1}(X)\otimes Q=\sp{\sim}\sb*(X;Q)$, as shown by the second author [Contemp. Math. 55, Part I, 89-115 (1986)]. The model (of cyclic homology and algebraic K-theory) under review is shown to be quite manageable, by performing an explicit computation of $HC\sp*(X;Q)$, including the extra structure coming from Connes' periodicity $HC\sp*\to\sp{S}HC\sp{*+2}$, in the case $H\sp*(X;Q)$ is a truncated algebra on one generator, and by showing that $H\sb{*+1}(X;Q)$ is a direct summand in $HC\sb*(X;Q)$, for a general X.
[ St.Papadima ]
Citations: Zbl.569.16021
Mots clés: Hochschild cohomology; 1-connected space; Sullivan minimal model; algebraic fibration; equivariant cohomology of the free loop space; Gysin sequence; cyclic cohomology; algebraic K-theory

Classification:: *55P62 Rational homotopy theory; 18F25 Algebraic K-theory, etc.; 55N91 Equivariant homology and cohomology; 55P45 H-spaces and duals

562.55011
Vigue-Poirrier, Micheline
Homotopie rationnelle et croissance du nombre de geodesiques fermees. (French)
[J] Ann. Sci. Ec. Norm. Super., IV. Ser. 17, 413-431 (1984).
Let X be a 1-connected finite complex and $\beta\sb i=\dim H\sp i(X\sp{S\sp 1};{\bbfQ})$ the ith Betti numbers of the space of free loops on X. The author proves that for a large class of spaces the sequence $(\beta\sb i)$ has a polynomial (resp. exponential) growth if $\dim (\Pi (X)\otimes {\bbfQ})<\infty$ (resp. dim $\Pi$ (X)$\otimes {\bbfQ}=\infty)$. As proved by {\it M. Gromov} [J. Differ. Geom. 13, 303-310 (1978; Zbl. 427.58010)] the sequence $\beta\sb i$ plays a crucial role in studying the number of closed geodesics with prescribed length.
[ J.C.Thomas ]
Citations: Zbl.427.58010
Mots clés: elliptic space; hyperbolic space; minimal model for the rational deRham algebra; Betti numbers of the space of free loops; number of closed geodesics

Classification:: *55P62 Rational homotopy theory; 53C22 Geodesics; 55P35 Loop spaces

554.55005
Vigue-Poirrier, Micheline
Homotopie rationnelle et croissance du nombre de geodesiques fermees. (French)
[J] Ann. Sci. Ec. Norm. Super., IV. Ser. 17, 413-431 (1984).
Soit X un espace topologique 1-connexe ayant le type d'homotopie rationnelle d'un C.W. complexe et tel que dim $H\sp*(X,{\bbfQ})<\infty$. On etudie la croissance de la suite des nombres de Betti $\beta\sb i=\dim H\sp i(X\sp{S\sp 1},{\bbfQ})$. Si dim $\Pi$ ${}\sb*(X)\otimes {\bbfQ}<\infty$, soit: $p=\dim \oplus\sb{n}\Pi\sb{2n+1}(X)\otimes {\bbfQ}$, on montre que si X possede un modele minimal de Sullivan ($\Lambda$ Z,d) ou $dZ\subset Z\sp{pair}.\Lambda Z$, alors on a, pour n assez grand, $A\sb 2n\sp p\le \sum\sp{n}\sb{i=0}\beta\sb i\le A\sb 1n\sp p$ ou $A\sb 1$ et $A\sb 2$ sont des constantes $>0$; et que sinon, on peut avoir $\sum \beta\sb i\le A\sb 1n\sp{p-h}$ ou h est un entier $>0$ donne d'avance. Si dim $\Pi$ ${}\sb*(X)\otimes {\bbfQ}=\infty$, on montre que la suite $(\beta\sb i)$ est a croissance exponentielle pour les classes d'espaces suivants: bouquets de spheres, varietes compactes 1-connexes conformelles X telles que l'algebre de cohomologie reelle $H=H\sp*(X,{\bbfR})$ ait un systeme de generateurs de meme degre, et $(H\sp+)\sp 4=0$. Si X est une variete riemannienne compacte 1-connexe munie d'une metrique generique, on sait que le nombre de geodesiques fermees de longueur $<n$ est minore par (c/n)$\sum\sp{n}\sb{i=1}\beta\sb i$ ou C est une constante $>0$; alors les resultats precedent se transposent immediatement en geometrie.
Mots clés: growth of Betti numbers; existence of infinitely many closed geodesics

Classification:: *55P62 Rational homotopy theory; 58E10 Appl. of bifurcation theory to geodesics; 57T25 Homology and cohomology of H-spaces

554.55006
Vigue-Poirrier, Micheline
Sur la croissance des nombres de Betti de l'espace des lacets libres sur un espace donne. (French)
[CA] Homotopie algebrique et algebre locale, Journ. Luminy/France 1982, Asterisque 113-114, 344-348 (1984).
[For the entire collection see Zbl. 535.00017.] \par The results of this paper are details from Ann. Sci. Ec. Norm. Super., IV. Ser. 17, 413-431 (1984; see the preceding review).
Citations: Zbl.554.55005; Zbl.535.00017
Mots clés: growth of Betti numbers; existence of infinitely many closed geodesics

Classification:: *55P62 Rational homotopy theory; 58E10 Appl. of bifurcation theory to geodesics; 57T25 Homology and cohomology of H-spaces

505.55006
Vigue-Poirrier, Micheline
Dans le fibre de l'espace des lacets libres, la fibre n'est pas, en general, totalement non cohomologue a zero. (French )
[J] Math. Z. 181, 537-542 (1982).
Citations: Zbl.475.55004
Mots clés: Sullivan's models; Halperin-Stasheff models of a topological space; 1-connected pointed space, whose rational cohomology is of finite type; space of free loops; space of based loops

Classification:: *55P35 Loop spaces; 55P62 Rational homotopy theory; 55P60 Localization and completion; 53C22 Geodesics

474.55009
Vigue-Poirrier, Micheline
Realisation de morphismes donnes en cohomologie et suite spectrale d'Eilenberg-Moore. (French )
[J] Trans. Am. Math. Soc. 265, 447-484 (1981).
Citations: Zbl.408.55009
Mots clés: relative version of the filtered model for a commutative differential graded algebra on a field of characteristic zero; deformation of the differentials; rational homotopy; realizations of a cohomology morphism between differential graded algebras; Eilenberg-Moore spectral sequence of a fibered square; Serre spectral sequence of a fibration; formal map

Classification:: *55P62 Rational homotopy theory; 55R20 Spectral sequences and homology of fiber spaces; 55T20 Eilenberg-Moore spectral sequences; 55R05 Fiber spaces; 55T10 Serre spectral sequences; 16W50 Graded associative rings and modules; 13N05 Differential algebra (commutative algebra)

423.55005
Vigue-Poirrier, Micheline
Formalite d'une application continue. (French)
[J] C. R. Acad. Sci., Paris, Ser. A 289, 809-812 (1979).
Mots clés: formality of a map; obstructions to formality; differential algebras

Classification:: *55P62 Rational homotopy theory

421.58007
Grove, Karsten; Halperin, Stephen; Vigue-Poirrier, Micheline
The rational homotopy theory of certain path spaces with applications to geodesics. (English)
[J] Acta Math. 140, 277-303 (1978).
Mots clés: minimal models; loop space of the complete Riemannian manifold; geodesics; rational homotopy theory

Classification:: *58E10 Appl. of bifurcation theory to geodesics; 55P62 Rational homotopy theory

361.53058
Vigue-Poirrier, Micheline; Sullivan, Dennis
The homology theory of the closed geodesic problem. (English)
[J] J. Diff. Geometry 11, 633-644 (1976).

Classification:: *53C99 Global differential geometry; 55P35 Loop spaces

328.12103
Heydemann-Tcherkez, Marie-Claude; Vigue-Poirrier, Micheline
Une consequence de la finitude de la cohomologie de certaines algebres differentielles. (French)
[J] C. r. Acad. Sci., Paris, Ser. A 279, 181-184 (1974).

Classification:: *12H05 Differential algebra; 13D05 Homological dimension (commutative rings); 12F05 Algebraic extensions; 13E10 Artinian rings and modules

321.13011
Heydemann-Tcherkez, Marie-Claude; Vigue-Poirrier, Micheline
Application de la theorie des polynomes de Hilbert-Samuel a l'etude de certaines algebres differentielles. (French)
[J] C. r. Acad. Sci., Paris, Ser. A 278, 1607-1610 (1974).

Classification:: *13C15 Dimension theory, etc. (commutative rings); 12H05 Differential algebra; 13D05 Homological dimension (commutative rings); 15A75 Exterior algebra, etc.

311.13004
Heydemann-Tcherkez, Marie-Claude; Vigue-Poirrier, Micheline
Etude de certaines algebres differentielles. (French)
[J] C. r. Acad. Sci., Paris, Ser. A 278, 741-744; 827-829 (1974).

Classification:: *13B10 Automorphisms, etc. (commutative rings); 12H05 Differential algebra; 15A75 Exterior algebra, etc.

311.53058
Sullivan, Dennis; Vigue-Poirrier, Micheline
Sur l'existence d'une infinite de geodesiques periodiques sur une variete riemannienne compacte. (French)
[J] C. r. Acad. Sci., Paris, Ser. A 281, 289-291 (1975).

Classification:: *53C20 Riemannian manifolds (global)

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