We consider a compact, oriented,smooth Riemannian manifold $M$ (with or without boundary) and wesuppose $G$ is a torus acting by isometries on $M$. Given $X$ in theLie algebra of $G$ and corresponding vector field $X_M$ on $M$, onedefines Witten's inhomogeneous coboundary operator $\d_{X_M} =\d+\iota_{X_M}: \Omega_G^\pm \to\Omega_G^\mp$ (even/odd invariantforms on $M$) and its adjoint $\delta_{X_M}$. First, Witten [35] showed that the resulting cohomology classeshave $X_M$-harmonic representatives (forms in the null space of$\Delta_{X_M} = (\d_{X_M}+\delta_{X_M})^2$), and the cohomologygroups are isomorphic to the ordinary de Rham cohomology groups ofthe set $N(X_M)$ of zeros of $X_M$. The first principal purpose isto extend Witten's results to manifolds with boundary. Inparticular, we define relative (to the boundary) and absoluteversions of the $X_M$-cohomology and show the classes haverepresentative $X_M$-harmonic fields with appropriate boundaryconditions. To do this we present the relevant version of theHodge-Morrey-Friedrichs decomposition theorem for invariant forms interms of the operators $\d_{X_M}$ and $\delta_{X_M}$; the proofinvolves showing that certain boundary value problems are elliptic.We also elucidate the connection between the $X_M$-cohomology groupsand the relative and absolute equivariant cohomology, followingwork of Atiyah and Bott. This connection is then exploited to showthat every harmonic field with appropriate boundary conditions on$N(X_M)$ has a unique corresponding an $X_M$-harmonic field on $M$to it, with corresponding boundary conditions. Finally, we define the interior and boundary portion of $X_M$-cohomology and then we definethe \emph{$X_M$-Poincar\' duality angles} between the interiorsubspaces of $X_M$-harmonic fields on $M$ with appropriate boundaryconditions.Second, In 2008, Belishev and Sharafutdinov [9] showed thatthe Dirichlet-to-Neumann (DN) operator $\Lambda$ inscribes into thelist of objects of algebraic topology by proving that the de Rhamcohomology groups are determined by $\Lambda$.In the second part of this thesis, we investigate to what extent is the equivariant topology of a manifold determined by a variant of the DN map?.Based on the results in the first part above, we define an operator$\Lambda_{X_M}$ on invariant forms on the boundary $\partial M$which we call the $X_M$-DN map and using this we recover the longexact $X_M$-cohomology sequence of the topological pair $(M,\partialM)$ from an isomorphism with the long exact sequence formed from thegeneralized boundary data. Consequently, This shows that for aZariski-open subset of the Lie algebra, $\Lambda_{X_M}$ determinesthe free part of the relative and absolute equivariant cohomologygroups of $M$. In addition, we partially determine the mixed cup product of$X_M$-cohomology groups from $\Lambda_{X_M}$. This shows that $\Lambda_{X_M}$ encodes more information about theequivariant algebraic topology of $M$ than does the operator$\Lambda$ on $\partial M$. Finally, we elucidate the connectionbetween Belishev-Sharafutdinov's boundary data on $\partial N(X_M)$and ours on $\partial M$.Third, based on the first part above, we present the(even/odd) $X_M$-harmonic cohomology which is the cohomology ofcertain subcomplex of the complex $(\Omega^{*}_G,\d_{X_M})$ and weprove that it is isomorphic to the total absolute and relative$X_M$-cohomology groups.

This thesis consists of two papers with somewhat different flavours. In Paper I we compute the étale cohomology ring H^*(X,Z/nZ) for X the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim. We also give examples of two distinct number fields whose rings of integers have isomorphic cohomology groups but distinct cohomology ring structures. In Paper II we define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2, 1)-categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne–Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme S is defined over a field, we give a criterion for when a stacky building datum comes from a ramified cover for a finite abelian group scheme over k, generalizing a result of Biswas–Borne. / Denna avhandling består av två artiklar som skiljer sig något i karaktär. I Artikel I beräknar vi den étala kohomologiringen H^*(X,Z/nZ) då X är ringen av heltal av en talkropp K. Som en tillämpning, ger vi ett kriterium i form av en formel för när en invariant definierad av Minhyong Kim är noll eller ej. Vi ger också exempel på två olika talkroppar vars ringar av heltal har isomorfa kohomologigrupper men olika kohomologiringstrukturer. I Artikel II definierar vi stackig byggnadsdata för stackiga övertäckningar i Pardinis anda och visar en ekvivalens av (2,1)-kategorier mellan kategorin av stackiga övertäckningar och kategorin av stackig byggnadsdata. Vi visar att varje stackig övertäckning är en platt rotstack i Olsson och Borne–Vistolis mening och vi ger en intrinsisk beskrivning av den som en rotstack med hjälp av stackig byggnadsdata. När basen S är definierad över en kropp, ger vi ett kriterium för när ett stackigt byggnadsdatum kommer från en ramifierad övertäckning för ett ändligt abelskt gruppschema över k. Detta generaliserar ett resultat av Biswas–Borne.

Dans cette thèse, nous décrivons explicitement la structure multiplicative et la structure d’algèbre de Lie graduée sur la cohomologie de l’algèbre enveloppante d’une algèbre de Lie de dimension finie. Dans un premier temps, nous introduisons une structure multiplicative de la cohomologie de l’algèbre de Lie. Ensuite, nous montrons explicitement qu’il existe un isomorphisme d’algèbres graduées commutatives entre l’algèbre de cohomologie de Hochschild de l’algèbre enveloppante munie du produit cup et l’algèbre de cohomologie de l’algèbre de Lie. Dans un deuxième temps, nous introduisons une structure d’algèbre de Lie graduée sur la cohomologie de l’algèbre de Lie. Ensuite, nous montrons qu’il existe un isomorphisme d’algèbres de Lie graduées entre l’algèbre de Lie de cohomologie de Hochschild de l’algèbre enveloppante munie du crochet de Gerstenhaber et l’algèbre de cohomologie de l’algèbre de Lie. Enfin, nous décrivons complètement le crochet de Gerstenhaber sur la cohomologie de Hochschild de l’algèbre enveloppante d’une algèbre de Lie de dimension _ 3. / In this thesis, we explicitly describe the multiplicative structure and the graded Lie algebra structure of the cohomology of finite-dimensional Lie algebras. In a first step, we introduce a multiplicative structure for the cohomology of Lie algebra. Then, we explicitly show that there exists an isomorphism of commutative graded algebras between the Hochschild cohomology algebra of the enveloping algebra provided with the cup product and the cohomology algebra of the Lie algebra. In a second step, we introduce a graded Lie algebra structure for the cohomology of Lie algebra. Then, we show that there exists an isomorphism of graded Lie algebras between the Hochschild cohomology Lie algebra of the enveloping algebra provided with the Gerstenhaber bracket and the cohomology algebra of the Lie algebra. Finally, we describe completely the Gerstenhaber bracket on the Hochschild cohomology of the enveloping algebra of a Lie algebra for dimension _ 3.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal