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We show that various methods for explicitly building resolutions of unbounded complexes in fact fail when applied to a rather simple and explicit complex. We show that one way to rescue these methods is to assume Roos (Ab.4$^*$)-$k$ axiom, which we adapt to encompass also resolutions in the framework of relative homological algebra. In the end we discuss the existence of model structures for relative homological algebra for unbounded complex under the relative (Ab.4$^*$)-$k$ condition, and present a variety of examples where our results apply., Comment: 34 pages

We study the behaviour of the Casson invariant $\lambda$, its square, and Othsuki's second invariant $\lambda_2$ as functions on the Johnson subgroup of the mapping class group. We show that since $\lambda$ and $d_2 = \lambda_2 - 18 \lambda^2$ are invariants that are morphisms on respectively the second and the third level of the Johnson filtration they never vanish on any level of this filtration. In contrast we prove that the invariant $\lambda_2-18\lambda^2 +3\lambda$ vanishes on the fifth level of the Johnson filtration, $\mathcal{M}_{g,1}(5)$, and as a consequence we prove that, for instance, the Poincar\'e homology sphere does not admit any Heegaard splitting with gluing map an element in $\mathcal{M}_{g,1}(5)$. Finally we determine a surgery formula for Othsuki's second invariant $\lambda_2$., Comment: 32 pages. Corrected a proof that needed a boundary argument. Added a section on the surgery formula for the new invariant. Corrected minor typos

Consider a finite connected $2$-complex $X$ endowed with a piecewise Riemannian metric and whose fundamental group is freely indecomposable, of rank at least $3$, and in which every $2$-generated subgroup is free. In this paper we show that we can always find a connected graph $\Gamma\subset X$ such that $\pi_1 \Gamma\simeq {\mathbb F}_2 \hookrightarrow\pi_1 X$ (in short, a $2$-incompressible graph) whose length satisfies the following curvature-free inequality: $\ell(\Gamma)\leq 4\sqrt{2\text{Area}(X)}$. This generalizes a previous inequality proved by Gromov for closed Riemannian surfaces with negative Euler characteristic. As a consequence we obtain that the volume entropy of such $2$-complexes with unit area is always bounded away from zero., Comment: Accepted version to be published in Revista Matem\'atica Iberoamericana

We prove that there is an action of the cyclic group $\mathbf{C}_2$ on the $10$-dimensional Floyd manifold which turns it into a conjugation manifold. The submanifold of fixed points is the $5$-dimensional Floyd manifold, whose cohomology is isomorphic to that of the large one, scaled down by dividing the cohomological degree by a factor two., Comment: 11 pages

It is a classical fact that Wall's index of a triplet of Lagrangians in a symplectic space over a field $k$ defines a $2$-cocycle $\mu_W$ on the associated symplectic group with values in the Witt group of $k$. Moreover, modulo the square of the fundamental ideal this is a trivial $2$-cocycle. In this work we revisit this fact from the viewpoint of the theory of Sturm sequences and Sylvester matrices developed by J.~Barge and J.~Lannes in teir book Suites de Sturm, indice de Maslov et p\'eriodicit\'e de Bott, volume 267 of Progress in Mathematics. Birkh\"auser Verlag, Basel, 2008. We define a refinement by a factor of $2$ of Wall's cocycle and use the technology of Sylvester matrices to give an explicit formula for the coboundary associated to the mod $I^2$ reduction of the cocycle which is valid for any field of characteristic different from $2$. Finally we explicitly compute the values of the coboundary on standard elements of the symplectic group., Comment: 30 pages. Extended overhaul, expanded proofs in particular of the key Shortcut Lemma

We study the relation between the set of oriented $\mathbb{Z}/d$-homology $3$-spheres and the level-$d$ mapping class groups, the kernels of the canonical maps from the mapping class group of an oriented surface to the symplectic group with coefficients in $\mathbb{Z}/d\mathbb{Z}$. We formulate a criterion to decide whenever a $\mathbb{Z}/d$-homology $3$-sphere can be constructed from a Heegaard splitting with gluing map an element of the level-$d$ mapping class group. Then we give a tool to construct invariants of $\mathbb{Z}/d$-homology $3$-spheres from families of trivial $2$-cocycles on the level-$d$ mapping class groups. We apply this tool to find all the invariants of $\mathbb{Z}/p$-homology $3$-spheres constructed from families of $2$-cocycles on the abelianization of the level-$p$ mapping class group with $p$ prime and to disprove the conjectured extension of the Casson invariant modulo a prime $p$ to rational homology $3$-spheres due B. Perron., Comment: 86 pages, minor typographical corrections, changed of vocabulary from "mod-p Torelli group" to the more accurate "level-p mapping class group"

We give alternative computations of the Schur multiplier of $Sp(2g,\mathbb Z/D\mathbb Z)$, when $D$ is divisible by 4 and $g\geq 4$: a first one using $K$-theory arguments based on the work of Barge and Lannes and a second one based on the Weil representations of symplectic groups arising in abelian Chern-Simons theory. We can also retrieve this way Deligne's non-residual finiteness of the universal central extension $\widetilde{Sp(2g,\mathbb Z)}$. We prove then that the image of the second homology into finite quotients of symplectic groups over a Dedekind domain of arithmetic type are torsion groups of uniformly bounded size. In contrast, quantum representations produce for every prime $p$, finite quotients of the mapping class group of genus $g\geq 3$ whose second homology image has $p$-torsion. We further derive that all central extensions of the mapping class group are residually finite and deduce that mapping class groups have Serre's property $A_2$ for trivial modules, contrary to symplectic groups. Eventually we compute the module of coinvariants $H_2(\mathfrak{sp}_{2g}(2))_{Sp(2g,\mathbb Z/2^k\mathbb Z)}=\mathbb Z/2\mathbb Z$., Comment: revised version, 44p. arXiv admin note: substantial text overlap with arXiv:1103.1855

Conjugation spaces are topological spaces equipped with an involution such that their fixed points have the same mod $2$ cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by two, generalizing the classical examples of complex projective spaces under complex conjugation. Spaces which are constructed from unit balls in complex Euclidean spaces are called spherical and are very well understood. Our aim is twofold. We construct "exotic" conjugation spaces and study the realization question: which spaces can be realized as real loci, i.e., fixed points of conjugation spaces. We identify obstructions and provide examples of spaces and manifolds which cannot be realized as such., Comment: 16 pages

Conjugation spaces are equipped with an involution such that the fixed points have the same mod 2 cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by 2, generalizing the classical examples of complex projective spaces under complex conjugation. Using tools from stable equivariant homotopy theory we provide a characterization of conjugation spaces in terms of purity. This conceptual viewpoint, compared to the more computational original definition, allows us to recover all known structural properties of conjugation spaces., Comment: 39 pages. This version corrected some misprints and a few misleading proofs

Let $M$ be a compact oriented three-manifold whose interior is hyperbolic of finite volume. We prove a variation formula for the volume on the variety of representations of $M$ in $\operatorname{SL}_n(\mathbb C)$. Our proof follows the strategy of Reznikov's rigidity when $M$ is closed, in particular we use Fuks' approach to variations by means of Lie algebra cohomology. When $n=2$, we get back Hodgson's formula for variation of volume on the space of hyperbolic Dehn fillings. Our formula also yields the variation of volume on the space of decorated triangulations obtained by Bergeron-Falbel-Guillou and Dimofte-Gabella-Goncharov., Comment: 31 pages

To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category A and fix a class I of "injective objects". We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category D(A; I), we need more: the split error term must vanish. This is the case when I is the class of all injective R-modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4*-n axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed., Comment: 42 pages (including a 10 page long appendix)

This work is NOT to be used as reference. First, because as C.F.~B\"odigheimer and M.~Korkmaz pointed to us the computation of the $\mathbf{Z}_2$ factor that remained undecided in M.~Korkmaz and A. Stipsicz, {\em The second homology groups of mapping class groups of orientable surfaces.} Math. Proc. Camb. Phil. Soc., was shown to exist by Skasai, see hi Theorem 4.9 and Corollary 4.10 in {\em Lagrangian mapping class groups from a group homological point of view.} Algebr. Geom. Topol. 12 (2012), no. 1, 267--291. Second, because one could obtain this result by gathering old results in the literature, first by noticing as Korkmaz kindly reminded me, that D.~Johnson, in \emph{Homeomorphisms of a surface which act trivially on homology} Porc. AMS Volume 75, Number 1, 1979. proved that the quotient of the Torelli group $\mathcal{T}_g/[\mathcal{T}_g,\mathcal{M}_g]$ is trivial for $g\geq 3$, the five term exact sequence then implies that the $\mathbf{Z}_2$ factor in Stein's computation of $H_2(Sp(6,\mathbf{Z});\mathbf{Z}) = \mathbf{Z}\oplus\mathbf{Z}_2$ (see his {\em The Schur Multipliers of $Sp_6(\mathbf{Z}), Spin_8(\mathbf{Z}), Spin_7(\mathbf{Z}),$ and $F_4(\mathbf{Z})$.} Math. Ann. 215 (1975), 173--193. ), detects the undecided $\mathbf{Z}_2$ factor in $H_2(\mathbf{M}_3;\mathbf{Z})$., Comment: 4 pages, 2 figures, some typos corrected, v3. Reference to previous work by Sakasai proving the result by different methods added, See the Abstract

For a commutative ring $R$, we exploit localization techniques and point-free topology to give an explicit realization of both the Zariski frame of $R$ (the frame of radical ideals in $R$) and its Hochster dual frame, as lattices in the poset of localizing subcategories of the unbounded derived category $D(R)$. This yields new conceptual proofs of the classical theorems of Hopkins-Neeman and Thomason. Next we revisit and simplify Balmer's theory of spectra and supports for tensor triangulated categories from the viewpoint of frames and Hochster duality. Finally we exploit our results to show how a coherent scheme $(X,\mathcal{O}_X)$ can be reconstructed from the tensor triangulated structure of its derived category of perfect complexes., Comment: v5:Minoir typos corrected the proof of tensor nilpotence is made totally point-free and self-contained; some simplifications and expository improvements; section on preliminaries shortened; 50pp. To appear in Trans. AMS

We prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further we show that the images of the mapping class groups have nontrivial 2-cohomology, at least for small levels. For this purpose we considered a series of quasi-homomorphisms on mapping class groups extending previous work of Barge and Ghys and of Gambaudo and Ghys. These quasi-homomorphisms are pull-backs of the Dupont-Guichardet-Wigner quasi-homomorphisms on pseudo-unitary groups along quantum representations., Comment: revised version, 24p

We show that the Schur multiplier of $Sp(2g,\mathbb Z/D\mathbb Z)$ is $\mathbb Z/2\mathbb Z$, when $D$ is divisible by 4., Comment: Bull. Soc. Math. France, to appear

We carry out the Cantor-Bendixson analysis of the space of all subgroups of any countable abelian group and we deduce a complete classification of such spaces up to homeomorphism., Comment: 22 pages, no figure

We show that H-spaces with finitely generated cohomology, as an algebra or as an algebra over the Steenrod algebra, have homotopy exponents at all primes. This provides a positive answer to a question of Stanley., Comment: 4 pages

For compact hyperbolic 3-manifolds we lift the Bloch invariant defined by Neumann and Yang to an integral class in K_3(C) Applying the Borel and the Bloch regulators, one gets back the volume and the Chern-Simons invariant of the manifold. We also discuss the non-compact case, in which there appears a Z/2-ambiguity., Comment: 16 pages

We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We give ncecessary and sufficient conditions for a function defined on the union of the Torelli groups to be an invariant of homology spheres. We apply this study to give a new purely algebraic construction of the Casson invariant and prove in this setting its surgery properties. As a by-product we get a new 2-torsion cohomology class in the second integral cohomology of the Torelli group., Comment: 25 pages, 12 figures

We investigate the isolated points in the space of finitely generated groups. We give a workable characterization of isolated groups and study their hereditary properties. Various examples of groups are shown to yield isolated groups. We also discuss a connection between isolated groups and solvability of the word problem., Comment: 30 pages, no figure. v2: minor changes, published version from March 2006

For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable 3-manifolds, including the ``piecewise geometric'' ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable 3-manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients holds. The non-oriented case is also discussed., Comment: 13 pages

We give a proof of the Jardine-Tillmann generalized group completion theorem. It is much in the spirit of the original homology fibration approach by McDuff and Segal, but follows a modern treatment of homotopy colimits, using as little simplicial technology as possible. We compare simplicial and topological definitions of homology fibrations., Comment: 13 pages