Abstract:
In this paper, we introduce the definition of the induced unfolding on the homology group. Some types of conditional foldings restricted on the elements of the homology groups are deduced. The effect of retraction on the homology group of a manifold is dicussed. The unfolding of variation curvature of manifolds on their homology group are represented. The relations between homology group of the manifold and its folding are deduced.

Abstract:
The smooth rational homology cobordism group of rational homology three spheres, T, contains subgroups T_p generated by 3-manifolds with first homology p-torsion, where p is a prime. Rochlin's theorem and gauge theoretic methods show that the inclusion of the direct sum of the T_p into T has infinitely generated kernel. We use Heegaard-Floer methods to show that the cokernel is infinitely generated. On the level of Witt groups of linking forms and conjecturally in the topological category, the inclusion is an isomorphism. One application is the demonstration of the failure in dimension three of an algebraic and higher dimensional theorem of Stoltzfus regarding primary splittings in the knot concordance group. We also build on work of the second author with Hedden and Ruberman to prove that the homology cobordism group of rational homology three spheres that bound topological rational homology 4-balls remains infinitely generated when modded out by the homology cobordism group of integral homology 3-spheres.

Abstract:
Let $X$ be a smooth manifold belonging to one of these three collections: (1) acyclic manifolds (compact or not, possibly with boundary), (2) compact manifolds (possibly with boundary) with nonzero Euler characteristic, and (3) homology spheres. We prove the existence of a constant $C$ such that any finite group acting effectively and smoothly on $X$ has an abelian subgroup of index at most $C$. The proof uses a result on finite groups by Alexandre Turull and the author which is based on the classification of finite simple groups. If $X$ is compact and its cohomology is torsion free and supported in even degrees, we also prove the existence of a constant $C'$ such that any finite abelian group $A$ acting on $X$ has a subgroup $A_0$ of index at most $C'$ such that $\chi(X^{A_0})=\chi(X)$.

Abstract:
Let $M$ be a complete oriented hyperbolic 3-manifold of finite volume. Using classifying spaces for families of subgroups we construct a class $\beta_P(M)$ in the Hochschild relative homology group $H_3([PSL_2(\C):\bar{P}];\Z)$, where $\bar{P}$ is the subgroup of parabolic transformations which fix infinity in the Riemann sphere. We prove that the group $H_3([PSL_2(\C):\bar{P}];\Z)$ and the Takasu relative homology group $H_3(PSL_2(\C),\bar{P};\Z)$ are isomorphic and under this isomorphism the class $\beta_P(M)$ corresponds to Zickert's fundamental class. This proves that Zickert's fundamental class is well-defined and independent of the choice of decorations by horospheres. We also construct a homomorphism from $H_3([PSL_(\C):\bar{P}];\Z)$ to the extended Bloch group $\hat{\mathcal{B}}(\C)$ which is isomorphic to $H_3(PSL_2(\C);\Z)$. The image of $\beta_P(M)$ under this homomorphism is the $PSL$-fundamental class constructed by Neumann and Zickert.

Abstract:
Simplicial homology manifolds are proposed as an interesting class of geometric objects, more general than topological manifolds but still quite tractable, in which questions about the microstructure of space-time can be naturally formulated. Their string orientations are classified by $H^3$ with coefficients in an extension of the usual group of D-brane charges, by cobordism classes of homology three-spheres with trivial Rokhlin invariant.

Abstract:
We introduce and investigate the notion of (strong) $K^n_G$-manifolds, where $G$ is an abelian group. One of the result related to that notion (Theorem 3.4) implies the following partial answer to the Bing-Borsuk problem \cite{bb}, whether any partition of a homogeneous metric $ANR$-space $X$ of dimension $n$ is cyclic in dimension $n-1$: If $X$ is a homogeneous metric $ANR$ compactum with $\check{H}^{n}(X;G)\neq 0$, then $\check{H}^{n-1}(M;G)\neq 0$ for every set $M\subset X$, which is cutting $X$ between two disjoint open subsets of $X$. Another implication of Theorem 3.4 (Corollary 3.6) provides an analog of the classical result of Mazurkiewicz \cite{ma} that no region in $\mathbb R^n$ can be cut by a subset of dimension $\leq n-2$. Concerning homology manifolds, it is shown that if $X$ is arcwise connected complete metric space which is either a homology $n$-manifold over a group $G$ or a product of at least $n$ metric spaces, then $X$ is a Mazurkiewicz arc $n$-manifold. We also introduce a property which guarantees that $H_k(X,X\setminus x;G)=0$ for every $x\in X$ and $k\leq n-1$, where $X$ is a homogeneous locally compact metric $ANR$.

Abstract:
We construct absolute and relative versions of Hamiltonian Floer homology algebras for strongly semi-positive compact symplectic manifolds with convex boundary, where the ring structures are given by the appropriate versions of the pair-of-pants products. We establish the absolute and relative Piunikhin-Salamon-Schwarz isomorphisms between these Floer homology algebras and the corresponding absolute and relative quantum homology algebras. As a result, the absolute and relative analogues of the spectral invariants on the group of compactly supported Hamiltonian diffeomorphisms are defined.

Abstract:
We define a model for the homology of manifolds and use it to describe the intersection product on the homology of compact oriented manifolds and to define homological quantum field theories which generalizes the notions of string topology introduced by Chas and Sullivan and homotopy quantum field theories introduced by Turaev.

Abstract:
Gromov's Conjecture states that for a closed $n$-manifold $M$ with positive scalar curvature the macroscopic dimension of its universal covering $\tilde M$ satisfies the inequality $\dim_{mc}\tilde M\le n-2$\cite{G2}. We prove this inequality for totally non-spin $n$-manifolds whose fundamental group is a virtual duality group with $vcd\ne n$. In the case of virtually abelian groups we reduce Gromov's Conjecture for totally non-spin manifolds to the vanishing problem whether $H_n(T^n)^+= 0$ for the $n$-torus $T^n$ where $H_n(T^n)^+\subset H_n(T^n)$ is the subgroup of homology classes which can be realized by manifolds with positive scalar curvature.