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Register a new userOn the second homotopy group of $SC(Z)$
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In our earlier paper (K. Eda, U. Karimov, and D. Repovv{s}, emph{A construction of simply connected noncontractible cell-like two-dimensional Peano continua}, Fund. Math. textbf{195} (2007), 193--203) we introduced a cone-like space $SC(Z)$. In the present note we establish some new algebraic properties of $SC(Z)$.
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We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,mathbb{Z})$-module.
We show that the homotopy type of a finite oriented Poincar{e} 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreck and Bauer, this applies in the case of smooth oriented 4-manifolds whose fundamental group is a finite subgroup of SO(3). An important class of examples are elliptic surfaces with finite fundamental group.
A pair $(alpha, beta)$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $alphacupbeta$ in $M_g$ are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In cite{Aou}, Aougab-Huang conjectured that the length of any filling pair on $M$ is at least $frac{m_{g}}{2}$, where $m_{g}$ is the perimeter of the regular right-angled hyperbolic $left(8g-4right)$-gon. In this paper, we prove a generalized isoperimetric inequality for disconnected regions and we prove the Aougab-Huang conjecture as a corollary.
Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homotopy coincides with strong shape. We attempt to simplify foundations of the theory and to clarify and improve some of its major results. Using geometric tools such as Milnors telescope compactification, comanifolds (=mock bundles) and the Pontryagin-Thom Construction, we obtain new simple proofs of results by Barratt-Milnor; Cathey; Dydak-Segal; Eda-Kawamura; Edwards-Geoghegan; Fox; Geoghegan-Krasinkiewicz; Jussila; Krasinkiewicz-Minc; Mardesic; Mittag-Leffler/Bourbaki; and of three unpublished results by Shchepin. An error in Lisitsas proof of the Hurewicz theorem in Steenrod homotopy is corrected. It is shown that over compacta, R.H.Foxs overlayings are same as I.M.James uniform covering maps. Other results include: - A morphism between inverse sequences of countable (possibly non-abelian) groups that induces isomorphisms on inverse and derived limits is invertible in the pro-category. This implies the Whitehead theorem in Steenrod homotopy, thereby answering two questions of A.Koyama. - If X is an LC_{n-1} compactum, n>0, its n-dimensional Steenrod homotopy classes are representable by maps S^nto X, provided that X is simply connected. The assumption of simply-connectedness cannot be dropped by a well-known example of Dydak and Zdravkovska. - A connected compactum is Steenrod connected (=pointed 1-movable) iff every its uniform covering space has countably many uniform connected components.
The main results of this paper are: (1) If a space $X$ can be embedded as a cellular subspace of $mathbb{R}^n$ then $X$ admits arbitrary fine open coverings whose nerves are homeomorphic to the $n$-dimensional cube $mathbb{D}^n$; (2) Every $n$-dimensional cell-like compactum can be embedded into $(2n+1)$-dimensional Euclidean space as a cellular subset; and (3) There exists a locally compact planar set which is acyclic with respect to v{C}ech homology and whose fine coverings are all nonacyclic.
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