Geometric topology of star spaces (II)
by
Usen Abdymanapov
Moscow Automobile Road Institute.Department of Mathematics.125829.Moscow,Russia

A metric theory of geometric topology of startoroidal neighborhood for everywhere dense atlas strata of star manifolding in star spaces is developed. In these star spaces demonstrated, star manifold for everywhere dense atlas strata, glued of the card of atlas strata plays the same role as density atlas strata and startoroidal mapping play in the star spaces. In the proofs of the received results were used author's well-known methods such as (compressible stars with special peculiarities, a web-method, an interlaced cobra, and a twisting compressible and untwisting uncompressible whirlwind star or a compressible and uncompressible tornado with contrary two-directions).

The main results are:

Theorem 1 Let S be a star manifold of star space ((SS), r) such that all components of atlas strata have dimension which never exceeds five, and let S^• ⊂ ((SS), r) be a pure subset of the star space ((SS), r).Then in a pure subset S^•((SS), r) of the  star space ((SS), r) there is a startoroidal neighborhood (NBHD)_{startor.(starsphere(s))} if and only if a startoroidal map startor.(•_•):(NBHD)_{startor.(starsphere(s))}→ {(NBHD)_{startor.(starsphere(s))}\S^•}∪_(••){S^•×((Z\bd.(-∞))∪

(Z\bd.(+∞)))} (where(•_•):S→S^•) is a star manifold for dense atlas strata.

Theorem 2 Let S be a star manifold of star space ((SS), r), and let
startor.(•_•):S→((Z\bd.(-∞))∪(Z\bd.(+∞))^m be such that

(•) startor.(•_•) is star proper,

(••) for each atlas stratum S_i ⊂ S and startor.(•_•):S_i→ ((Z\bd.(-∞))∪(Z\bd.(+∞)))^m

is a topological submersion,

(•••) for each t ∈ ((Z\bd.(-∞))∪(Z\bd.(+∞)))^m the filtration of S restricts to a filtration

of startor.(•_•)^-1(t) giving startor.(•_•)^-1(t) the structure of a star manifold S of the star space ((SS), r) such that all components of atlas strata have dimension which neve exceeds five. Then startor.(•_•) is a star space ((SS), r) and can be trivialized by atlas stratum preserving homeomorphism

H:startor.(•_•)^-1(0)×((Z\bd.(-∞))∪(Z\bd.(+∞)))^m→ S

such that startor.(•_•)○H is a projection.

Theorem 3 Let S be a compact star manifold of star space ((SS), r) such that all components of atlas strata have dimension which never exceeds five. Then the group of all atlas stratum preserving self-homeomorphisms of S is locally contractible in the compact-open topology.

Theorem 4 Let S be a star manifold of star space ((SS), r) such that all components of atlas strata have dimension which never exceeds five. Then the induction construction of teardrop of expanded startoroidal neighborhood corresponds one-to-one the regulation of the atlas stratum, preserving homeomorphism of star manifold S over S×((Z\bd.(-∞))∪(Z\bd.(+∞))) to the teardrop of the startoroidal neighborhood of the star manifold S , where the startoroidal mapping

startor.(•_•):(NBHD)_{startor.(starsphere(s))}→
{(NBHD)_{startor.(starsphere(s))}\S^•}∪_(••){S^•×((Z\bd.(-∞))

∪(Z\bd.(+∞)))} is a star manifold S of the star space ((SS), r).

Date received: June 3, 2008