LET (H, O) BE A HYPERGROUP. CONSIDER THE FUNDAMENTAL RELATION B*, AS THE SMALLEST EQUIVALENCE RELATION ON H, SUCH THAT THE QUOTIENT ALGEBRAIC STRUCTURES (FORMULA), THE FUNDAMENTAL GROUP OF H, IS A GROUP. IN THIS PAPER WE INVESTIGATE SOME CONDITIONS SUCH THAT FOR A GIVEN FINITE HYPER GROUP H, ITS FUNDAMENTAL GROUP (FORMULA) IS A TRIVIAL GROUP.

IN THE PRESENT WORK A GENERALIZATION OF RIEMANNIAN YAMABE SOLITONS FOR INEQUALITIES IS STUDIED AND AMONG THE OTHERS IT IS SHOWN THAT EVERY RIEMANNIAN COMPLETE NONTRIVIAL SHRINKING YAMABE SOLITON HAS FINITE FUNDAMENTAL GROUP AND ITS FIRST COHOMOLOGY GROUP VANISHES, WHENEVER THE SCALAR CURVATURE IS BOUNDED. AS WELL THE FUNDAMENTAL GROUP OF THE SPHERE BUNDLE, AND ITS COHOMOLOGY GROUP VANISHES.

IN THIS TALK, AFTER REVIEWING CONCEPTS OF COMPACT-OPEN TOPOLOGY, WHISKER TOPOLOGY AND LASSO TOPLOGY ON FUNDAMENTAL GROUPS, WE PRESENT SOME TOPOLOGICAL PROPERTIES FOR THE WHISKER TOPOLOGY ON A FUNDAMENTAL GROUP.

IN THIS TALK, AFTER REVIEWING CONCEPTS OF COVERING, SEMICOVERING AND GENERALIZED COVERING SUBGROUPS INTRODUCED BY J. BRAZAS, WE GIVE A NEW CRITERION FOR A SUBGROUP H≤ P1 (X, X0) TO BE A GENERALIZED COVERING SUBGROUP.

IN THIS TALK, AFTER REVIEWING THE CONCEPT OF LOCAL HOMEOMORPHISMS, COVERING MAPS AND FUNDAMENTAL GROUPS, FIRST WE MENTION A RESULT ON UNIQUENESS OF LIFTING FOR LOCAL HOMEOMORPHISM. SECOND, WE PROVE SOME OF THE WELL-KNOWN PROPERTIES OF COVERING MAPS FOR LOCAL HOMEOMORPHISMS WITH UNIQUE PATH LIFTING AND PATH LIFTING PROPERTIES.

IN THIS TALK, WE INTEND TO INTRODUCE TWO SUBGROUPS OF TOPOLOGIZED FUNDAMENTAL GROUPS WITH RESPECT TO SMALL LOOP TRANSFER SPACES SPECIALLY SLT LOOPS AND THEIR INFLUENCE ON THE FUNDAMENTAL GROUP EQUIPPED WITH THE WHISKER TOPOLOGY. WE ALSO GIVE SOME RELATIONSHIP BETWEEN THESE INTRODUCED SUBGROUPS AND THE OTHER WELL KNOWN SUBGROUPS OF THE FUNDAMENTAL GROUP.