Abstract. The main purpose of this work is to study fixed points of fiber-preserving maps over the circle S 1 for spaces which are fibrations over S 1 and the fiber is the torus T . For the case where the fiber is a surface with nonpositive Euler characteristic, we establish general algebraic conditions, in terms of the fundamental group and the induced homomorphism, for the existence of a deformation of a map over S 1 to a fixed point free map. For the case where the fiber is a torus, we classify all maps over S 1 which can be deformed fiberwise to a fixed point free map.

Abstract. The main purpose of this work is to study fixed points of fiber-preserving maps over the circle S 1 for spaces which are fiber bundles over S 1 and the fiber is the Klein bottle K. We classify all such maps which can be deformed fiberwise to a fixed point free map. The similar problem for torus fiber bundles over S 1 has been solved recently.Introduction. Given a fiber bundle E → B and a fiber-preserving map f : E → E over B, the question whether f can be deformed over B (by a fiberwise homotopy) to a fixed point free map has been considered by many, Fadell and Husseini showed that the above problem can be stated in terms of obstructions (including higher ones). This was done under the hypothesis that the base space, the total space and the fiber F are manifolds, and the dimension of F is greater than or equal to 3. The case where the fiber has dimension 2 was considered in [GPV04], where a few generalities were discussed and the fixed point problem over B as defined above was completely solved for any torus fiber bundles over the circle S 1 . In the present work we study the fixed point problem over B for Klein bottle fiber bundles over S 1 .Recall that a Klein bottle bundle over S 1 has as total space the mapping torus M (φ) where φ : K → K is a homeomorphism. A relevant step in solving the problem is to determine, for each fiber bundle M (φ) → S 1 , the set of homotopy classes of maps f over S 1 such that f restricted to the fiber can be deformed to a fixed point free map. This is done in Theorem 2.4. The main result of the paper is Theorem 6.26, which gives a classification of the homotopy classes of fiber-preserving maps given by Theorem 2.4 which can be deformed over S 1 to a fixed point free map. Our method is to study solutions of a system of equations in a free group either by providing an explicit solution or by considering the system in some quotients of this group.

Surface functionalization of liposomes can play a key role in overcoming the current limitations of nanocarriers to treat solid tumors, i.e., biological barriers and physiological factors. The phospholipid vesicles (liposomes) containing anticancer agents produce fewer side effects than non-liposomal anticancer formulations, and can effectively target the solid tumors. This article reviews information about the strategies for targeting of liposomes to solid tumors along with the possible targets in cancer cells, i.e., extracellular and intracellular targets and targets in tumor microenvironment or vasculature. Targeting ligands for functionalization of liposomes with relevant surface engineering techniques have been described. Stimuli strategies for enhanced delivery of anticancer agents at requisite location using stimuli-responsive functionalized liposomes have been discussed. Recent approaches for enhanced delivery of anticancer agents at tumor site with relevant surface functionalization techniques have been reviewed. Finally, current challenges of functionalized liposomes and future perspective of smart functionalized liposomes have been discussed.

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