Double point self-intersection surfaces of immersions

A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may have odd Euler characteristic if and only if k is congruent to 1 modulo 4 or k+1 is a power of 2. This corrects a previously published result by Andras Szucs. The method of proof is to evaluate the Stiefel-Whitney numbers of the double point self-intersection surface. By earier work of the authors these numbers can be read off from the Hurewicz image h(\alpha ) in H_{2k+2}\Omega ^{\infty }\Sigma ^{\infty }MO(k) of the element \alpha in \pi _{2k+2}\Omega ^{\infty }\Sigma ^{\infty }MO(k) corresponding to the immersion under the Pontrjagin-Thom construction.Comment: 22 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper4.abs.htm

Topological Complexity and LS-Category of Certain Manifolds

The Lusternik–Schnirelmann category and topological complexity are important invariants of topological spaces. In this paper, we calculate the Lusternik–Schnirelmann category and topological complexity of products of real projective spaces and their wedge products by using cup and zero-cup length. Also, we will find the topological complexity of RP2k+1 by using the immersion dimension of RP2k+1

Determining the Characteristic Numbers of Self-Intersection Manifolds

The bordism class of a self-transverse immersion f : M n\Gammak # R n corresponds to an element ff of the homotopy group n\Omega 1 \Sigma 1 MO(k). We explain how the Z=2 Hurewicz image h(ff) 2 Hn(\Omega 1 \Sigma 1 MO(k);Z=2) may be used to determine the characteristic numbers of the self-intersection manifolds \Delta r (f) of the immersion f . 1 Introduction Given a self-transverse immersion f : M n\Gammak # R n of a compact closed smooth manifold M of dimension n \Gamma k in n-dimensional Euclidean space and a positive integer r, the r-fold intersection set I r (f) is defined as follows: I r (f) = f f(x 1 ) = f(x 2 ) = : : : = f(x r ) j x i 2 M; i 6= j ) x i 6= x j g: The self-transversality of f implies that this subset of R n is itself the image of an immersion (not necessarily self-transverse) ` r (f ): \Delta r (f) # R n of a manifold \Delta r (f) of dimension n \Gamma kr called the r-fold self-intersection manifold of f . (This means in particular that fo..