Hypersurface representation and the image of the double S^3-transfer

TL;DR

This paper investigates the image of a transfer homomorphism in stable homotopy groups, demonstrating that a specific element of order 8 is represented by a framed hypersurface and determining the transfer's image in a lower dimension.

Contribution

It proves that an element of order 8 in the 18-dimensional stable stem is in the image of a double transfer, confirming its representation by a framed hypersurface, and determines the transfer's image in dimension 16.

Findings

Element of order 8 in 18-dimensional stable stem is in the image of double transfer.

Reproves that this element is represented by a framed hypersurface.

Determines the image of the transfer homomorphism in 16-dimensional stable stem.

Abstract

We study the image of a transfer homomorphism in the stable homotopy groups of spheres. Actually, we show that an element of order 8 in the 18 dimensional stable stem is in the image of a double transfer homomorphism, which reproves a result due to P J Eccles that the element is represented by a framed hypersurface. Also, we determine the image of the transfer homomorphism in the 16 dimensional stable stem.