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abstract: 'Triple linking numbers were defined for $3$-component oriented surface-links in $4$-space using signed triple points on projections in $3$-space. In this paper we give an algebraic formulation using intersections of homology classes (or cup products on cohomology groups). We prove that spherical links have trivial triple linking numbers and that triple linking numbers are link homology invariants.'
author:
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    J. Scott Carter\
    University of South Alabama\
    Mobile, AL 36688\
    carter@mathstat.usouthal.edu
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    Seiichi Kamada\
    Osaka City University\
    Osaka 558-8585, JAPAN\
    kamada@sci.osaka-cu.ac.jp\
    skamada@mathstat.usouthal.edu
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    Masahico Saito\
    University of South Florida\
    Tampa, FL 33620\
    saito@math.usf.edu
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    Shin Satoh\
    Osaka City University\
    Osaka 558-8585, JAPAN\
    susato@sci.osaka-cu.ac.jp
title: 'Triple Linking of Surfaces in $4$-Space[[^1]]{}'
---

Introduction
============

A [*surface-link*]{} is a closed surface $F$ embedded in ${\bf R}^4$ locally flatly. In this paper, we always assume that $F$ is [*oriented*]{}, that is, each component of $F$ is orientable and given a fixed orientation. For a 3-component surface-link $F=K_1\cup K_2\cup K_3$, a linking number was defined in [@CJKLS] using its projection in ${\bf R}^3$ in a way that is analogous to the linking number in classical knot theory. In that paper, it was introduced as an example of non-triviality of 