Geodesic
Bordism Classes Represented by Multiple Point Manifolds of Immersed Manifolds
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 55-60

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We present a geometrical version of Herbert's theorem determining the homology classes represented by the multiple point manifolds of a self-transverse immersion. Herbert's theorem and generalizations can readily be read off from this result. The simple geometrical proof is based on ideas in Herbert's paper. We also describe the relationship between this theorem and the homotopy theory of Thom spaces. 
@article{TM_2006_252_a5,
     author = {P. J. Eccles and M. Grant},
     title = {Bordism {Classes} {Represented} by {Multiple} {Point} {Manifolds} of {Immersed} {Manifolds}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {55--60},
     publisher = {mathdoc},
     volume = {252},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2006_252_a5/}
}
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P. J. Eccles; M. Grant. Bordism Classes Represented by Multiple Point Manifolds of Immersed Manifolds. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 55-60. http://geodesic.mathdoc.fr/item/TM_2006_252_a5/