Celebratio Mathematica

incollection M. Freedman and R. Kirby: “A geometric proof of Rochlin’s theorem,” pp. 85–97 in Algebraic and geometric topology (Stanford University, CA, August 2–21, 1976), part 2. Edited by R. J. Milgram. Proceedings of Symposia in Pure Mathematics XXXII. American Mathematical Society (Providence, RI), 1978. MR 0520525 Zbl 0392.57018

In 1974 Andrew Casson outlined to us a proof of Rochlin’s Theorem on the index of a smooth, closed 4-manifold \( M^4 \). His proof involved the Arf invariant of a certain quadratic form defined on the first homology group of a surface in \( M^4 \) which is dual to the second Stiefel–Whitney class of \( M^4 \). Our proof was derived from Casson’s; it is the same in principle but differs considerably in detail. After this manuscript was written, we discovered that Rochlin had already in 1971 given a short sketch of this proof; it appears in a paper [Rochlin 1972] about real algebraic curves in \( \mathbb{R}P^2 \).

In addition we obtain a “stable” converse to the Kervaire–Milnor nonimbedding theorem [Kervaire and Milnor 1961], and in Section 2, by relaxing some orientability assumptions, we prove a new (but unspectacular) nonimbedding theorem and find an obstruction to approximating unoriented simplicial 3-chains in a 5-manifold by an immersed 3-manifold.

Richard James Milgram	Related
M. H. Freedman	Related	Volume

D. Gay and R. Kirby: “Reconstructing 4-manifolds from Morse 2-functions,” pp. 103–114 in Proceedings of the Freedman Fest (Santa Barbara, CA, 15–17 April 2011 and Berkeley, CA, 6–10 June 2011). Edited by R. Kirby, V. Krushkal, and Z. Wang. Geometry & Topology Monographs 18. Geometry & Topology Publications (Coventry, UK), 2012. Proceedings of a conference on low-dimensional manifolds and high-dimensional categories and a symposium dedicated to Mike Freedman on the occasion of his 60th birthday. MR 3084235 Zbl 1267.57020 ArXiv 1202.3487 incollection

Zhenghan Wang	Related
V. Slava Krushkal	Related
M. H. Freedman	Related	Volume
David T. Gay	Related

@incollection {key3084235m,
	AUTHOR = {Gay, David and Kirby, Robion},
	TITLE = {Reconstructing 4-manifolds from {M}orse
		 2-functions},
	BOOKTITLE = {Proceedings of the {F}reedman {F}est},
	EDITOR = {Kirby, Rob and Krushkal, Vyacheslav
		 and Wang, Zhenghan},
	SERIES = {Geometry \& Topology Monographs},
	NUMBER = {18},
	PUBLISHER = {Geometry \& Topology Publications},
	ADDRESS = {Coventry, UK},
	YEAR = {2012},
	PAGES = {103--114},
	DOI = {10.2140/gtm.2012.18.103},
	NOTE = {(Santa Barbara, CA, 15--17 April 2011
		 and Berkeley, CA, 6--10 June 2011).
		 Proceedings of a conference on low-dimensional
		 manifolds and high-dimensional categories
		 and a symposium dedicated to Mike Freedman
		 on the occasion of his 60th birthday.
		 ArXiv:1202.3487. MR:3084235. Zbl:1267.57020.},
	ISSN = {1464-8989},
}

R. Kirby, P. Melvin, and P. Teichner: “Cohomotopy sets of 4-manifolds,” pp. 161–190 in Proceedings of the Freedman Fest (Santa Barbara, CA, 15–17 April 2011 and Berkeley, CA, 6–10 June 2011). Edited by R. Kirby, V. Krushkal, and Z. Wang. Geometry & Topology Monographs 18. Geometry & Topology Publications (Coventry, UK), 2012. MR 3084237 Zbl 1273.57012 ArXiv 1203.1608 incollection

Zhenghan Wang	Related
V. Slava Krushkal	Related
M. H. Freedman	Related	Volume
Peter Teichner	Related
Paul Michael Melvin	Related

Proceedings of the Freedman Fest (Santa Barbara, CA, 15–17 April 2011 and Berkeley, CA, 6–10 June 2011). Edited by R. Kirby, V. Krushkal, and Z. Wang. Geometry & Topology Monographs 18. Geometry & Topology Publications (Coventry, UK), 2012. Conference on low-dimensional manifolds and high-dimensional categories and symposium dedicated to Mike Freedman on the occasion of his 60th birthday. MR 3137657 Zbl 1253.00022 book

Zhenghan Wang	Related
V. Slava Krushkal	Related
M. H. Freedman	Related	Volume

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Robion C. Kirby

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