User andrew ranicki - MathOverflow

Answer by Andrew Ranicki for Homotopy equivalences preserving structure

2013-05-09T09:20:30Z

Proposition 2.7 of my 1990 Mathematische Zeitschrift paper with Michael Weiss Chain complexes and assembly gives the corresponding result for projective chain complexes. The reference in the proof to "Proposition 2.5 of [17]", my 1985 Math. Scand. paper The algebraic theory of the finiteness obstruction, should have been to "Proposition 1.1 of [17]". This is just the well-known result that a chain map is a chain equivalence if and only if the algebraic mapping cone is chain contractible, the analogue of the well-known result that a map of CW complexes $f:X\to Y$ is a homotopy equivalence if and only if $X$ is a deformation retract of the mapping cylinder.

Answer by Andrew Ranicki for Is there a notion of a chain complex with corners?

2013-05-08T05:47:22Z

The chain complex n-ads in my 1992 CUP book Algebraic L-theory and topological manifolds are chain complexes with corners. They are the chain complex analogues of Wall's n-ads (which hark back to J.H.C. Whitehead).

Answer by Andrew Ranicki for Are the stiefel-Whitney classes of the tangent bundle determined by the mod 2 cohomology?

2013-05-05T05:19:46Z

The most conceptual way of understanding the relation between the mod 2 Wu and Stiefel-Whitney classes of a manifold and the action of the Steenrod algebra on the mod 2 cohomology is to use the homotopy theory of Poincare duality spaces and the Spivak normal fibration, and also the chain homotopy theory of chain complexes with symmetric Poincare complexes and the normal chain bundle expounded in my 1980 paper The algebraic theory of surgery (Part I, Part II). A map $f:L\to N$ of $n$-dimensional manifolds which induces isomorphisms in $Z_2$-coefficient cohomology also induces a chain equivalence of $n$-dimensional symmetric Poincare complexes over $Z_2$. Such a chain equivalence automatically preserves the Spivak normal chain bundles. The mod 2 Wu and Stiefel-Whitney classes of the manifolds are preserved by $f$ because they only depend only on the underlying chain homotopy structure. It is also worth reminding ourselves that Atiyah's 1960 paper Thom complexes established the $S$-duality between the Thom space of the normal bundle of a manifold $X$ and $X_+=X \cup {*}$, and so proved a conjecture of Milnor and Spanier: the stable fibre homotopy type of the tangent sphere bundle of a differentiable manifold $X$ depends only on the homotopy type of $X$.

Answer by Andrew Ranicki for Definition of "simplicial complex"

2013-04-30T08:07:56Z

Abstract simplicial complexes have had quite a renaissance recently. Simplicial complexes were originally used to describe pre-existing topological spaces such as manifolds, as in the question. But now they are the key tool in constructing discrete models for topological spaces. The nerve of a covering of a set is a simplicial complex - if the set is a topological space and the subspaces are contractible (plus some technical conditions) the nerve has the same homotopy type as the space. The Wikipedia article http://en.wikipedia.org/wiki/Rips_complex is a good introduction.

Answer by Andrew Ranicki for Definition of "simplicial complex"

2013-04-27T20:56:44Z

I used simplicial complexes of the Wikipedia kind in my 1992 CUP book Algebraic L-theory and Topological Manifolds to construct the algebraic L-theory assembly map. The construction was extended to $\Delta$-sets (in the sense of Rourke and Sanderson - not to be confused with Allen Hatcher's $\Delta$-complexes) in my 2012 joint paper with Michael Weiss The Algebraic L-theory of $\Delta$-sets.

Answer by Andrew Ranicki for Why are cup-i products and Steenrod Squares often (always?) unary?

2012-08-10T07:32:50Z

I use $\cup_i$ products as binary products in my work on the algebraic theory of surgery

http://www.maths.ed.ac.uk/~aar/papers/ats2.pdf

They give the higher symmetry properties {$\phi_s|s \geq 0$} of the Poincare duality chain equivalence $$\phi_0=[M] \cap - : C(M)^{m-*} \to C(M)$$ of an $m$-dimensional manifold $M$, with $$d\phi_s+\phi_sd^*+\phi_{s-1}+\phi_{s-1}^*=~0~(\phi_{-1}=0)$$ up to sign.

Answer by Andrew Ranicki for Existence of a chain map lifting the identity; Alexander-Whitney/Eilenberg-Zilber maps

2012-06-03T05:11:14Z

Chapter XI "Products" of Cartan and Eilenberg's 1956 classic book "Homological Algebra" includes an explicit formula on page 219.

Answer by Andrew Ranicki for Mapping torus of a homotopy equivalence

2011-12-19T07:42:02Z

Check out

Homotopy equivalences and mapping torus projections D. S. Coram, P. F. Duvall Fund. Math. 109 (1980), 1-7

http://matwbn.icm.edu.pl/ksiazki/fm/fm109/fm10911.pdf

Answer by Andrew Ranicki for Higher dimensional Heegaard splittings?

2011-11-16T07:15:23Z

Every odd-dimensional manifold has an open book decomposition (T. Lawson, Topology 17, 189-192 (1978)) and is thus a twisted double. In the even dimensions there is an asymmetric Witt group obstruction to the existence of an open book decomposition (F. Quinn, Topology 18, 55-73 (1979)), which is also the obstruction to being a twisted double. In the simply-connected 4k-dimensional case this is the original Winkelnkemper signature obstruction. There is an account of open books and twisted doubles in Chapters 29,30 of my book "High-dimensional knot theory" (Springer Monograph, 1998)

Answer by Andrew Ranicki for What are some examples of colorful language in serious mathematics papers?

2011-10-22T13:27:23Z

I am rather fond of Sylvester's "Aspiring to these wide generalizations, the analysis of quadratic functions soars to a pitch from whence it may look proudly down on the feeble and vain attempts of geometry proper to rise to its level or to emulate it in its flights." (1850)

Answer by Andrew Ranicki for Intuition behind Alexander duality

2011-10-16T06:10:44Z

Of course Alexander duality is most exciting for arbitrary compact subspaces $X \subset S^n$, Cech cohomology etc. But there is a very direct combinatorial proof for subcomplexes $X \subset \partial \Delta^{n+1}$, due to Blakers and Massey "The homotopy groups of a triad II" (Ann. of Maths. 55, 192-201 (1952)), which harks back to Poincare's proof of his duality using dual cells. For a simplicial complex $K$ let $K'$ denote the barycentric subdivision, and let $\sigma^* \subset K'$ denote the dual cell of a simplex $\sigma \in K$. Define the supplement of the subcomplex $X\subset \partial \Delta^{n+1}$ to be the subcomplex $\overline{X} \subset \Sigma^n=(\partial \Delta^{n+1})'$ consisting of the dual cells $\sigma^* \subset \Sigma^n$ of the simplices $\sigma \in \partial \Delta^{n+1} \backslash X$. The relative cellular chain complex $C^{cell}(\Sigma^n,\bar{X})$ is isomorphic to the $n$-dual ${C^{simp}(X)}^{n-*}$ of the simplicial chain complex $C^{simp}(X)$, inducing the Alexander duality isomorphisms

$H_* (\Sigma^n,\overline{X}) \cong H^{n-*}(X)$ .

I used this combinatorial approach to Alexander duality in my book "Algebraic L-theory and topological manifolds" (CUP (1992)) to construct a combinatorial model for the generalized homology theory with algebraic surgery spectrum coefficients.

Answer by Andrew Ranicki for motivation of surgery

2010-04-17T22:41:43Z

The Wikipedia article on surgery theory explains this! In addition, the Edinburgh Surgery Theory Study Group provides all kinds of surgery-related materials, including YouTube videos (not for the squeamish). There is all kinds of surgery bric-a-brac on Surgery Bits and Pieces and also here.

Answer by Andrew Ranicki for First appearance of Novikov's additivity theorem

2011-10-01T15:42:45Z

Novikov's 1970 paper Pontrjagin classes, the fundamental group and some problems of stable algebra is available online. A proof of Novikov additivity was first published on pages 587-589 of the Atiyah-Singer paper "The index of elliptic operators: III." (Annals of Maths. 87, 546-604 (1968)) mentioned in the question. There is a footnote on page 587 "We are indebted to Hirzebruch for drawing our attention to Novikov's result."

Answer by Andrew Ranicki for What is the best paper or book studying the P homomorphism, J homomorphism and Hopf invariant in Homotopy theory?

2011-09-22T05:59:55Z

The lecture I gave in Bonn in 2008 http://www.maths.ed.ac.uk/~aar/slides/bonn3.pdf is an introduction to the Hopf invariant and its applications.

When is a compact topological 4-manifold a CW complex?

2011-08-22T18:32:18Z

Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres". Kirby showed that a compact 4-manifold has a handlebody structure if and only if it is smoothable: 1 and 2. When is a compact topological 4-manifold a CW complex?

Answer by Andrew Ranicki for Where to start with research regarding maslov index/class

2011-08-09T18:52:42Z

May I recommend self-study with http://www.maths.ed.ac.uk/~aar/maslov.htm ?

Answer by Andrew Ranicki for knot complement

2011-08-01T05:41:15Z

Papakyriakopoulos, C. D. On Dehn's lemma and the asphericity of knots. Ann. of Math. (2) 66 (1957), 1–26.

Answer by Andrew Ranicki for German mathematical terms like "Nullstellensatz"

2011-04-19T12:26:16Z

And what about the Wiedersehen metric?

Answer by Andrew Ranicki for German mathematical terms like "Nullstellensatz"

2011-04-19T11:31:01Z

Viergeflechte, the original German name for 2-bridge knots, still occasionally used in an English context. In his Mathematical Review of Schubert's 1956 paper "Knoten mit 2 Bruecken" Fox explicitly notes that "Viergeflecht" is untranslatable.

Answer by Andrew Ranicki for Regular homotopy invariance of Wall's self-intersection form.

2011-04-14T16:37:08Z

The double points of a generic immersion $f:P^p \to Q^q$ of closed manifolds without triple points constitute a closed $(2p-q)$-dimensional manifold $S_2[f]$ (defined as in the question). This is also true for a generic immersion $(f,\partial f):(P,\partial P) \to (Q,\partial Q)$ of manifolds with boundary: in this case the double points constitute a $(2p-q)$-dimensional manifold with boundary $(S_2[f],S_2[\partial f])$. In particular, this is the case for the trace immersion of the regular homotopy $$(f,\partial f):(P,\partial P)=N^n \times (I,\partial I) \to (Q,\partial Q)=M^{2n}\times (I,\partial I)$$ for $p=n+1$, $q=2n+1$, with $(S_2[f],S_2[\partial f])$ a 1-dimensional manifold with boundary. Following Wall work in the universal cover $\tilde{M}$ : the count of the double points of $\partial f$ is $$\mu(\partial f)=\delta\mu(f)-(-)^n\overline{\delta\mu(f)}\in Z[\pi_1(M)]$$ with $\delta\mu(f)\in Z[\pi_1(M)]$ the count of the double point arcs of $f$. The double point circles of $f$ can be ignored here.

Answer by Andrew Ranicki for Which manifolds are homeomorphic to simplicial complexes?

2011-03-21T08:28:48Z

For a discussion of the 4-dimensional case see http://www.map.him.uni-bonn.de/Questions_about_surgery_theory

Answer by Andrew Ranicki for Reference for monkeying with the topology of a mapping cylinder

2011-01-16T18:09:02Z

This is not an answer to the question, but a reference to a more recent appearance of such a trick. The teardrop topology is defined for a map $p:X \to Y \times {\mathbb R}$ to be the smallest topology on the disjoint union $X \sqcup Y$ such that the inclusion $X \subseteq X \sqcup Y$ is an open embedding and the collapse map $c:X \sqcup Y \to Y \times (-\infty,\infty]$ is continuous. This is used in the 2000 Topology paper Neighborhoods in stratified spaces with two strata by Hughes, Taylor, Weinberger and Williams.

Answer by Andrew Ranicki for Definition of the Kervaire invariant for normal maps (as in Browder's book)

2011-01-16T09:47:02Z

My algebraic theory of surgery gives the following approach to the definition of the Kervaire invariant. Let $Q_n(C,\gamma)$ be the Weiss twisted quadratic $Q$-group defined for any chain bundle $(C,\gamma)$. A spherical fibration $\nu:X \to BG(k)$ determines a chain bundle $(C(X),\gamma(\nu))$ with a Hurewicz-style group morphism $$h~:~\pi^S_{n+k}(T(\nu)) \to Q_n(C(X),\gamma(\nu))$$ from the stable homotopy groups of the Thom space $T(\nu)$. The image $h(\rho)$ of a stable homotopy class $\rho$ relates the evaluations on the Hurewicz-Thom image fundamental homology class $$[X]~=~[\rho] \in H_{n+k}(T(\nu))~=~H_n(X)$$ of the Steenrod squares of $X$ and the cup products with the Wu classes $v_r(\nu) \in H^r(X)$, verifying on the chain level the formula of Wu and Thom $$\langle v_r(\nu) \cup y,[X] \rangle = \langle Sq^r(y),[X] \rangle~(y\in H^{n-r}(X))~.$$ An $n$-dimensional geometric Poincare complex $X$ (e.g. an $n$-dimensional manifold) has a canonical class of pairs $(\nu_X:X \to BG(k),\rho_X:S^{n+k} \to T(\nu_X))$ with $\nu_X$ the Spivak normal fibration (= sphere bundle of the normal bundle of an embedding $X \subset S^{n+k}$ for a manifold $X$). A fibre homotopy trivialization $b:\nu_X \simeq *:X \to BG(k)$ (e.g. one determined by a framing of a manifold) determines a morphism $Q_n(C(X),\gamma(\nu_X)) \to {\mathbb Z}_2$ such that the image of $h(\rho_X)$ is the Kervaire invariant $K(X,b)\in {\mathbb Z}_2$. More generally, a Wu-orientation $b$ of $X$ (for which Browder's 1969 Annals paper The Kervaire invariant of framed manifolds and its generalization is a good reference) determines a morphism $Q_n(C(X),\gamma(\nu_X)) \to {\mathbb Z}_8$ such that the image of $h(\rho_X)$ is the Brown generalized Kervaire invariant $K(X,b)\in {\mathbb Z}_8$. Most of this is already explained in my paper Algebraic Poincare cobordism.

Answer by Andrew Ranicki for Poincare duality and the $A_\infty$ structure on cohomology

2010-12-31T14:33:42Z

What Nathaniel wrote "This statement is over ${\mathbb Z}$. I don't know of a rational version of Ranicki's total surgery obstruction, and Ranicki told me he doesn't either." is strictly true in the sense that I had forgotten that I had a manifold interpretation of the vanishing of the rational total surgery obstruction in Proposition 7.7.5 (page 763) of my 1980 book on exact sequences in the algebraic theory of surgery (reference 2 below)! So I have now edited my original answer accordingly

The total surgery obstruction $s^R(X) \in {\mathbb S}_n(X;R)$ of an $R$-coefficient Poincare duality space $X$ can be defined for any ring $R$ with ${\mathbb Z} \subseteq R \subseteq {\mathbb Q}$. In the 1970's Quinn (and others) developed surgery obstruction theories for $R$-coefficient Poincar\'e duality spaces, but these theories have languished both for theoretical and practical reasons (e.g. a lack of examples). Proposition 7.7.5 of 2 does give a manifold interpretation of $s^R(X)=0 \in {\mathbb S}_n(X;R)$, but I am not sure if it is sufficiently geometric for practical purposes.

If anyone is interested in reading more about the total surgery obstruction, here are some documents which are available online:

1 "The total surgery obstruction" 1978 paper

2 "Exact sequences in the algebraic theory of surgery" 1980 book

3 "Algebraic L-theory and topological manifolds" 1992 book

4 "The total surgery obstruction" 2010 MPIM lecture

Answer by Andrew Ranicki for Ranicki symmetric L-groups of finite fields?

2010-12-05T12:06:56Z

The symmetric $L$-group $L^*(F)$ of a field $F$ are 4-periodic, $$L^n(F)=L^{n+4}(F)$$ by Proposition 7.1 of http://www.maths.ed.ac.uk/~aar/papers/ats1.pdf

$L^{2i}(F)$ is the Witt group of $(-)^i$-symmetric forms: see Milnor and Husemoller!

$L^{2i+1}(F)=0$, see http://www.maths.ed.ac.uk/~aar/papers/simple.pdf (my shortest paper).

Answer by Andrew Ranicki for Failure of smoothing theory for topological 4-manifolds

2010-09-14T21:23:10Z

I have always been mystified about handlebody structures on topological 4-manifolds. Already in 1970 Kirby and Siebenmann had established that topological n-manifolds have a handlebody structure for n>5 (see Essay III.2 in the 1976 K-S book), and Quinn proved this for n=5 in Ends of Maps III (1982). Finally I just sent an email to Kirby, who gave a simple argument that a topological 4-manifold has a handlebody structure if and only if it is smoothable. I have posted his email on the surgery pages of the Manifold Atlas Project.

Answer by Andrew Ranicki for Intutive interpretation about Linking forms

2010-08-31T05:22:08Z

This may be a good place to explain the well-known principle $$\text{intersection in the interior = linking in the boundary}$$ in an oriented $m$-dimensional manifold with boundary $(M,\partial M)$. Let $$f~:~(K,\partial K)\subset (M,\partial M)~,~g~:~(L,\partial L) \subset (M,\partial M)$$ be embeddings of oriented manifolds with boundary, such that $${\rm dim}(K)~=~k~,~{\rm dim}(L)~=~\ell~,~k+\ell~=~m~,~ f(\partial K) \cap g(\partial L)~=~\emptyset \subset \partial M~.$$ Assume there exists an isotopy (= homotopy through embeddings) rel $\partial K$ $$f_t~:~K \to M~~(0 \leqslant t \leqslant 1)$$ such that $f_0=f$ and $f_1(K)\subset \partial M$, with each $f_t(K), g(L)$ intersecting transversely in $M$, so that $f_t(K) \cap g(L)\subset M$ is a finite set with an intersection index $I(x)\in {\pm 1}$ at each point $x \in f_t(K) \cap g(L)$ according to the orientations. A continuity argument shows that the function $$\lambda~:~[0,1] \to {\mathbb Z}~:~t\mapsto \lambda(t)= \sum\limits_{x \in f_t(K) \cap g(L)}I(x)$$ is constant, so that $${\rm intersection}(f(K),g(L)\subset M)~=~\lambda(0)~=~\lambda(1)$$ $$=~{\rm linking}(f(\partial K),g(\partial L) \subset \partial M) \in {\mathbb Z}~.$$ This is best seen by drawing pictures for $(M,\partial M)=(D^2,S^1)$.

The localization exact sequence in algebraic $L$-theory is based on an abstract homological version of this principle.

Answer by Andrew Ranicki for How can you tell if a space is homotopy equivalent to a manifold?

2009-11-03T17:54:23Z

The main result of the Browder-Novikov-Sullivan-Wall surgery theory (1962-1969) is that for n>4 a space X is homotopy equivalent to a compact n-dimensional topological (resp. differentiable) manifold if and only if X is homotopy equivalent to a finite CW complex M with n-dimensional Poincaré duality, and there is a topological (resp. vector) bundle over M for which the corresponding normal map (f,b):N--> M from an n-dimensional manifold N has zero surgery obstruction in the Wall L-group of quadratic forms over Z[π₁(X)]. Thus there are two obstructions, a primary topological K-theory obstruction to the existence of a bundle, and (depending on the vanishing of the primary one, and a choice of reason) a secondary obstruction in algebraic L-theory. The original theory was for differentiable manifolds: the extension to topological manifolds due to Kirby and Siebenmann (1970) remains a major success of surgery theory. All this is explained (at some length) in Wall's own book Surgery on compact manifolds (1970/1998) and also in my own books Algebraic L-theory and topological manifolds (1992) and Algebraic and geometric surgery (2002), as well as many other references (such as Wolfgang Lück's notes listed in a previous post). I have made available a large number of surgery-related resources on my website.

Answer by Andrew Ranicki for understanding Steenrod squares

2009-10-19T11:59:26Z

Section 4L of Hatcher's Algebraic Topology is a downloadable reference.

Comment by Andrew Ranicki

2012-07-26T17:56:45Z

3 reviews of "Rites of Love and Math" maths.ed.ac.uk/~aar/baked/ritesrev.pdf maths.ed.ac.uk/~aar/baked/kirbyreview.pdf ritesofloveandmath.com/RITES-Tangente-English.pdf

Comment by Andrew Ranicki

2011-11-24T21:33:11Z

Rado's original triangulation paper is now also available from the website, along with sundry other items.

Comment by Andrew Ranicki

2011-10-16T06:30:51Z

Alexander duality brings to mind not only the Alexander horned sphere but also the Alexander horned giraffe maths.ed.ac.uk/~aar/giraffe.pdf (courtesy of the New Yorker).

Comment by Andrew Ranicki

2011-09-08T20:45:38Z

In any case, all ICM proceedings are available online at mathunion.org/ICM

Comment by Andrew Ranicki

2011-09-08T20:43:53Z

maths.ed.ac.uk/~aar/haupt/siebicm.pdf

Comment by Andrew Ranicki

2011-08-22T21:56:43Z

I have good reason to believe that it is an open question! Apologies - I hadn't seen the earlier posting mathoverflow.net/questions/36838/…

Comment by Andrew Ranicki

2011-06-03T14:15:06Z

And readers like Annals papers which do not cause problems!

Comment by Andrew Ranicki

2010-12-04T06:28:53Z

I have put together some of the source material on knots and their applications on my home page maths.ed.ac.uk/~aar/knots

Comment by Andrew Ranicki

2010-09-15T05:59:44Z

PS. I am still mystified about CW structures: when is a topological 4-manifold a CW complex? Again, if and only if smoothable?

Comment by Andrew Ranicki

2010-03-24T03:45:23Z

Inside a high-dimensional Euclidean space. Every manifold can be embedded in such a space, by Whitney.