Sergey Melikhov

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 Name Sergey Melikhov Member for 2 years Seen Apr 24 at 22:04 Website Location Steklov Math Institute (Moscow) Age 32
 Mar14 comment Covering maps in real life that can be demonstrated to studentsG.C.: The question is not equivalent to asking whether the mapping cylinder embeds in $\Bbb R^3$. Also, there are no problems with point-set topology because quotient topology wasn't mentioned. In fact, the question is equivalent to asking whether the mapping cylinder has a level-preserving embedding in $\Bbb R^3\times [0,1]$, where the mapping cylinder is endowed with the (metrizable) topology of quotient uniformity. If the domain is compact, this is the same as quotient topology, but if it's not, quotient topology is non-metrizable. Mar14 answered Covering maps in real life that can be demonstrated to students Mar14 comment How should one think about pushforward in cohomology?It turns out that some people seemingly unaware of B-R-S have developed a similar (but not identical) approach to geometric homology, which involves additional choices, but on the other hand may be more suitable for K-theory: mathoverflow.net/questions/119872/… Mar14 comment How should one think about pushforward in cohomology?$f_!$ can well be non-zero for negative $n$ (in ordinary cohomology). For example, even though the projection $f:S^5\times S^7\to S^5$ represents $[f]=0\in H_{-7}(S^5)$, the pushforward map $f_!:H^7(S^5\times S^7)\to H^0(S^5)$ is an isomorphism. It sends the class of a 7-comanifold $g:S^5\to S^5\times S^7$ to the class of the $0$-comanifold $fg:S^5\to S^5\times S^7\to S^5$. Mar9 revised Making CW-complexes metrizableadded 1084 characters in body Mar9 comment Seifert surfaces via Alexander duality(see also ncbi.nlm.nih.gov/pmc/articles/PMC1076924) Mar9 answered Where should I learn about immersion theory? Mar9 comment Constructing a deformation neighborhoodIt is easier 1) to find a neighborhood of $X$ in the standard cubical grid with side $1/2^m$, for some $m$, that retracts (but not deformation retracts) onto $X$; 2) to find a neighborhood of $X$ that is a cubical complex (but not a subcomplex of the standard grid) and deformation retracts onto $X$. To get (1), use 4.30 and 4.31 in arxiv.org/abs/1106.3249v4 . To get (2), use 4.3 in arxiv.org/abs/1109.0346v2 , or alternatively (this is slightly less effective) use Misha's answer and apply the canonical subdivision to cut simplices into cubes. Mar9 comment Seifert surfaces via Alexander duality"Serre's theorem" that $H^1(X)=[X,S^1]$ was proved by Bruschlinsky, Math. Ann. 109 (1934), 525-537. Mar9 revised Making CW-complexes metrizableadded 65 characters in body Mar9 answered Making CW-complexes metrizable Mar7 comment Is there a conceptual reason why topological spaces have quotient structures while metric spaces don’t?Eric, do you mean 1-Lipschitz (non-expansive) maps, or is it strictly distance-decreasing and/or strictly preserving (isometries)? It seems that what you're saying is easy to prove for 1-Lipshitz maps, but anyway are there any good references for this sort of questions? Mar7 comment Is there a conceptual reason why topological spaces have quotient structures while metric spaces don’t?Yes, your answer is related to Eric's. The category of metric spaces and uniformly continuous maps has many quotients (though not all) defined by the formula in Eric's answer. (In other words, that formula gives a well-defined metric, in many cases, if the original metric is changed without changing the uniform structure.) For example, you can define mapping cylinder and join in this category (which you can't do for metric spaces and continuous maps). This is done in arxiv.org/abs/1106.3249 Mar5 comment The role of ANR in modern topologyBreadth of applicability is very good; I'm all for model categories (and homotopy type theory). I just don't see what this all has to do with ANRs (and topological manifolds). As you explain, ANRs are not really needed to do homotopy theory; on the other hand, model categories haven't yet helped anyone to do ANRs (and hence topological manifolds), AFAIK. Mar5 comment The role of ANR in modern topologyAlso, by "topology literature" and "early topology papers" you probably mean algebraic topology? Things like Freedman's proof of the topological 4D Poincare Conjecture, Quinn's proof of the Annulus Conjecture, and Edwards and Cannon's proof of Milnor's Double Suspension Conjecture are very much about ANRs. For the record, these include 2 Annals papers from 1979, and a 1986 Fields medal; a further 1975 Annals paper mentions "ANRs" in its title (it's the main ingredient of West's proof that ANRs have finite types). Mar5 comment The role of ANR in modern topologyDoes "more scaleable" mean more axiomatic? But I don't know any nontrivial model category where all objects, or all (co)fibrant objects are ANRs. (One problem is that the cone over a non-compact space is non-metrizable.) So I don't see how any talk about ANRs (or even regular neighborhoods) could be made implicit by cofibrations. If your point is that using ANRs looks dated in some AT textbooks, it's a question of presentation. The essential feature of ANRs is, of course, that they include topological manifolds and are similar enough to them, but more easily manageable. Mar5 answered The role of ANR in modern topology Mar2 comment How to see the quaternionic hopf map generates the stable 3-stem?A geometric construction of a generator of $\pi_3^{st}$ is discussed in this paper by Eckholm and Takase: arxiv.org/abs/0903.0238 Note also that the composition of the 8-fold covering $S^3\to S^3/Q$, where $Q$ is the quaternion group, and a standard embedding $S^3/Q\to S^4$ represents a generator of $3\pi_3^{st}$. Mar1 comment A uniformity with a countable base is a pseudometric uniformity.François, uniform spaces have some (pre)history independent of topological groups (see the chapter by Bentley, Herrlich, and Husek in "Handbook of the History of General Topology, Vol. 2"). In particular your proof above is originally from a paper by Alexandroff and Urysohn (C. R. Acad. Sci., Paris 177 (1923), 1274-1276) Mar1 comment is there any fibration $\mathbb{R}^n\to \mathbb{S}^n$?This makes it much simpler, thanks! Feb16 revised Original proof of the existence of Seifert surfacesedited body Feb16 revised Original proof of the existence of Seifert surfacesadded 9 characters in body Feb16 answered Original proof of the existence of Seifert surfaces Feb11 accepted When is the Freudenthal compactification an ANR?