Ricardo Andrade
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Member for 12 years, 10 months
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Last seen more than 9 years ago
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isotopy inverse embeddings vs. diffeomorphisms
@Sergey: Thanks for the clarification.
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isotopy inverse embeddings vs. diffeomorphisms
@Agol: You are right about my remark (1)... It seems I cannot count... Sorry.
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isotopy inverse embeddings vs. diffeomorphisms
... or at least make it clear that is how you obtain the differential structure on $Y_t$.
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isotopy inverse embeddings vs. diffeomorphisms
@Agol: Thanks for the answer. It looks good to me. I will wait a little while to see what others say. It is certainly a beautiful idea. I was trying to make an analogy with the Schroeder-Bernstein theorem but kept coming short because the manifolds were not compact (so I could not apply some isotopy extension). Your method seems to cleverly realize that vague intuition. Just a couple of silly remarks: (1) I think you meant $(K_{2k-2i},k-2i)$ at the end of the fifth paragraph. (2) Perhaps you should redefine $Y_t$ to be the colimit of the interiors of $K_{2i}$, ...
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isotopy inverse embeddings vs. diffeomorphisms
@Sergey and @Zack: I have a perhaps silly question. How do you guarantee that non-local knots are preserved under isotopies (through embeddings, not necessarily diffeomorphisms) of the identity?
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isotopy inverse embeddings vs. diffeomorphisms
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isotopy inverse embeddings vs. diffeomorphisms
@Ben: Thank you for pointing out that my edit introduced an obvious counter-example. I partially reversed the edit to avoid your counter-examples.
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isotopy inverse embeddings vs. diffeomorphisms
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isotopy inverse embeddings vs. diffeomorphisms
@Zack: I was actually thinking of simply using the Whitehead manifold together with its embedding into ${\mathbb R}^3$, but I know next to nothing about the space of self-embeddings of the Whitehead manifold.
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isotopy inverse embeddings vs. diffeomorphisms
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Explicit constructions of K(G,2)?
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Homomorphisms of Topological Groups which are Automatically Fiber Bundles?
The article of Palais unfortunately discusses only groups which are locally compact Hausdorff. Moreover, the results on local triviality seem to be only for Lie groups.
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General gluing theorem for adjunction spaces
The result you seek is given as propositions 5.3.2 and 5.3.3 of tom Dieck's book "Algebraic topology".
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Does the bordism homology theory satisfy the weak equivalence axiom?
I was looking at the section 17.3 on mixed model structures in the book "More concise algebraic topology", and it seems that the mixed cofibrations are precisely the Strom cofibrations $f:A\to B$ which are homotopy equivalent rel $A$ to Serre cofibrations $g:A\to C$ (theorem 17.3.5). In fact, proposition 17.3.4 proves what I stated above: a Strom cofibration between mixed cofibrant objects (i.e. with the homotopy type of a CW-complex) is a mixed cofibration, which was sufficient for our purposes.
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When do basepoints matter in homotopy theory?
$\cdots =K(G,2)^2\times H^2(X,G)\times H^2(Y,G)$ would have $\pi_2=G\times G$ which would contradict the previous statement in the case $G=\Bbb{Z}$.
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When do basepoints matter in homotopy theory?
@Tom: I don't quite understand what you are trying to ask. Nevertheless, I will leave the following remark, just in case. The quasi-category of 1-connected spaces (full sub-quasi-category of the quasi-category of spaces) does not have any binary coproducts. To see this, note that for any 1-connected space $X$ the space of maps $\text{Map}(X,K(G,2))\simeq K(G,2)\times H^2(X,G)$ has $\pi_2=G$ (for $G$ an abelian group). However, if a binary coproduct $X\coprod Y$ existed in 1-connected spaces, the space of maps $\text{Map}(X\coprod Y,K(G,2))=\text{Map}(X,K(G,2))\times\text{Map(Y,K(G,2))}=\cdots$