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* the $2$-sphere: $S^2 := \{ (x, y) \in \Rr^2 | x^2 + y^2 = 1 \}$,
The case $g=0$ refers to the 2-[[sphere]] $S^2$. The number $g$ is called the [[Wikipedia:Genus_(mathematics)#Orie
11 KB (1,636 words) - 15:10, 27 February 2022
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...hat each $i(r_i) \in H_2(M_{(F, \bar w)})$ is represented by an embedded 2-sphere with trivial normal bundle. Let $N_{(G, w)}$ be the manifold obtained by a
19 KB (2,940 words) - 21:07, 12 November 2016
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...fy $S^1 \subset \Cc$ with the unit complex numbers and recall that the $3$-sphere, $S^3 = \{(z_1, z_2)\, | \, |z_1|^2 + |z_1|^2 = 1 \} \subset \Cc^2$, admits
...he other hand the smooth Hirzebruch surfaces are the total spaces of the 2-sphere bundle of a 3-dimensional vector bundle over $S^2$ and these bundles are cl
9 KB (1,415 words) - 15:57, 5 April 2011
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...hich is still open in general, holds up to connected sum with a homotopy 8-sphere.
11 KB (1,840 words) - 04:28, 7 January 2020
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.... By the Generalised Poincaré Conjecture proven by Smale, every homotopy sphere in dimension $n \geq 5$ is homeomorphic to $S^n$: this statement holds in d
...pheres. We review four such constructions: plumbing, Brieskorn varieties, sphere-bundles and twisting.
21 KB (3,384 words) - 23:04, 22 November 2022
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...t is irreducible and has infinite fundamental group. This follows from the Sphere Theorem <nowiki>[</nowiki>[[#{{anchorencode:Hempel1976}}|Hempel1976]], Theo
...1$ and $M_2$ of dimension $n \ge 3$ which are not homotopy equivalent to a sphere, then $M$ is not aspherical.
59 KB (7,971 words) - 14:39, 27 September 2012
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...f $B = PBO$ is the path fibration over $BO$, then $MB$ is homotopic to the sphere spectrum $S$ and $\pi_n(S) = \pi_n^S$ is the [[Wikipedia:Stable_homotopy_gr
18 KB (3,039 words) - 20:14, 11 September 2019
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...orresponding to the path fibration over $BO$ is homotopy equivalent to the sphere spectrum $S$ since the path space is contractible. Thus we get $$\Omega_n^
3 KB (381 words) - 09:40, 8 July 2011
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vector bundle $M\times\mathbb C^k$). The sphere $S^n$ provides an example
such a way that the half-sphere $S^n_\le\times 0$ is identified
18 KB (2,836 words) - 19:52, 28 March 2013
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...tring} \cong \Zz \oplus \Zz/2$, generated by the [[Exotic spheres|exotic 8-sphere]] for the 2-torsion and a certain [[Bott manifold]]: see \cite{Laures2004}.
...ega_{10}^{String} \cong \Zz/6$, generated by an [[Exotic spheres|exotic 10-sphere]].
10 KB (1,694 words) - 09:53, 13 July 2017
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...classifying embeddings of closed manifolds $N$ into Euclidean space or the sphere up to isotopy (i.e., to the Knotting Problem of Remark \ref{r:zee} for embe
{{beginthm|Remark|(Embeddings into Euclidean space and the sphere)}}\label{spheu}
34 KB (5,424 words) - 07:01, 6 November 2021
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Together with [[3-manifolds_in_6-space#Examples|the Haefliger knotted sphere]] \cite[Example 2.1]{Skopenkov2016t}, \cite[Example 3.4]{Skopenkov2006}, th
... said to be obtained by linked embedded connected sum of $f_0$ with an $n$-sphere representing the `homology Alexander dual' $A:=\widehat{A_{f_0}}a\in H_n(C_
33 KB (5,787 words) - 04:46, 1 September 2021
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...ubes to form an embedded 3-sphere and taking any embedding with image this sphere. , and the tubes are chosen so that the connected sum has an orientation co
...Corollary}}\label{co8} (a) If $H_1(N)=0$ (i.e. $N$ is an integral homology sphere), then the Kreck invariant $E^6_D(N)\to\Zz$ is a 1-1 correspondence.
18 KB (2,941 words) - 11:02, 25 June 2019
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vector bundle $M\times\mathbb C^k$). The sphere $S^n$ provides an example
25 KB (4,167 words) - 15:46, 8 May 2012
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...$\eta:S^3\to S^2$ is the Hopf fibration and $S^2$ is identified with the 2-sphere formed by unit length quaternions of the form $ai+bj+ck$.
...ed onto the arc in $S^6$ joining $x$ to $\eta(x)$. Clearly, the boundary 3-sphere of $\Cc P^2_0$ is standardly embedded into $S^6$. Hence $f$ extends to an e
18 KB (3,056 words) - 07:10, 4 April 2020
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... a surgery presentation for $N$. Suppose that $N$ is a rational homology 3-sphere. Let $A$ be the matrix of (self-) linking numbers of the surgery presentat
...on of smooth $(q-1)$-connected $(2q+1)$ manifolds with boundary a homotopy sphere, see \cite[Theorem 7]{Wall1967}.
7 KB (1,201 words) - 11:31, 29 March 2019
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Take the stable vector bundle $\xi$ over the $4i$-sphere corresponding to a generator of $\pi_{4i}(BO) = \mathbb{Z}$. By defintion,
540 B (89 words) - 23:00, 2 June 2014
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... is the orbit spaces of a free linear action of a finite cyclic group on a sphere. The importance of lens spaces stems from the fact that they provide exampl
...o be the orbit space of the free action of the cyclic group $\Zz_m$ on the sphere $S^{2d-1} = S (\Cc^d)$ given by the formula
4 KB (618 words) - 06:25, 22 July 2014
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A homotopy sphere of dimension $n$ is an oriented closed smooth
...y sphere is called an exotic sphere if it not diffeoemorphic to a standard sphere.
4 KB (627 words) - 08:56, 8 June 2010
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... is a manifold again. Given a free tame action of the circle on a $(2n+1)$-sphere, the orbit space is a fake $\Cc P^n$. On the other hand, if $M$ is a closed
9 KB (1,587 words) - 11:57, 3 April 2014
