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...omorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2$. Thus, the set of homeomorphism classes of surfaces is a commutative monoid with respect
...in $\CP^2$ whose gradient vector does not vanish in any point of the zero set) is a compact, connected, orientable, real surface of genus
11 KB (1,636 words) - 15:10, 27 February 2022
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...seminar 2012: Program#The structure set and Wall realisation|The structure set and Wall realisation]]: \cite{Lück2001|5.1}; [[User:Münster|Münster]], [
8 KB (1,111 words) - 15:44, 9 May 2012
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Let $\mathcal{M}_{5}$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]], [[wikipe
...hat $2^r \cdot x = 0$ for some $x \in w_2^{-1}(1)$. If $M$ is spinable we set $h(M) = 0$.
19 KB (2,940 words) - 21:07, 12 November 2016
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... is [[Wikipedia:Second_countable|second countable]] (i.e., its topological structure has a countable base), satisfies the [[Wikipedia:Hausdorff_space|Hausdorff
...m ''curve'' may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion of a smooth manifold of dimension 1 to Euc
54 KB (8,541 words) - 08:32, 18 July 2013
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to be the set of oriented $h$-cobordism classes of homotopy spheres. [[Wikipedia:Connect
Let $G$ be a graph with vertices $\{v_1, \dots, v_n\}$ such that the edge set between $v_i$ and $v_j$, is non-empty only if $p_i = q_j$. We form the man
21 KB (3,384 words) - 23:04, 22 November 2022
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... topological structure: orientation, spin-structure, weakly almost complex structure etc. The situation for [[Wikipedia:Piecewise_linear_manifold|piecewise lin
The formulation of the general set-up for B-Bordism dates back to {{cite|Lashof1963}}. There are detailed tre
18 KB (3,039 words) - 20:14, 11 September 2019
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For a multi-index $J=(j_1, \dots, j_n)$, we set $\pi^J=\pi^{j_1}\dots \pi^{j_n}$ and $n(J)=\sum_i j_i$.
A spin structure on a closed $n$-manifold $M$ induces a KO-orientation $[M]\in KO_n(M)$, so
9 KB (1,462 words) - 06:17, 3 February 2021
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$\varOmega_n^O$ the set of bordism classes of $n$-dimensional
Zero is represented by the bordism class of an empty set (which is
18 KB (2,836 words) - 19:52, 28 March 2013
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The additive structure of the bordism groups is not fully determined yet. It is known that only 2
...opies of $KO$. However, it is not an algebra over $KO$. Its multiplicative structure for $p=2$ can be read off the formula
10 KB (1,694 words) - 09:53, 13 July 2017
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...o\Rr^m$ is `readily calculable' if it involves a 1-1 correspondence with a set or a group which is `easily' calculated from the given number $m$ and the m
...and the action of the group of homotopy self-equivalences on the structure set
34 KB (5,424 words) - 07:01, 6 November 2021
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... unitary struture $\bar \nu$ on a manifold $M$ is a choice of weak complex structure on the stable normal bundle of $M$. By the [[B-Bordism#Pontrjagin-Thom isom
...lds]]. It is one of the most important theories of bordism with additional structure, or [[B-Bordism|B-bordism]].
25 KB (4,167 words) - 15:46, 8 May 2012
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... 7-space and for the corresponding action of the group $E^7_D(S^4)$ on the set $E^7_D(N)$, see e.g. \cite[$\S$4]{Skopenkov2016c}.
...{Milnor&Stasheff1974}. Since also any PL 4-manifold admits a unique smooth structure \cite[$\S$1.2]{Mandelbaum1980}, we may consider a smooth 4-manifold as a PL
18 KB (3,056 words) - 07:10, 4 April 2020
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...cted sum was used in \cite{Ajala1984} and \cite{Ajala1987} to describe the set of smooth structures on the product of spheres $\Pi_{i=1}^r S^{n_i}$. This
* to define, for $m\ge 2p+q+3$, a group stucture on the set $E^m(S^p \times S^q)$ of (smooth or PL) isotopy classes of embeddings $S^p
5 KB (721 words) - 00:04, 8 April 2020
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...\#$, and for the corresponding action of the group $E^m_D(S^{p+q})$ on the set $KT^m_{p,q,D}$, see e.g. \cite[$\S$4]{Skopenkov2016c}.
...rametric_connected_sum#Applications|$S^p$-parametric connected sum]] group structure on $KT^m_{p,q}$ is constructed for $m\ge2p+q+3$ in \cite{Skopenkov2006}, \c
12 KB (2,194 words) - 09:28, 31 August 2021
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$\frac{du}{\omega(u)}=\frac{dF(u,v)}{\omega(F(u,v))}$. Set
The set of Chern characteristic numbers of a manifold $M$ defines
18 KB (3,231 words) - 15:59, 8 May 2012
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Let $\mathcal{M}_6^\Cat$ be the set of $\Cat$ isomorphism classes of closed oriented simply connected 6-dimensi
..._6$ for short). Our first two invariants determine the (additive) homology structure of $M$:
21 KB (3,625 words) - 08:56, 19 October 2014
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...n addition certain fake complex projective spaces are part of a generating set of the topological oriented cobordism groups.
The surgery structure set of $\Cc P^n$ is in bijection to the set of free tame circle actions on $S^{2n+1}$ modulo equivariant homeomorphism.
9 KB (1,587 words) - 11:57, 3 April 2014
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...e a manifold outside of a compact 'singularity set', while the singularity set has a neighborhood that looks like the product of manifold and a cone.
...e approach to manifolds with singularities would be to remove the singular set and to define an equivalence relation on the remaining manifold that 'remem
8 KB (1,233 words) - 15:57, 10 December 2010
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# a set $C \subseteq X$ is closed if and only if $C \cap X_n$ is closed for every $
# $\mathbb{R}$ with the $\mathbb{Z}$-CW structure described above is a model for the classifying space of $\mathbb{Z}$ for th
9 KB (1,492 words) - 16:00, 17 January 2013
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...ting an element in the [[Wikipedia:Surgery structure set|surgery structure set]] $\mathcal{S}^s (L(\alpha_k))$ the so-called '''splitting invariants''':
...$\mathbf{s}_{4i-2} (h)$ are obtained by passing to the associated manifold structure on the real projective space $\Rr P^{2d-1}$ (alias restricting the action t
24 KB (3,914 words) - 15:18, 25 April 2013
