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• 5-manifolds with fundamental group of order 2
  ...oup in advance and consider the classification of manifolds with the given fundamental group. ...ke into account concerning the classification of manifolds with nontrivial fundamental groups $\pi_1$ is that the higher homotopy groups $\pi_i$ ($i \ge 2$) are m
  5 KB (834 words) - 00:36, 7 February 2013
• Fundamental class
  == The integral fundamental class == ... the orientability by the existence of a certain homology class called the fundamental class. The background is the following theorem. Recall that
  5 KB (846 words) - 15:25, 6 March 2014
• Classifying Poincaré complexes via fundamental triples
  ... this note we survey the classification of Poincaré duality complexes via fundamental triples. -->Let '''$\textup{CW}_0$''' be the category of reduced CW-complex ...homomorphism $\omega_X: \pi_1 X \rightarrow \mathbb Z / 2 \mathbb Z$ and a fundamental class $[X] \in {\rm H}_n(X; \mathbb Z^{\omega})$, such that
  9 KB (1,565 words) - 14:29, 22 July 2014
• Fundamental groups of 3-dimensional spherical space forms
  ...forms: a history | this article]]. Today the list of groups which arise as fundamental groups of $3$-dimensional spherical space forms is known. These are exactly Groups $Q(8n,k,l)$ were excluded from the list of fundamental groups of $3$-manifolds only after resolution of the Geometrization Conject
  8 KB (1,267 words) - 21:45, 11 August 2022
• Non-prime solvable fundamental groups (Ex)
  ...orphism all orientable closed $3$-manifolds, which are not prime and whose fundamental groups are solvable.
  206 B (26 words) - 05:05, 8 January 2019
• Fundamental groups of surfaces (Ex)
  ... $2$-dimensional manifolds, possibly with boundary, do not have hyperbolic fundamental groups?
  192 B (21 words) - 07:13, 6 January 2019

Page text matches

• 2-manifolds
  ... a_gb_ga_1^{-1}b_1^{-1}\cdots a_g^{-1}b_g^{-1}\rangle$. In particular, the fundamental group of the 2-torus is isomorphic to the abelian group $\Zz^2$. ** The fundamental group of $R_h$ is $\pi_1(R_h,*)\cong \langle a_1,\ldots, a_h \mid a_1^2\cdo
  11 KB (1,636 words) - 15:10, 27 February 2022
• Regensburg Surgery Blockseminar 2012: General information
  ...the surgery exact sequence. Starting with exotic spheres we see the spaces fundamental spaces $O$, $PL$, $TOP$ and $G$ and their quotients, compute their homotopy
  8 KB (1,111 words) - 15:44, 9 May 2012
• Hirzebruch surfaces
  * The first Pontrjagin class vanishes as its evaluation on the fundamental class of $H_n$ is an oriented bordism invariant \cite{Milnor&Stasheff1974|L
  9 KB (1,415 words) - 15:57, 5 April 2011
• Complete intersections
  $[X_n({\underline{d}})] \in H_{2n}(X_n({\underline{d}}))$ denote the fundamental class of $X_n({\underline{d}})$ and let $d = d_1 d_2 \dots d_k$ be the prod
  11 KB (1,840 words) - 04:28, 7 January 2020
• 1-manifolds
  ... desire. Surprisingly, many Topology textbooks manage not to mention this fundamental result. An orientation of a 1-manifold $X$ gives rise to the fundamental class of $X$, which belongs to $H_1^{BM}(X,\partial X)$.
  54 KB (8,541 words) - 08:32, 18 July 2013
• Aspherical manifolds
  ...R}\mathbb{P}^n$, irreducible closed orientable $3$-manifolds with infinite fundamental groups, locally symmetric spaces arising from almost connected Lie groups a * Which groups occur as fundamental groups of aspherical closed manifolds?
  59 KB (7,971 words) - 14:39, 27 September 2012
• B-Bordism
  We list some fundamental examples with common notation and also indicate the fibration B. ...$(M, \bar \nu)$ is given a local orientation. This amounts to a choice of fundamental class of $M$ which is a generator
  18 KB (3,039 words) - 20:14, 11 September 2019
• Spin bordism
  ...ifford-linear) Dirac operator can be considered as a representative of the fundamental class $[M]\in KO_n(M)$,
  9 KB (1,462 words) - 06:17, 3 February 2021
• Oriented bordism
  ...ipedia:Signature_(topology)|signature]] of a closed oriented manifold is a fundamental bordism invariant defining a ring homomorphism ...is the k-the Pontrjagin of $M$ and $[M]$ its [[Wikipedia:Fundamental_class|fundamental class]]. The [[Wikipedia:Stiefel-Whitney class#Stiefel–Whitney numbers|St
  7 KB (1,048 words) - 09:26, 22 July 2019
• Complex bordism
  equivalent to orienting $M$. The image of the fundamental class of M\rangle\in H_{2n}(M)$ the fundamental homology class.
  25 KB (4,167 words) - 15:46, 8 May 2012
• 4-manifolds in 7-space
  ...surface' for $f$ is the image $A_{f,n}[N]\in H_{n+1}(C_f,\partial)$ of the fundamental class $[N]$.
  18 KB (3,056 words) - 07:10, 4 April 2020
• Linking form
  of $z \cup PD(y)$ with the fundamental class of $N$ yields $L_N(x,y)$.
  7 KB (1,201 words) - 11:31, 29 March 2019
• Simplicial volume
  is a fundamental cycle of $M$} \bigr\} \in \mathbb{R}_{\geq 0}, $$ where $[M] \in H_n(M;\mathbb{R})$ is the fundamental class of $M$ with real coefficients.
  31 KB (4,564 words) - 13:22, 23 March 2012
• Formal group laws and genera
  ... and then calculate the value of the resulting characteristic class on the fundamental class $\langle M\rangle\in H_{2n}(M)$.
  18 KB (3,231 words) - 15:59, 8 May 2012
• 6-manifolds: 1-connected
  the ``abstract fundamental class´´, incorporates all possible information about multiplication and o
  21 KB (3,625 words) - 08:56, 19 October 2014
• Intersection form
  ...the [[Poincare isomorphism|Poincaré duals]] of $x$, $y$, and $[N]$ is the fundamental class of the manifold $N$. We can also define the ''cup (cohomology interse Fundamental invariants are rank, signature and being odd or even (aka '''type''').
  14 KB (2,390 words) - 10:32, 16 December 2023
• Classifying spaces for families of subgroups
  ...ups play an important role in the classification of manifolds with a given fundamental group $G$. The [[Farrell-Jones Conjecture]] relates the $K$- and $L$-theory ...sifying space $BG=G\backslash EG$. (Recall that $BG$ is a CW complex whose fundamental group is $G$ and whose higher homotopy groups are all zero; it is unique up
  9 KB (1,492 words) - 16:00, 17 January 2013
• Rho-invariant
  ... an expression obtained by evaluating certain cohomological classes on the fundamental classes of the $g$-fixed point submanifolds $Y^g$ of $Y$. In particular if ...s needed which says that for an odd-dimensional manifold $X$ with a finite fundamental group $G$ there always exists a $k \in \Nn$ and a manifold with boundary $(
  5 KB (766 words) - 09:51, 13 June 2013
• Fake lens spaces
  ...e respective simple homotopy types. The classification of lens spaces with fundamental groups of order $N$ with $N$ odd and $N = 2$, was one of the first spectacu ...fication of fake lens spaces of a given dimension $(2d-1) \geq 5$ with the fundamental group $G \cong \Zz_N$ which is stated in the two theorems below.
  24 KB (3,914 words) - 15:18, 25 April 2013
• 4-manifolds: 1-connected
  Any finitely presentable group may occur as the fundamental group of a smooth closed 4-manifold. On the other hand, the class of simply ...putes the Chern classes of $S_d$. Evaluating the second Chern class on the fundamental class $[S_d]$ yields the Euler characteristic and therefore the rank of $H^
  9 KB (1,307 words) - 13:58, 19 April 2011