I am trying to understand whether the finite cyclic group of order $n$, $C_n$ is a Poincare duality group, i.e. whether it's classifying space $K(C_n,\,1)$ is a Poincare complex. I know that the classifying space of a cyclic group is a lens space, but am unsure whether a lens space is necessarily a Poincare complex.
2026-03-25 19:11:20.1774465880
Is a finite cyclic group a Poincare duality group?
140 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ALGEBRAIC-TOPOLOGY
- How to compute homology group of $S^1 \times S^n$
- the degree of a map from $S^2$ to $S^2$
- Show $f$ and $g$ are both homeomorphism mapping of $T^2$ but $f$ is not homotopy equivalent with $g.$
- Chain homotopy on linear chains: confusion from Hatcher's book
- Compute Thom and Euler class
- Are these cycles boundaries?
- a problem related with path lifting property
- Bott and Tu exercise 6.5 - Reducing the structure group of a vector bundle to $O(n)$
- Cohomology groups of a torus minus a finite number of disjoint open disks
- CW-structure on $S^n$ and orientations
Related Questions in CYCLIC-GROUPS
- Confusing step in proof of property of cyclic group automorphisms
- If $G=\langle x\rangle$ is cyclic group and order of $G$ is $40$ then how many order of $x^3$
- How to arrange $p-1$ non-zero elements into $A$ groups of $B$ where $p$ is a prime number
- $e^{n/e}$ estimate of the maximum order of permutation group element: proof
- Intuitive understanding of $g^iH=(gH)^i$ factor groups
- Exams exercise involving the permutation group $S_5$
- Find the order of 5 in $\mathbb Z_{12}$
- The commutator of two subgroup in a finite group
- Show that, for every $x\ \epsilon \ C_{m}$, we have that $ord(f(x))$ is a divisor of d.
- Why are $-1$ and $1$ generators for the Set of integers under addition?
Related Questions in DUALITY-THEOREMS
- Computing Pontryagin Duals
- How to obtain the dual problem?
- Optimization problem using Fenchel duality theorem
- Deriving the gradient of the Augmented Lagrangian dual
- how to prove that the dual of a matroid satisfies the exchange property?
- Write down the dual LP and show that $y$ is a feasible solution to the dual LP.
- $\mathrm{Hom}(\mathrm{Hom}(G,H),H) \simeq G$?
- Group structure on the dual group of a finite group
- Proving that a map between a normed space and its dual is well defined
- On the Hex/Nash connection game theorem
Related Questions in CLASSIFYING-SPACES
- References for classifying spaces and cohomology of classifying spaces
- explicit map of classifying spaces
- Classifying space of a category / classifying space of a group
- Is the classifying space of a non-compact Lie group equal the classifying space of the maximal compact subgroup?
- How to compute sheaf cohomology of a classifying space?
- Eilenberg–MacLane space for explicit group examples
- Classifying spaces [related to Eilenberg–MacLane] for explicit group examples
- "Eilenberg-MacLane property" for the classifying space of a groupoid
- Classifying space $B$SU(n)
- Eilenberg–MacLane space $K(\mathbb{Z}_2,n)$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
No. A Poincaré duality space must in particular have vanishing cohomology above some degree, but a nontrivial finite cyclic group has nonvanishing cohomology in arbitrarily high degrees.