1
vote
0answers
35 views

A technical step in the proof of Atiyah-Bott fixed point formula

From John Roe, Elliptic operators, topology and asymptotic methods, page 135. Here Roe tried to prove that $$ \DeclareMathOperator{\Tr}{Tr} \sum_{q}(-1)^{q} ...
0
votes
0answers
45 views

How to compute the Lefschetz number

Given a continuous function $f: X \to X$ how do you compute: $$ \Lambda_f = \sum_{k \geq 0} (-1)^k \mathrm{Tr}(f_*|H_k(X,\mathbb{Q})) $$ which is known as the Lefschetz number. For instance let $X: ...
4
votes
0answers
77 views

A question on fixed point property

Assume that $0<k<n-1$, Note that $\mathbb{C}P^{k}$ can be considered as a closed subset of $\mathbb{C}P^{n}$, in a natural way. We collapse $\mathbb{C}P^{k}$ to a point. The resulting space is ...
2
votes
1answer
95 views

Proof for the existence of a second fixed point in Poincaré's last geometric theorem

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem: "An area-preserving transformation of an annulus that moves the boundary circles in opposite ...
3
votes
0answers
64 views

Reference request: generalized Lefschetz-Hopf fixed point theorem

Let $X$ be a smooth projective variety over $\mathbb{F}_p$. A basic (and very important) theorem is that we have equality $$ \# X(\mathbb{F}_{p^n}) = \sum_{i=0}^{2\dim X} (1)^i ...
2
votes
0answers
73 views

On the Proof of the Perron-Frobenius Theorem.

The Perron-Frobenius theorem states that a square matrix with nonnegative entries has a real nonnegative eigenvalue. One possible proof uses the Brouwer fixed point theorem, and every proof I've seen ...
4
votes
0answers
127 views

algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy ...
2
votes
1answer
180 views

Homotopy equivalence an retractions

I have some questions about homotopy. Before starting here a definition: A topological space $X$ is called contractible if $X$ is homotopy equivalent with a one-point-space Suppose $X$ a toplogical ...
4
votes
1answer
207 views

Can a finite group act freely (as homeomorphisms) on $R^n$

And what for a single finite order element $f$, i.e. $f:R^n\rightarrow R^n$ is a homeomorphism such that $f^d=id_{R^n}$, must $f$ have a fixed point?
2
votes
2answers
383 views

Brouwer's fixed point theorem for free?

I think I found a proof of Brouwer's fixed point theorem which is much simpler than any of the proofs in my books. One part is standard: Suppose there is an $f:D^n \rightarrow D^n$ with no fixed ...
1
vote
1answer
146 views

The wedge sum of two circles has fixed point property?

The wedge sum of two circles has fixed point property? I'm trying to find a continuous map from the wedge sum to itself, that this property fails, I couldn't find it, I need help. Thanks
22
votes
1answer
325 views

Does every continuous map from $\mathbb{H}P^{2n+1}$ to itself have a fixed point?

This question is motivated by Fixed point property of Cayley plane and idle curiosity. In the link, it is shown that every continuous map from the Cayley plane to itself has a fixed point and the ...
4
votes
1answer
115 views

Fixed point property of Cayley plane

I want to know whether the Cayley plane has fixed point property or not. I think it is so but I am not able to prove this. It certainly does not admit maps of period two without fixed points. By fixed ...
4
votes
2answers
565 views

What can Lefschetz fixed point theorem tell us about the group of automorphisms of a compact Riemann surface?

Let $X$ be a compact Riemann surface of genus $g>1$, $f\in Aut(X)$, a biholomorphism of $X$ onto itself, $x\in X$ a fixed point of $f$. Since tangent map of a holomorphic map (on the real tangent ...