Fixed-point index

<h2 id="the-lefschetzhopf-theorem">The Lefschetz–Hopf theorem</h2>
<p>The importance of the fixed-point index is largely due to its role in the <a href="/facts/Solomon_Lefschetz/Ut2VFwGr">Lefschetz</a>–<a href="/facts/Heinz_Hopf/PMks10NV">Hopf</a> theorem, which states:
</p>

∑
          
            x
            ∈
            
              F
              i
              x
            
            (
            f
            )
          
        
        
          i
          n
          d
        
        (
        f
        ,
        x
        )
        =
        
          Λ
          
            f
          
        
        ,
      
    
    {\displaystyle \sum _{x\in \mathrm {Fix} (f)}\mathrm {ind} (f,x)=\Lambda _{f},}

<p>where Fix(<i>f</i>) is the set of fixed points of <i>f</i>, and <i>Λ</i><i>f</i> is the <a href="/facts/Lefschetz_number/TEjN2DQ8">Lefschetz number</a> of <i>f</i>.
</p><p>Since the quantity on the left-hand side of the above is clearly zero when <i>f</i> has no fixed points, the Lefschetz–Hopf theorem trivially implies the <a href="/facts/Lefschetz_fixed-point_theorem/TEjN2DQ8">Lefschetz fixed-point theorem</a>.
</p>
<h2 id="notes">Notes</h2>

<ul><li>Robert F. Brown: <i>Fixed Point Theory</i>, in: I. M. James, <i>History of Topology</i>, Amsterdam 1999, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-444-82375-1, 271–299.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>A. Katok and B. Hasselblatt(1995), Introduction to the modern theory of dynamical systems, Cambridge University Press, Chapter 8. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Fixed-point index open-in-new

Fixed-point index