<p>The Development of Intersection Homology Theory </p><p>Steven L. Kleiman </p><p>Contents Foreword </p><p>Received October 9, 2006. </p><p>226 </p><p>Steven L. Kleiman </p><p>Preface </p><p>The Development of Intersection Homology Theory </p><p>227 </p><p>Discovery </p><p>228 </p><p>Steven L. Kleiman </p><p>∈</p><ul style="display: flex;"><li style="flex:1">∈</li><li style="flex:1">∈</li></ul><p></p><ul style="display: flex;"><li style="flex:1">⊃</li><li style="flex:1">⊃</li></ul><p>⊃</p><ul style="display: flex;"><li style="flex:1">⊃</li><li style="flex:1">⊃ · · · ⊃ </li><li style="flex:1">⊃</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">−1 </li><li style="flex:1">−2 </li><li style="flex:1">−3 </li><li style="flex:1">1</li><li style="flex:1">0</li></ul><p></p><ul style="display: flex;"><li style="flex:1">−1 </li><li style="flex:1">−2 </li></ul><p></p><p>−</p><p>−1 </p><p>−</p><p>−1 </p><p>×<br>⊃ · · · ⊃ </p><p>−1 </p><p>∩<br>×</p><p>−</p><p>−2 </p><p>The Development of Intersection Homology Theory </p><p>229 </p><p>−</p><p>−</p><p>−</p><p>−</p><p>++ − </p><p></p><ul style="display: flex;"><li style="flex:1">×</li><li style="flex:1">−→ </li></ul><p>−</p><p>230 </p><p>Steven L. Kleiman </p><p>2<br>−2 </p><p></p><ul style="display: flex;"><li style="flex:1">b</li><li style="flex:1">c</li></ul><p></p><p>2</p><p></p><ul style="display: flex;"><li style="flex:1">≤</li><li style="flex:1">≤</li></ul><p></p><p>The Development of Intersection Homology Theory </p><p>231 </p><p>A fortuitous encounter </p><p>2 − </p><p></p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">}</li></ul><p><sub style="top: 0.1481em;">2 ∗ </sub>· · · </p><p></p><ul style="display: flex;"><li style="flex:1">2∗ </li><li style="flex:1">−</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">≤ (2 ) </li><li style="flex:1">≤ (2) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">−2 </li></ul><p></p><p>−</p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">−2 </li></ul><p></p><p>−</p><p>2 − </p><p>−</p><p>2 − +1 </p><p>≤</p><p>232 </p><p>Steven L. Kleiman </p><p></p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">}</li></ul><p>−</p><p>−1 </p><p>The Development of Intersection Homology Theory </p><p>233 </p><p>§<br>§</p><p>§<br>§</p><p>→</p><p>∗</p><p>§§</p><ul style="display: flex;"><li style="flex:1">≥</li><li style="flex:1">≥</li></ul><p></p><p>−2 </p><p></p><ul style="display: flex;"><li style="flex:1">b</li><li style="flex:1">c</li></ul><p></p><p>2<br>2 − </p><p>§</p><p>234 </p><p>Steven L. Kleiman </p><p>|</p><p>0</p><p>|<br>→</p><p>−1 </p><p></p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">∈</li><li style="flex:1">|</li><li style="flex:1">≥ } </li></ul><p>⊗</p><p>∗</p><p>×</p><p>+ = </p><p>§<br>§</p><p>§</p><p></p><ul style="display: flex;"><li style="flex:1">→</li><li style="flex:1">→</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li><li style="flex:1">2</li><li style="flex:1">2</li></ul><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">2</li></ul><p></p><p>§</p><ul style="display: flex;"><li style="flex:1">∩</li><li style="flex:1">→</li></ul><p></p><ul style="display: flex;"><li style="flex:1">∩</li><li style="flex:1">−→ </li></ul><p></p><p>∗</p><p>The Development of Intersection Homology Theory </p><p>235 </p><p></p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">−</li></ul><p></p><p>∗</p><p>≤</p><p>∗</p><p></p><ul style="display: flex;"><li style="flex:1">∩</li><li style="flex:1">→</li><li style="flex:1">−</li></ul><p>−</p><ul style="display: flex;"><li style="flex:1">∩</li><li style="flex:1">→</li><li style="flex:1">−</li></ul><p></p><p>∗</p><p>−<br>§</p><p>§<br>§</p><p>The Kazhdan–Lusztig conjecture </p><p>236 </p><p>Steven L. Kleiman </p><p>≤</p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)− ( ) </li></ul><p></p><p>−</p><p></p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">−</li></ul><p>−</p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">−</li></ul><p>≤</p><p>−<br>≤</p><p>§</p><p></p><ul style="display: flex;"><li style="flex:1">3</li><li style="flex:1">4</li><li style="flex:1">5</li><li style="flex:1">3</li><li style="flex:1">3</li><li style="flex:1">4</li><li style="flex:1">4</li><li style="flex:1">4</li><li style="flex:1">3</li><li style="flex:1">5</li></ul><p></p><ul style="display: flex;"><li style="flex:1">6</li><li style="flex:1">4</li></ul><p></p><p>The Development of Intersection Homology Theory </p><p>237 </p><p>→<br>∈</p><p>2</p><p>§</p><p>238 </p><p>Steven L. Kleiman </p><p>∼</p><p>2<br>2</p><p>∈</p><p>−</p><p>+</p><p>∼</p><p></p><ul style="display: flex;"><li style="flex:1">∩</li><li style="flex:1">−→ </li></ul><p></p><p>2 +1 <br>2</p><p>2</p><p>-modules </p><p>∗<br>∗</p><p>The Development of Intersection Homology Theory </p><p>239 </p><p>∞</p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">−</li><li style="flex:1">· · · </li><li style="flex:1">−</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">1</li></ul><p></p><ul style="display: flex;"><li style="flex:1">( ) </li><li style="flex:1">( −1) </li></ul><p></p><p>· · · </p><p>1</p><p>−</p><p>240 </p><p>Steven L. Kleiman </p><p>−</p><p>1</p><p></p><ul style="display: flex;"><li style="flex:1">−→ </li><li style="flex:1">−→ </li><li style="flex:1">−→ </li></ul><p>·</p><p></p><ul style="display: flex;"><li style="flex:1">−→ </li><li style="flex:1">−→ </li><li style="flex:1">−→ </li></ul><p></p><p>The Development of Intersection Homology Theory </p><p>241 </p><p>∗<br>∗</p><p>∈</p><p>∗</p><p>Ω</p><ul style="display: flex;"><li style="flex:1">Ω</li><li style="flex:1">−1 </li></ul><p></p><p>⊗</p><p></p><ul style="display: flex;"><li style="flex:1">∗</li><li style="flex:1">∗∗ </li><li style="flex:1">∗</li></ul><p>∗</p><p>∗</p><p>→</p><p>242 </p><p>Steven L. Kleiman </p><p>1</p><p></p><ul style="display: flex;"><li style="flex:1">→</li><li style="flex:1">→</li><li style="flex:1">⊗</li><li style="flex:1">→ · · · → </li><li style="flex:1">⊗</li><li style="flex:1">→</li></ul><p></p><p>∗</p><p>12 </p><p>→</p><p>The Development of Intersection Homology Theory </p><p>243 triv </p><p>∈</p><ul style="display: flex;"><li style="flex:1">→</li><li style="flex:1">⊗</li></ul><p></p><p>triv </p><p>→</p><p></p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">−</li></ul><p></p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">−</li></ul><p></p><p>⊗<br>⊗<br>−<br>−<br>−</p><p>∈</p><p>triv </p><p>⊗</p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">⊗</li></ul><p></p><p>C</p><p>244 </p><p>Steven L. Kleiman </p><p>(</p><p>)− </p><p>−</p><p>− ( ) </p><p>−</p><p>Perverse sheaves </p><p>§<br>∼</p><p>The Development of Intersection Homology Theory </p><p>245 </p><p></p><ul style="display: flex;"><li style="flex:1">≥</li><li style="flex:1">≥</li></ul><p></p><ul style="display: flex;"><li style="flex:1">≥</li><li style="flex:1">≥</li></ul><p>§</p><p>∗</p><p>14 </p><p>∼</p><p>15 17 </p><p>−<br>−</p><p>246 </p><p>Steven L. Kleiman </p><p>−<br>§<br>∈</p><p>−1 </p><p>→</p><p>18 </p><p></p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">−</li></ul><p>→</p><p>∗<br>∗</p><p></p><ul style="display: flex;"><li style="flex:1">∈</li><li style="flex:1">→</li></ul><p></p><p>The Development of Intersection Homology Theory </p><p>247 </p><p>∗</p><p></p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">→</li><li style="flex:1">−</li></ul><p>∈<br>→<br>∈</p><p>∗</p><p>∼</p><p>∗</p><p>·</p><p>248 </p><p>Steven L. Kleiman </p><p>−</p><p>−1 </p><p></p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">−</li></ul><p></p><p>+1 </p><p>§</p><p>triv </p><p>Purity and decomposition </p><p>∗</p><p>→</p><p>2</p><p>−<br>−<br>−</p><p>The Development of Intersection Homology Theory </p><p>249 </p><p>§<br>≥<br>→</p><p>∗</p><p>−</p><p>i</p><p>∗</p><p>1<br>250 </p><p>Steven L. Kleiman </p><p>§<br>−</p><p>i</p><p>§<br>→</p><p>∗</p><p>§</p><p>The Development of Intersection Homology Theory </p><p>251 <br>2( − ) </p><p></p><ul style="display: flex;"><li style="flex:1">≤</li><li style="flex:1">≤</li><li style="flex:1">≤</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">1</li></ul><p></p><p>−</p><p></p><ul style="display: flex;"><li style="flex:1">[</li><li style="flex:1">2] </li></ul><p>252 </p><p>Steven L. Kleiman </p><p>11<br>22<br>3</p><p>−−</p><ul style="display: flex;"><li style="flex:1">− · · · </li><li style="flex:1">∩ · · · ∩ </li></ul><p>∩ · · · ∩ </p><p>1</p><p>≥</p><p>3</p><p>− · · · </p><p>1</p><p>§<br>→<br>≥ }&nbsp;≥ </p><p>−1 </p><p></p><ul style="display: flex;"><li style="flex:1">{</li><li style="flex:1">∈</li><li style="flex:1">|</li></ul><p>§</p><p>∈<br>−</p><p></p><ul style="display: flex;"><li style="flex:1">§</li><li style="flex:1">→</li></ul><p></p><p>∗</p><p>−</p><p>i</p><p></p><ul style="display: flex;"><li style="flex:1">≥</li><li style="flex:1">≥</li></ul><p></p><p>∗</p><ul style="display: flex;"><li style="flex:1">∗</li><li style="flex:1">∗</li></ul><p></p><p>The Development of Intersection Homology Theory </p><p>253 </p><p>0</p><p>→</p><p>∗<br>∗</p><p>→</p><p>∗</p><ul style="display: flex;"><li style="flex:1">∗</li><li style="flex:1">0</li></ul><p></p><p>0</p><p>∗</p><p>§</p><p>0</p><p></p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">⊗</li></ul><p></p><p>∗∗</p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><ul style="display: flex;"><li style="flex:1">2dim( </li><li style="flex:1">)</li></ul><p></p><p>0</p><p>|</p><p>0</p><p>∗</p><p>⊗</p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><p>§<br>§</p><p>254 </p><p>Steven L. Kleiman </p><p>0</p><p>Other work and open problems </p><p>2<br>2<br>(2) </p><p></p><ul style="display: flex;"><li style="flex:1">∧ ∗ </li><li style="flex:1">∞</li><li style="flex:1">∧ ∗ </li><li style="flex:1">∞</li></ul><p>∗</p><p>2</p><p>∗</p>