The finite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf You could've invented tmf . Aaron Mazel-Gee University of California, Berkeley [email protected] April 21, 2013 Aaron Mazel-Gee You could've invented tmf . The finite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf Overview Aaron Mazel-Gee You could've invented tmf . 2 Chromatic homotopy theory 3 Topological modular forms 4 Fun with tmf The finite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf Overview 1 The finite stable homotopy category Aaron Mazel-Gee You could've invented tmf . 3 Topological modular forms 4 Fun with tmf The finite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf Overview 1 The finite stable homotopy category 2 Chromatic homotopy theory Aaron Mazel-Gee You could've invented tmf . 4 Fun with tmf The finite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf Overview 1 The finite stable homotopy category 2 Chromatic homotopy theory 3 Topological modular forms Aaron Mazel-Gee You could've invented tmf . The finite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf Overview 1 The finite stable homotopy category 2 Chromatic homotopy theory 3 Topological modular forms 4 Fun with tmf Aaron Mazel-Gee You could've invented tmf . The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf 1. The finite stable homotopy category Aaron Mazel-Gee You could've invented tmf . stable finite CW complexes This is called the finite stable homotopy category, denoted SHCfin. Let's do some topology! So, what is topology? Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the homotopy category of topological spaces. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Aaron Mazel-Gee You could've invented tmf . stable finite CW complexes This is called the finite stable homotopy category, denoted SHCfin. So, what is topology? Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the homotopy category of topological spaces. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! Aaron Mazel-Gee You could've invented tmf . stable finite CW complexes This is called the finite stable homotopy category, denoted SHCfin. Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the homotopy category of topological spaces. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Aaron Mazel-Gee You could've invented tmf . stable finite CW complexes This is called the finite stable homotopy category, denoted SHCfin. But this is hard. Let's make things a bit easier for ourselves, and study the homotopy category of topological spaces. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Topology is the study of topological spaces. Aaron Mazel-Gee You could've invented tmf . stable finite CW complexes This is called the finite stable homotopy category, denoted SHCfin. Let's make things a bit easier for ourselves, and study the homotopy category of topological spaces. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Topology is the study of topological spaces. But this is hard. Aaron Mazel-Gee You could've invented tmf . stable finite CW complexes This is called the finite stable homotopy category, denoted SHCfin. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the homotopy category of topological spaces. Aaron Mazel-Gee You could've invented tmf . stable finite CW complexes This is called the finite stable homotopy category, denoted SHCfin. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the homotopy category of topological spaces. Aaron Mazel-Gee You could've invented tmf . stable finite This is called the finite stable homotopy category, denoted SHCfin. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the homotopy category of topological spaces. CW complexes Aaron Mazel-Gee You could've invented tmf . stable finite This is called the finite stable homotopy category, denoted SHCfin. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the homotopy category of topological spaces. CW complexes Aaron Mazel-Gee You could've invented tmf . stable This is called the finite stable homotopy category, denoted SHCfin. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the homotopy category of topological spaces. finite CW complexes Aaron Mazel-Gee You could've invented tmf . stable This is called the finite stable homotopy category, denoted SHCfin. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the homotopy category of topological spaces. finite CW complexes Aaron Mazel-Gee You could've invented tmf . This is called the finite stable homotopy category, denoted SHCfin. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the stable homotopy category of topological spaces. finite CW complexes Aaron Mazel-Gee You could've invented tmf . This is called the finite stable homotopy category, denoted SHCfin. The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the stable homotopy category of topological spaces. finite CW complexes Aaron Mazel-Gee You could've invented tmf . The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin Let's do some topology! So, what is topology? Topology is the study of topological spaces. But this is hard. Let's make things a bit easier for ourselves, and study the stable homotopy category of topological spaces. finite CW complexes This is called the finite stable homotopy category, denoted SHCfin. Aaron Mazel-Gee You could've invented tmf . Recall that the suspension of a space X is given by (0; x1) (0; x2) ΣX = [0; 1] X ∼ × (1; x1) (1; x2): ∼ Given a map X Y , we can suspend it to a map ΣX ΣY . ! ! This gives us a system [X ; Y ] [ΣX ; ΣY ] [Σ2X ; Σ2Y ] [Σ3X ; Σ3Y ] : ! ! ! !··· The Freudenthal suspension theorem tells us that this system always stabilizes. So for two finite CW complexes X and Y , we define n n Hom fin (X ; Y ) = lim [Σ X ; Σ Y ]: SHC n!1 The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf What does it mean to stabilize? Aaron Mazel-Gee You could've invented tmf . Given a map X Y , we can suspend it to a map ΣX ΣY . ! ! This gives us a system [X ; Y ] [ΣX ; ΣY ] [Σ2X ; Σ2Y ] [Σ3X ; Σ3Y ] : ! ! ! !··· The Freudenthal suspension theorem tells us that this system always stabilizes. So for two finite CW complexes X and Y , we define n n Hom fin (X ; Y ) = lim [Σ X ; Σ Y ]: SHC n!1 The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf What does it mean to stabilize? Recall that the suspension of a space X is given by (0; x1) (0; x2) ΣX = [0; 1] X ∼ × (1; x1) (1; x2): ∼ Aaron Mazel-Gee You could've invented tmf .